Large \(N\) scaling and factorization in \({\mathrm {SU}}(N)\) Yang–Mills gauge theory
- 70 Downloads
Abstract
The large \(N\) limit of \({\mathrm {SU}}(N)\) gauge theories is well understood in perturbation theory. Also non-perturbative lattice studies have yielded important positive evidence that ’t Hooft’s predictions are valid. We go far beyond the statistical and systematic precision of previous studies by making use of the Yang–Mills gradient flow and detailed Monte Carlo simulations of \({\mathrm {SU}}(N)\) pure gauge theories in 4 dimensions. With results for \(N=3,4,5,6,8\) we study the limit and the approach to it. We pay particular attention to observables which test the expected factorization in the large \(N\) limit. The investigations are carried out both in the continuum limit and at finite lattice spacing. Large \(N\) scaling is verified non-perturbatively and with high precision; in particular, factorization is confirmed. For quantities which only probe distances below the typical confinement length scale, the coefficients of the \(1/N\) expansion are of \(\mathrm{O}(1)\), but we found that large (smoothed) Wilson loops have rather large \(\mathrm{O}(1/N^2)\) corrections. The exact size of such corrections does, of course, also depend on what is kept fixed when the limit is taken.
1 Introduction
An interesting approach to study quantum chromodynamics (QCD) is to consider the order of the gauge group \(N\) as a free parameter. As shown by ’t Hooft [1], by taking the large \(N\) limit of the perturbative weak coupling expansion, the theory simplifies in many ways, and in fact one can treat theories at finite \(N\) as corrections in the “small” parameter \(1/N\). Moreover, the large \(N\) expansion predicts that quark loop effects are suppressed by a power of \(1/N\), so that the weak coupling expansion of large \(N\) QCD is dominated by planar diagrams with purely gluonic internal loops. All of these rather remarkable properties of large \(N\) QCD, make it an interesting theory to study not only from the theoretical perspective, but also from a practical point of view, as results for real world QCD could be obtained by considering corrections to the \(N= \infty \) theory which are parametrized by powers of \(1/N\).
Although this \(1/N\) scaling is obtained perturbatively, lattice computations provide evidence that it also holds at the non-perturbative level, both in \(D=4\) space-time dimensions [2, 3, 4, 5, 6, 7, 8, 9] and in \(D=3\) [9, 10, 11, 12, 13, 14]. The evidence is usually based on complicated observables, where typically one needs to project onto ground states by large Euclidean times. It is then difficult to obtain high precision at various \(N\) in order to verify ’t Hooft scaling with good confidence. Let us stress the fact that the validity of the \(1/N\) scaling, beyond the weak coupling expansion, is not a trivial statement. Hence, it is desirable to test it by means of lattice simulations and with statistically and systematically very precise observables.
One more aspect where Eq. (1.1) plays a crucial role has to do with the idea of volume independence, which starting from the work of the authors in Ref. [22], has been used in the lattice formulation to study the large \(N\) limit of the Yang–Mills theory by performing simulations in small spacetime volumes [23, 24], and even in single site lattices, provided a clever choice of boundary conditions [25, 26, 27] is made.
The above indicates that factorization is not only relevant in the theoretical context, but also on the practical level, as it is a requirement for the single site lattice simulations to be valid. To be more precise, the equivalence is expected to hold between the single site and the infinite volume theory in the \(N\rightarrow \infty \) limit. The equivalence is argued for on the basis of the Makeenko–Migdal loop equations [28] on the lattice. As originally shown in Ref. [22], the loop equations in both theories are equivalent, provided that the product of the expectation value of the Wilson loops factorize as stated in Eq. (1.1). Let us mention that shortly after volume reduction was put forward, it was clear the situation is more complicated, and phase transitions spoil the reduction in its simplest form [23, 24, 25, 29, 30, 31]. Workarounds this issue have been presented in the literature [23, 25, 26, 27] and show that either full or partial volume reduction are a possible way to make simulations of \({\mathrm {SU}}(N)\) Yang–Mills theory at large \(N\) more accessible, as there is a significant compensation of the extra cost for increasing \(N\) by the much smaller number of lattice sites.
