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We compute the first four perturbative coefficients of the internal energy for the twisted reduced principal chiral model (TRPCM) using numerical stochastic perturbation theory (NSPT). This matrix model has the same large $N$ limit as the ordinary principal chiral model (PCM) at infinite volume. Indeed, we verify that the first three coefficients match the analytic result for the PCM coefficients at large $N$ with a precision of three to four significant digits. The fourth coefficient also matches our own NSPT calculation of the corresponding PCM coefficient at large $N$. The finite-$N$ corrections to all coefficients beyond the leading order are smaller for TRPCM than for PCM. We analyze the variance to determine the feasibility of extending the calculations to higher orders., Comment: 27 pages, 12 figures, using ws-ijmpa.cls style, using siunitx version 3

We present our preliminary results on the determination of the low lying meson spectrum for pure gauge theory in the large $N$ limit. Some results are also shown for the theory with two flavours of quarks in the adjoint representation., Contribution to Lattice 2019 (Wuhan)

We present the results of an exploratory study of the numerical stochastic perturbation theory (NSPT) applied to the four dimensional twisted Eguchi-Kawai (TEK) model. We employ a Kramers type algorithm based on the Generalized Hybrid Molecular Dynamics (GHMD) algorithm. We have computed the perturbative expansion of square Wilson loops up to $O(g^8)$. The results of the first two coefficients (up to $O(g^4)$) have a high precision and match well with the exact values. The next two coefficients can be determined and even extrapolated to large $N$, where they should coincide with the corresponding coefficients for ordinary Yang-Mills theory on an infinite lattice. Our analysis shows the behaviour of the probability distribution for each coefficient tending to Gaussian for larger $N$. The results allow us to establish the requirements to extend this analysis to much higher order., pdflatex 30 pages, version appeared in JHEP

Motivated by our previous study of the Twisted Eguchi-Kawai model for non minimal twists, we re-examined the behaviour of the reduced version of the two dimensional principal chiral model. We show that this single matrix model reproduces the same features as the standard lattice model. In particular, scaling towards the continuum limit, the correct value of the internal energy, the magnetic susceptibility and the mass gap. Given our capacity to reach larger values of $N$, we use the reduced model to study the nature and properties of its large $N$ phase transition existing at intermediate coupling. We conclude that the transition is of first order, pdflatex 21 pages, 10 figures

Journal of high energy physics 1807(07), 169 (2018). doi:10.1007/JHEP07(2018)169, We compute and analyse the low-lying spectrum of 2+1 dimensional SU(N) Yang-Mills theory on a spatial torus of size l × l with twisted boundary conditions. This paper extends our previous work [1]. In that paper we studied the sector with non-vanishing electric flux and concluded that the energies only depend on the parameters through two combinations: x = λN l/(4π) (with λ the ’t Hooft coupling) and the twist angle $ \tilde{\theta} $ defined in terms of the magnetic flux piercing the two-dimensional box. Here we made a more complete study and we are able to condense our results, obtained by non-perturbative lattice methods, into a simple expression which has important implications for the absence of tachyonic instabilities, volume independence and non-commutative field theory. Then we extend our study to the sector of vanishing electric flux. We conclude that the onset of the would-be large-volume glueball states occurs at an approximately fixed value of x, much before the stringy torelon states have become very massive., Published by Springer, Berlin

We evaluate the $\Lambda$-parameter in the $\overline{\mathrm{MS}}$ scheme for the pure SU(3) gauge theory with the twisted gradient flow (TGF) method. A running coupling constant $g_{\mathrm{TGF}}^2(1/L)$ is defined in a finite volume box with size of $L^4$ with the twisted boundary condition. This defines the TGF scheme. Using the step scaling method for the TGF coupling with lattice simulations, we can evaluate the $\Lambda$-parameter non-perturbatively in the TGF scheme. In this paper we determine the dimensionless ratios, $\Lambda_{\mathrm{TGF}}/\sqrt{\sigma}$ and $r_{0}\Lambda_{\mathrm{TGF}}$ together with the $\Lambda$-parameter ratio $\Lambda_{\mathrm{SF}}/\Lambda_{\mathrm{TGF}}$ on the lattices numerically. Combined with the known ratio $\Lambda_{\overline{\mathrm{MS}}}/\Lambda_{\mathrm{SF}}$, we obtain $\Lambda_{\overline{\mathrm{MS}}}/\sqrt{\sigma} = 0.517(10)(^{+8}_{-7})$ and $r_{0}\Lambda_{\overline{\mathrm{MS}}}=0.593(12)(^{+12}_{-9})$, where the first error is statistical one and the second is our estimate of systematic uncertainty., Comment: 23 pages, 11 figures, LaTeX

We derive the perturbative expansion of Wilson loops to order g^4 in a SU(N) lattice gauge theory with twisted boundary conditions. Our expressions show that the thermodynamic limit is attained at infinite N for any number of lattice sites and allow to quantify the deviations from volume independence at finite large N as a function of the twist., Comment: 7 pages, one figure, contribution to the 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton, UK

Lattice results are presented for the meson spectrum of 1+1 dimensional gauge theory at large $N$, using the Twisted Eguchi-Kawai model. Comparison is made to the results obtained by `t Hooft in the light cone gauge., pdflatex 7 pages

We estimate the $\Lambda$-parameter in the $\overline{\mathrm{MS}}$ scheme for the SU(3) pure gauge theory with the twisted gradient flow method non-perturbatively. We obtain $\Lambda_{\overline{\mathrm{MS}}}/\sqrt{\sigma}=0.527(13)(10)$ and $r_{0}\Lambda_{\overline{\mathrm{MS}}}=0.605(15)(5)$ which are consistent with the known values.This demonstrates the validity of the present method., Comment: 7 pages, 6 figures, uses PoS.cls, presented at the 34th International Symposium on Lattice Field Theory (Lattice 2016), 24 - 30 July 2016, University of Southampton, UK, PoS(LATTICE2016) 185

Recently, we proposed a new method to calculate meson propagators in the large $N$ limit from twisted space-time reduced model. In this note, we give simulation details for obtaining meson spectra and discuss the smearing technique which should improve the signal of meson propagators in future works., 7 pages, 4 figures, Proceedings of the 33rd International Symposium on Lattice Field Theory (Lattice 2015), 14-18 July 2015, Kobe International Conference Center, Kobe, Japan

Journal of High Energy Physics 2015.8 (2015): 034 reproduced by permission of Scuola Internazionale Superiore di Studi Avanzati (SISSA), In this work we consider the SU(N) gauge theory with two Dirac fermions in the adjoint representation, in the limit of large N. In this limit the infinite-volume physics of this model can be studied by means of the corresponding twisted reduced model defined on a single site lattice. Making use of this strategy we study the reduced model for various values of N up to 289. By analyzing the eigenvalue distribution of the adjoint Dirac operator we test the conformality of the theory and extract the corresponding mass anomalous dimension, We acknowledge financial support from the MCINN grants FPA2012-31686 and FPA2012-31880, and the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249. M. O. is supported by the Japanese MEXT grant No 26400249 and the MEXT program for promoting the enhancement of research universities. Calculations have been done on Hitachi SR16000 supercomputer both at High Energy Accelerator Research Organization(KEK) and YITP in Kyoto University, and the HPC-clusters at IFT. Work at KEK is supported by the Large Scale Simulation Program No.14/15-03

We measure the running of the twisted gradient flow coupling in the Twisted Eguchi-Kawai (TEK) model, the SU(N) gauge theory on a single site lattice with twisted boundary conditions in the large N limit., 7 pages, 3 figures, contribution to the 32nd International Symposium on Lattice Field Theory, 23-28 June, 2014. Columbia University New York, NY

Recently, we proposed a new method to extract the string tension from 4-dimensionally smeared Wilson loops. In this talk, we first show that the results obtained using this smearing method are identical to those obtained by Wilson flow, once the time step is sufficiently small. We then demonstrate the practical advantage of our method by applying it to the calculation of string tension in SU(3) Yang-Mills theory., 7 pages, 10 figures, Contribution to the 32nd International Symposium on Lattice Field Theory (Lattice 2014), 23-28 June 2014, Columbia University, New York, NY, USA