Additionally, we would like to point out that important physics is contained in the corrections to factorization. The most obvious one is that glueball masses are obtained from the connected correlation functions of Wilson loops.
The previous discussion motivates the search for a non-perturbative proof, beyond the realm of weak coupling perturbation theory. Several authors have investigated factorization beyond perturbation theory [32, 33, 34, 35]; and as mentioned earlier, lattice simulations also suggest this result to be valid. In particular, the results presented in Ref. [5], Sect. 6, provide strong evidence for factorization. Here we go beyond this. We consider several high precision normalized observables by using the gradient flow to study the large \(N\) scaling, and address the important issue of the size of the correction to \(N= \infty \) in a non-perturbative lattice computation.
This paper is organized as follows, in Sect. 2 we present the observables that are used both to check the large \(N\) scaling, as well as factorization. In Sect. 3 we discuss different ways of defining the large \(N\) limit and in particular the two choices we made for our investigation. In Sect. 4 we describe the ensembles and lattice parameters used for the simulations and in Sect. 5 we present our results, both at finite lattice spacing, and in the continuum limit. We finish with a short summary of the results.
2 Observables
2.1 The gradient flow coupling at large \(N\)
2.2 Smooth Wilson loops
The favourable properties of smooth Wilson loops have already been exploited in the literature, as for example to estimate the string tension at small values of t in Refs. [6, 44], or to study the large \(N\) phase transition in the eigenvalue spectrum of the Wilson loop matrices [45]. For our purpose, the limit of small t is not required, as the smooth loops are used to test factorization and the large \(N\) limit for well defined renormalized observables, regardless of their relation to the operators at \(t=0\).
In the end, we study the large \(N\) limit of square Wilson loops, i.e. for loops where the path C in Eq. (2.1) is given by a square of size \(R\times R\). In order to take the large \(N\) limit, the loops are matched at different \(N\) relating their size to the scale \(t_0\) introduced in the previous section. More precisely, the large \(N\) and continuum limits are taken for loops of size \(R_c = \sqrt{8 c t_0}\) (see Fig. 1), where the smoothing parameter \(t = c t_0\), and c is a constant parameter.
2.3 Observables to test factorization
Parameters of the simulations. For each of the gauge groups \({\mathrm {SU}}(N)\) we give the inverse lattice coupling \(\beta =2N^2/\lambda _0\), the dimensions of the lattice, the approximate lattice spacing using \(\sqrt{t_0}=0.166 \, {\mathrm {fm}}\) followed by the number \(N_{{\mathrm {meas}}}^\mathrm {W}\) of measurements used for the computation of the smooth Wilson loops, and \(N_{{\mathrm {meas}}}^\mathrm {E}\) for the action density, Eq. (2.4). In the second to last column we present the values of \(t_0/a^2\):\(~^{*}\) taken from Ref. [53] and\(~^{**}\) taken from Ref. [2]
#run | \(N\) | \(\beta \) | T / a | L / a | \(a[{\mathrm {fm}}]\) | \(N_{{\mathrm {meas}}}^\mathrm {W}\) | \(N_{{\mathrm {meas}}}^\mathrm {E}\) | \(t_0/a^2\) | \(L/\sqrt{8t_0}\) |
---|---|---|---|---|---|---|---|---|---|
\(A(3)_2\) | 3 | 6.11 | 80 | 20 | 0.078 | 320 | 6720 | \(4.5776(15)^{*}\) | 3.3050(5) |
\(A(3)_3\) | 3 | 6.24 | 96 | 24 | 0.064 | 280 | 280 | 6.783(23) | 3.258(6) |
\(A(3)_4\) | 3 | 6.42 | 96 | 32 | 0.050 | 252 | 252 | 11.19(4) | 3.382(6) |
\(A(4)_1\) | 4 | 10.92 | 64 | 16 | 0.096 | 248 | 17,341 | \(2.9900(7)^{**}\) | 3.2714(4) |
\(A(4)_2\) | 4 | 11.14 | 80 | 20 | 0.078 | 300 | 35,960 | \(4.5207(8)^{**}\) | 3.3257(3) |
\(A(4)_3\) | 4 | 11.35 | 96 | 24 | 0.065 | 312 | 15,460 | \(6.4849(16)^{**}\) | 3.3321(4) |