Journal of High Energy Physics 2015.6 (2015): 193 reproduced by permission of Scuola Internazionale Superiore di Studi Avanzati (SISSA), We investigate Monte Carlo updating algorithms for simulating SU(N) YangMills fields on a single-site lattice, such as for the Twisted Eguchi-Kawai model (TEK). We show that performing only over-relaxation (OR) updates of the gauge links is a valid simulation algorithm for the Fabricius and Haan formulation of this model, and that this decorrelates observables faster than using heat-bath updates. We consider two different methods of implementing the OR update: either updating the whole SU(N) matrix at once, or iterating through SU(2) subgroups of the SU(N) matrix, we find the same critical exponent in both cases, and only a slight difference between the two, We acknowledge financial support from the MCINN grants FPA2012-31686 and FPA2012-31880, and the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249. M.O. is supported by the Japanese MEXT grant No. 26400249 and the MEXT program for promoting the enhancement of research universities

Journal of High Energy Physics 2015.1 (2015): 038 reproduced by permission of Scuola Internazionale Superiore di Studi Avanzati (SISSA), We measure the running of the SU(∞) ’t Hooft coupling by performing a step scaling analysis of the Twisted Eguchi-Kawai (TEK) model, the SU(N) gauge theory on a single site lattice with twisted boundary conditions. The computation relies on the conjecture that finite volume effects for SU(N) gauge theories defined on a 4-dimensional twisted torus are controlled by an effective size parameter ˜l = l√ N, with l the torus period. We set the scale for the running coupling in terms of ˜l and use the gradient flow to define a renormalized ’t Hooft coupling λ(˜l). In the TEK model, this idea allows the determination of the running of the coupling through a step scaling procedure that uses the rank of the group as a size parameter. The continuum renormalized coupling constant is extracted in the zero lattice spacing limit, which in the TEK model corresponds to the large N limit taken at fixed value of λ(˜l). The coupling constant is thus expected to coincide with that of the ordinary pure gauge theory at N = ∞. The idea is shown to work and permits us to follow the evolution of the coupling over a wide range of scales. At weak coupling we find a remarkable agreement with the perturbative two-loop formula for the running coupling, We acknowledge financial support from the MCINN grants FPA2009-08785, FPA2009-09017, FPA2012-31686 and FPA2012-31880, the Comunidad Autónoma de Madrid under the program HEPHACOS S2009/ESP-1473, the European Union under Grant Agreement PITN-GA-2009-238353 (ITN STRONGnet), and the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249. M. O. is supported by the Japanese MEXT grant No 26400249

For a pure SU(N) gauge theory on the lattice we test if the expectation values of small Wilson loops become volume independent in the large N limit., 7 pages. Contribution to Lattice 2014

In this paper we present our results concerning the dependence of Wilson loop expectation values on the size of the lattice and the rank of the SU(N) gauge group. This allows to test the claims about volume independence in the large N limit, and the crucial dependence on boundary conditions. Our highly precise results provide strong support for the validity of the twisted reduction mechanism and the TEK model, provided the fluxes are chosen within the appropriate domain., Comment: 33 pages, latex, 10 figures

We present our results for the K+ → π+ π0 decay amplitude with the Wilson quark action in the quenched approximation at β = 6.1. The amplitude is extracted from the ratio of K → ππ 3-point function divided either by K and π meson 2-point functions or by K meson 2-point function and I = 2 ππ 4-point function, which yield results in good agreement. Finite size effects are examined with calculations made on 243 × 64 and 323 × 64 lattices. We convert the lattice amplitude into the continuum value employing a recent one-loop calculation of chiral perturbation theory by Golterman and Leung. The result is consistent with the experimental value if extrapolated to the chiral limit. Various uncertainties inherent in the present methods for calculating the decay amplitude are discussed.

We report on our effort toward a determination of the heavy-light decay constant in quenched QCD. Simulation results are analyzed in terms of the non-relativistic interpretation of relativistic fermions fully incorporating oneloop corrections in the mass and current renormalization factors for massive quark. For the Wilson action for which the analysis is completed, we obtain ƒ d = 192(30)MeV, ƒ d = 214(33)MeV, ƒ b = 171(29)MeV and ƒ b s =193(32)MeV in the continuum limit.

We study the scaling behavior of the step scaling function for SU(3) gauge theory, employing the Iwasaki gauge action and the Luescher-Weisz gauge action. In particular, we test the choice of boundary counter terms and apply a perturbative procedure for removal of lattice artifacts for the simulation results in the extrapolation procedure. We confirm the universality of the step scaling functions at both weak and strong coupling regions. We also measure the low energy scale ratio with the Iwasaki action, and confirm its universality.

We determine the nonperturbative anisotropic parameter of the gauge action in the quenched approximation with less than 1% accuracy using the Sommer scale measured by the L\"uscher-Weisz algorithm or smearing technique. We also study the nonperturbative O(a)-improvement of the quark action. The bare quark anisotropy is determined using the masses from the temporal and spatial directions. For the determination of the $O(a)$ improvement coefficients, we apply the Schr\"odinger functional method., Comment: 3 pages, 5 figures, Lattice2004(improved)

The twisted reduced model of large $N$ QCD with two adjoint Wilson fermions is studied numerically using the Hybrid Monte Carlo method. This is the one-site model, whose large $N$ limit (large volume limit) is expected to be conformal or nearly conformal. The string tension calculated at $N$=289 approaches zero as we decrease quark mass and the preliminary value of the mass anomalous dimension $\gamma_*$ is close to one if we assume that the theory is governed by an infrared fixed point. We also discuss the twisted reduced model with single adjoint Wilson fermion. The string tension remains finite as the quark mass decreases to zero, supporting that this is the confining theory., Comment: 7 pages, 5 figures; Contribution to SCGT12 "KMI-GCOE Workshop on Strong Coupling Gauge Theories in the LHC Perspective", 4-7 Dec. 2012, Nagoya University

We discuss the perturbative expansion of SU(N) Yang-Mills theories defined on a d-dimensional torus of linear size l with twisted boundary conditions, generalizing previous results in the literature. For a specific class of twist tensors depending on a single integer flux value k, we show that perturbative results to all orders depend on the combination lN^(2/d) and a flux-dependent angle theta. This implies a new kind of volume independence that holds at finite N and for fixed values of theta. Our results also provide interesting information about the possible occurrence of tachyonic instabilities at one-loop order. We support the prescription that instabilities are avoided, if the large N limit is taken keeping theta > theta_c, and appropriately scaling the magnetic flux k with N. Numerical results in 2+1 dimensions provide a test of how these ideas extend into the non-perturbative regime., Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

In this work we consider the SU(N) gauge theory with two Dirac fermions in the adjoint representation, in the limit of large N. Taking advantage of large N twisted volume reduction we do this on a single site lattice, but we should still get infinite-volume physics in the large N limit. We describe our progress in extracting the mass anomalous dimension from the eigenvalue distribution of the adjoint Dirac operator, using data for N up to 289., 7 pages, 5 figures, presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, Germany

We study the 2+1 dimensional SU(N) Yang-Mills theory on a finite two-torus with twisted boundary conditions. Our goal is to study the interplay between the rank of the group N, the length of the torus L and the Z_N magnetic flux. After presenting the classical and quantum formalism, we analyze the spectrum of the theory using perturbation theory to one-loop and using Monte Carlo techniques on the lattice. In perturbation theory, results to all orders depend on the combination x=\lambda NL and an angle defined in terms of the magnetic flux (\lambda\ is 't Hooft coupling). Thus, fixing the angle, the system exhibits a form of volume independence (NL dependence). The numerical results interpolate between our perturbative calculations and the confinement regime. They are consistent with x-scaling and provide interesting information about the k-string spectrum and effective string theories. The occurrence of tachyonic instabilities is also analysed. They seem to be avoidable in the large N limit with a suitable scaling of the magnetic flux., Comment: 62 pages, 7 figures

Artículo escrito por un elevado número de autores, solo se referencian el que aparece en primer lugar, el nombre del grupo de colaboración, si le hubiere, y los autores pertenecientes a la UAM, The space-time reduced model of large N QCD with two adjoint Wilson fermions is constructed by applying the symmetric twist boundary conditions with nonvanishing flux k. For large but finite N = L2, the model should behave as the large N version of the ordinary lattice gauge model on a V = L4 spacetime volume. We perform a comparison of the N dependence of several quantities in this model and in the k = 0 model (corresponding to periodic boundary conditions). Although the Z4(N) symmetry seems unbroken in all cases, the N-dependence analysis favors the use of the same values of k and L for which the symmetry is also unbroken in the pure gauge case. In particular, the k = 0 model, studied recently by several authors, shows a large and irregular dependence on N within our region of parameters. This makes this reduced model very impractical for extracting physical information about the large N lattice theory. On the contrary, the model for N = 289 = 172 and large enough k gives consistent results, even for extended observables as Wilson loops W(R, T) up to R, T = 8, matching the expected behavior for the lattice model with a 174 space-time volume, A. G.-A. is supported by Spanish Grants No. FPA2009-08785, No. FPA2009- 09017, No. CSD2007-00042, No. HEPHACOS S2009/ ESP-1473, No. PITN-GA-2009-238353 (ITN STRONGnet), No. CPAN CSD2007-00042, and the Spanish MINECO Centro de excelencia Severo Ochoa Program under Grant No. SEV-2012-0249. M.O. is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (No. 23540310). The calculation has been done on the Hitachi SR16000-M1 computer at High Energy Accelerator Research Organization (KEK) supported by the Large Scale Simulation Program No. 12/13-01 (FY2012-13)