\(A(4)_4\) | 4 | 11.65 | 96 | 32 | 0.049 | 320 | 640 | 11.55(3) | 3.329(4) |
\(A(5)_1\) | 5 | 17.32 | 64 | 16 | 0.095 | 320 | 9871 | \(3.0636(7)^{**}\) | 3.2319(4) |
\(A(5)_2\) | 5 | 17.67 | 80 | 20 | 0.077 | 240 | 21,680 | \(4.6751(8)^{**}\) | 3.2703(3) |
\(A(5)_3\) | 5 | 18.01 | 96 | 24 | 0.064 | 248 | 8007 | \(6.8151(18)^{**}\) | 3.2504(4) |
\(A(5)_4\) | 5 | 18.21 | 96 | 32 | 0.049 | 328 | 328 | 11.51(3) | 3.334(4) |
\(A(6)_1\) | 6 | 25.15 | 64 | 16 | 0.095 | 320 | 19,360 | \(3.0824(4)^{**}\) | 3.2220(2) |
\(A(6)_2\) | 6 | 25.68 | 80 | 20 | 0.076 | 264 | 11,392 | \(4.8239(9)^{**}\) | 3.2195(3) |
\(A(6)_3\) | 6 | 26.15 | 96 | 24 | 0.063 | 288 | 6704 | \(6.9463(13)^{**}\) | 3.2195(3) |
\(A(8)_2\) | 8 | 32.54 | 20 | 80 | 0.076 | 320 | 320 | 4.782(5) | 3.2336(17) |
2.4 Finite volume
For a numerical test, we need to choose a finite volume. We chose our parameters such that \(L/\sqrt{8t_0}\approx 3.3\). Table 1 shows the actual values used in our simulations. Since L is thus approximately constant, it is omitted as an argument of the observables. We note that the large \(N\) limit and factorization can be tested in infinite or in finite volume. To be on the safe side, we chose the latter, even though we are not far from the infinite volume limit for most observables.
3 Defining the approach to the large N limit
The complete definition of a quantum field theory involves a regularization (here Wilson’s lattice theory) as well as a non-trivial renormalisation before the regulator can be removed. Although this is usually not discussed, quantitative statements about the approach to the large \(N\) limit, such as the ones we are seeking here, do depend on the renormalisation scheme if the renormalisation scheme defines which quantity is held fixed as we take \(N\rightarrow \infty \).
Due to the existence of the limit Eq. (3.2), we may also scale distances with respect to any one particular reference scale (choice of \({\mathcal {O}}\)). In our numerical work we have chosen \(t_0\), Eq. (2.6), because of its high precision.
The preceding discussion is about the continuum theory. It thus saliently assumes that first we take the continuum limit at finite \(N\) and then we perform \(N\rightarrow \infty \). However, we may also proceed in the opposite order: first take the large \(N\) limit at fixed lattice spacing and then send the lattice spacing to zero.^{2} Let us briefly discuss that this order of limits is indeed the same as above; the limits are interchangeable.
3.1 Large \(N\) limit at fixed lattice spacing
In general, taking the large \(N\) limit at fixed lattice spacing has to be followed by the \(a\rightarrow 0\) limit at \(N=\infty \). However, when we investigate factorization, the second step is not expected to be necessary. This is because the perturbative proof of factorization holds in the lattice regularization [48] at finite a. If factorization holds non-perturbatively we thus also expect Eq. (2.13) at any fixed a. In any case, verifying Eq. (2.13) at arbitrary finite lattice spacing implies that it holds in the continuum limit.
Note also that even the large \(N\) limit of divergent quantities, such as Wilson loops at \(t=0\), is expected to exist. A high precision numerical test has recently been performed [8].
4 Lattice details
In this section we give the details of our lattice simulations. We simulate the pure gauge theory with \(N=3,\, 4,\, 5,\, 6,\, 8\) at several lattice spacings. The lattice action is the Wilson gauge action and we use open boundary conditions in the time direction [49]. The simulations are performed using a combination of heatbath and overrelaxation local updates using the Cabibbo–Marinari strategy [50] to refresh the \({\mathrm {SU}}(N)\) matrices. The ratio of overrelaxation to heatbath updates is fixed to L / (2a).