Statistical properties of effective mass are analyzed. We show from a general ground that effective mass as a function of time should not exhibit long plateaux whatever high statistics simulations are made: the mass should fluctuate beyond the one standard deviation of error bars after a few time slices for large times where the ground state dominates. This explains the difficulty of obtaining long plateaux experienced in previous simulations. Implications of the observation for global $\chi^2$ fits are discussed, and results for hadron masses are presented., Comment: contribution to Lattice 95 (presented by A. Ukawa), 4 pages, 4 postscript figures (file name for figures modified)

Status report is made of our quenched study of heavy-light matrix elements employing the Wilson quark action for heavy quark. Results obtained up to now with 200 configurations at β = 6.1 on a 24 3 × 64 lattice and with 100 configurations at β = 6.3 on a 32 3 × 80 lattice suggest that the pseudoscalar decay constant varies little over this range of β in both charm and bottom regions. Results for the B parameter are also reported.

We study the anisotropic lattice QCD for precision computations of heavy-light matrix elements. Our previous study in which the lattices are calibrated with a few percent accuracy has already given results comparable to the existing calculations. This suggests that even higher precision may be achieved by a more precise calibration of anisotropic lattices. We describe our strategy to tune the gauge and quark parameters with accuracies much less than 1 % in the quenched approximation., 3 papes, 2 figures, Lattice2003(heavy)

The non-perturbative cSW determined by the Schrodinger functional (SF) method with the RG-improved gauge action in dynamical Nf = 3 QCD shows a finite volume effect when the numerical simulations are carried out at a constant lattice size L a . We remove the unwanted finite volume effect by keeping physical lattice extent L at a constant. The details of the method and the result obtained for non-perturbative cSW with a constant L are reported.

We report our calculation of the heavy-light decay constants f B and f D on a 20 3 × 48 lattice at β = 5.2 with n f = 2 sea quarks by the fully 0(a)-improved Wilson fermion and heavy quarks in Fermilab formalism. We find that the B meson decay constants are consistent with those obtained with the NRQCD formalism. It is shown that the quark mass dependence almost cancels in the B/D ratio of the decay constant so that the systematic errors in the chiral extrapolation can be reduced. Taking the Grinstein ratio, remaining uncertainties from the perturbation and discretization also cancel to give a precise prediction of ( f B s f B d ) ( f D s f D d ) of 1% accuracy.

Using smeared Creutz ratios we extract the string tension for SU(N) pure gauge theory and $N$=3,4,5,6,8. We employ these results to extrapolate to large N. The same methodology is applied to the single-site Twisted Eguchi Kawai model. The corresponding string tension matches perfectly within errors with the extrapolated one, providing strong evidence in favour of the twisted reduction framework. Interesting results are also obtained on the behaviour of Creutz ratios for large sizes., Comment: 7 pages and 3 figures. Contribution to Lattice 2012 in Cairns

We study the chiral behavior of the ratio of heavy-light decay constants on unquenched lattices with two flavor dynamical quarks using the plaquette gauge action and the nonperturbativelyO(a)-improved Wilson fermion action at β = 5.2. The heavy quarks are described by the clover action and the NRQCD action. We find that the sea quark mass dependence of the decay constants largely cancels in the double ratioR1 =fBs/fBd/fDs/fDd and the systematic error in the chiral limit can be drastically reduced. Our preliminary result with heavy clover isR1 = 1.018 ± 0.006 ± 0.010, where the first and the second errors are the statistical and the systematics errors.

The twisted reduced model of large $N$ QCD with two adjoint Wilson fermions is studied numerically using the Hybrid Monte Carlo method. This is the one-site model, whose large $N$ limit (large volume limit) is expected to be conformal or nearly conformal. The symmetric twist boundary condition with flux $k$ is used. $k$=0 corresponds to periodic boundary conditions. It is shown that the quark mass and $N$ dependencies of the model with non-vanishing $k$ differ significantly from those of the $k$=0 model. A preliminary result for the string tension calculated at $N$=289 is presented. The string tension seems to vanish as the physical quark mass decreases to zero in a way consistent with the theory being governed by an infrared fixed point with $\gamma_* = 0.8 \sim 1.2$., Comment: 7 pages, 8 figures, Proceedings of the 30th International Symposium on Lattice Field Theory, June 24-29, 2012, Cairns, Australia

We present the results of $1+1+1$ flavor $\mathrm{QCD}+\mathrm{QED}$ simulation at the physical point, in which the dynamical quark effects in QED and the up-down quark mass difference are incorporated by the reweighting technique. The physical quark masses together with the lattice spacing are determined with ${m}_{{\ensuremath{\pi}}^{+}}$, ${m}_{{K}^{+}}$, ${m}_{{K}^{0}}$ and ${m}_{{\ensuremath{\Omega}}^{\ensuremath{-}}}$ as physical inputs. Calculations are carried out using a set of $2+1$ flavor QCD configurations near the physical point generated by the nonperturbatively $O(a)$-improved Wilson quark action and the Iwasaki gauge action at $\ensuremath{\beta}=1.9$ on a ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ lattice. We evaluate the values of the up, down and strange quark masses individually with nonperturbative QCD renormalization.

We review the work of the PACS-CS Collaboration, which aimed to realize lattice quantum chromodynamics (QCD) calculations at the physical point, i.e., those with quark masses set at physical values. This has been a long-term goal of lattice QCD simulation since its inception in 1979. After reviewing the algorithmic progress, which played a key role in this development, we summarize the simulations that explored the quark mass dependence of hadron masses down to values close to the physical point. In addition to allowing a reliable determination of the light hadron mass spectrum, this work provided clues on the validity range of chiral perturbation theory, which is widely used in phenomenology.We then describe the application of the technique of quark determinant reweighting, which enables lattice QCD calculations exactly on the physical point. The physical quark masses and the strong coupling constants are fundamental constants of the strong interaction. We describe a non-perturbative Schrodinger functional approach to figure out the non-perturbative renormalization needed to calculate them. There are a number of physical applications that can benefit from lattice QCD calculations carried out either near or at the physical point. We take up three illustrative examples: calculation of the physical properties of the ρ meson as a resonance, the electromagnetic form factor and charge radius of the pion, and charmed meson spectroscopy. Bringing single hadron properties under control opens up a number of new areas for serious lattice QCD research. One such area is electromagnetic effects in hadronic properties.We discuss the combined QCD plus QED simulation strategy and present results on electromagnetic mass difference. Another area is multi-hadron states, or nuclei. We discuss the motivations and difficulties in this area, and describe our work for deuteron and helium as our initial playground. We conclude with a brief discussion on the future perspective of lattice QCD.

We present the results of an analysis of a 2+1 dimensional pure SU(N) Yang-Mills theory formulated on a 2-dimensional spatial torus with non-trivial magnetic flux. We focus on investigating the dependence of the electric-flux spectrum, extracted from Polyakov loop correlators, with the spatial size l, the number of colours N, and the magnetic flux m. The size of the torus acts a parameter that allows to control the onset of non-perturbative effects. In the small volume regime, where perturbation theory holds, we derive the one-loop self-energy correction to the single-gluon spectrum, for arbitrary N and m. We discuss the transition from small to large volumes that has been investigated by means of Monte-Carlo simulations. We argue that the energy of electric flux e, for the lowest gluon momentum, depends solely on e/N and on the dimensionless variable x=lambda N l, with lambda the 't Hooft coupling. The variable x can be interpreted as the dimensionless 't Hooft coupling for an effective box size given by Nl. This implies a version of reduction that allows to trade l by N without modifying the electric-flux energy., Comment: 7 pages, 3 figures. Proceedings of the 30th International Symposium on Lattice Field Theory, June 24 - 29, 2012, Cairns, Australia. Minor change: Fig. 1 modified to correctly account for the sign convention in Eq. (2.5)