For convenience, we present the values of the lattice spacing, as well as lattice sizes in physical units by assigning a value to \(t_0\) such that \(\sqrt{t_0}=0.166 \, {\mathrm {fm}}\). This choice is motivated by the result in \({\mathrm {SU}}(3)\) for \(\sqrt{8t_0}/r_0 = 0.941(7)\) [51] and the value of the reference scale \(r_0 \approx 0.5 \, {\mathrm {fm}}\) [52]. Notice that this choice is somewhat arbitrary, as apart from the missing quark loops, for \(N\ne 3\) the theory cannot be directly identified with Nature.
The parameters of the simulations are displayed in Table 1. The configurations used for the measurements are a subset of those reported in Ref. [2] for all ensembles except for those at \(N=3, 8\), and for the finest lattice spacings in the case of \(N=4, 5\). As announced above, all the lattices considered in Table 1 are of approximately the same spatial size \(L \approx 1.55 \, {\mathrm {fm}}\). In addition, we have used two additional ensembles with \(L \approx 2.35 \, {\mathrm {fm}}\) at the coarsest lattice spacing (\(a \approx 0.1 \, {\mathrm {fm}}\)) for \(N= 4, 5\) in order to check for effects due to small variations in the volume. Notice that for the ensembles which have been reported in Ref. [2], we have a very large number of measurements for the Yang–Mills action density.
The flow equations are integrated using a third order Runge–Kutta integrator [42] and the observables are measured at intervals \(\Delta t\) of t of \(\Delta t/a^2 \approx 2-3 \times 10^{-2}\). Afterwards, they are interpolated using a second order polynomial in order to obtain their values at arbitrary t. The action density is defined exactly as in [42], using the clover discretization and it is measured from \(t=0\) up to \(t \approx 1.2 \, t_0\). The loops, W(c), are measured only in the vicinity of \(t=ct_0\), with \(c={1/2,1,9/4}\), and then interpolated to the exact value of t. For the loops, one has to do an additional interpolation to \(R_c\), and since their statistical precision is very high, one has to be careful with small potential systematic effects. The details of this interpolation were already presented in Ref. [54].
5 Results
5.1 Large \(N\) scaling
In order to test and provide a precise verification of \(1/N^2\) scaling, we analyse our results for W(c) and for the gradient flow coupling \(\bar{\lambda }_{{\mathrm {GF}}}\). Let us first discuss our results for the latter.
5.1.1 The gradient flow coupling
As can be observed in Fig. 2, cut-off effects are large at small t. At \(t= 0.1 \, t_0\) the relative difference between the results at the finest lattice spacing (\(a \approx 0.05 \, {\mathrm {fm}}\)) and at the coarsest (\(a \approx 0.1 \, {\mathrm {fm}}\)) one, is around \(20\%\); while the errors in the measurements themselves is at the per-mill level. The situation is better at larger values of t, so let us first focus on values of \(t/t_0 \ge 0.3\), where the relative size of cut-off effects is reduced tenfold, when compared to the case at \(t/t_0=0.1\). In Fig. 3 we show a plot of the continuum extrapolation of \(\bar{\lambda }_{{\mathrm {GF}}}\) at \(t/t_0 = 0.8\) and the large \(N\) extrapolation both at finite lattice spacing and in the continuum. In order to be able to use the dataset at \(N=8\), in addition to the continuum limit extrapolations, we consider \(a^2/t_0=0.2091\), the value on ensemble A(8)\(_2\). We then interpolated the results for all the other gauge groups to that lattice resolution.
Parameters of the large \(N\) extrapolations, eq. (5.2), of \(\bar{\lambda }_{{\mathrm {GF}}}(1/\sqrt{8ct_0})\) and W(c) at finite lattice spacing (L) and in the continuum (C)
Obs. | c | Fit | \(a_0\) | \(a_1\) | \(a_2\) | \(\chi ^2/{{\mathrm {dof}}}\) | \(\eta (1/9)\) | \(\delta (1/9)\) |
---|---|---|---|---|---|---|---|---|
\(\bar{\lambda }_{{\mathrm {GF}}}\) | 0.2 | L | 15.916(6) | \(-\,0.472(15)\) | \(-\,0.05(12)\) | 0.79 | 0.05 | 0.03 |
\(\bar{\lambda }_{{\mathrm {GF}}}\) | 0.4 | L | 23.410(6) | \(-\,0.3567(90)\) | \(-\,0.043(70)\) | 1.07 | 0.04 | 0.04 |
\(\bar{\lambda }_{{\mathrm {GF}}}\) | 0.8 | L | 39.011(3) | \(-\,0.1233(30)\) | \(-\,0.011(25)\) | 1.02 | 0.014 | 0.06 |
\(\bar{\lambda }_{{\mathrm {GF}}}\) | 0.2 | C | 16.2(5) | \(-\,0.08(96)\) | \(-\,1.9(65)\) | 0.44 | 0.03 | 0.02 |
\(\bar{\lambda }_{{\mathrm {GF}}}\) | 0.4 | C | 23.6(2) | \(-\,0.15(26)\) | \(-\,1.0(18)\) | 0.01 | 0.03 | 0.03 |
\(\bar{\lambda }_{{\mathrm {GF}}}\) | 0.8 | C | 39.05(6) | \(-\,0.045(45)\) | \(-\,0.50(32)\) | 0.02 | 0.011 | 0.05 |
W | 1 / 2 | L | 0.7760(7) | 0.355(33) | \(-\,0.09(24)\) | 0.49 | 0.04 | – |
W | 1 | L | 0.6575(3) | 0.449(22) | 0.66(22) | 0.49 | 0.06 | – |
W | 9 / 4 | L | 0.4228(7) | 0.626(85) | 2.67(82) | 0.51 | 0.10 | – |
W | 1 / 2 | C | 0.792(4) | 0.17(19) | 0.8(13) | 0.11 | 0.03 | – |
W | 1 | C | 0.666(3) | 0.54(14) | \(-\,0.5(10)\) | 0.01 | 0.05 | – |
W | 9 / 4 | C | 0.426(9) | 1.27(69) | \(-\,3.4(46)\) | 0.24 | 0.10 | – |
As an example of results at smaller t, we show our analysis at \(t/t_0=0.4\) in Fig. 4. In this case, the magnitude of the cut-off effects is larger, but the same analysis as before can be carried out.
As mentioned earlier, dealing with \(\bar{\lambda }_{{\mathrm {GF}}}\) at values of \(t/t_0 \le 0.3\) presents a bigger challenge, so one cannot reach the same level of accuracy as the results presented in this section. However, we have proceeded to do a similar analysis for such small values of t, including corrections of higher order in \(a^2/t_0\). Details are found in Appendix B.
The magnitude of the \(1/N^2\) corrections can be read off from the coefficients \(a_1\) and \(a_2\) collected in Table 2, together with those of the smooth Wilson loops which we discuss next.
5.1.2 Smooth Wilson loops
Once again, to quantify the magnitude of the finite \(N\) corrections, we collect in Table 2 the values of \(a_1\) and \(a_2\) from the fit to Y. We observe that the relative magnitude of them grow at larger values of c (or t equivalently). Similarly, the deviation between \({\mathrm {SU}}(3)\) and \({\mathrm {SU}}(\infty )\) also grows up to a value of \(\eta (1/9)=0.1\) when \(c=9/4\). In all cases we find an excellent fit to Y (the values of \(\chi ^2/{{\mathrm {dof}}}\) are reported in Table 2).
5.2 Factorization
Parameters of the large \(N\) extrapolations of \(H_E\), \(G_W\) and \(H_W\). We present the results for three different cases, L: at finite lattice spacing, \(\hbox {L}_4\): at finite lattice spacing excluding \({\mathrm {SU}}(3)\), and C: in the continuum. Additionally, we fit the data to a function with \(b_0 =0\), so that factorization is imposed at finite lattice spacing L* and in the continuum C*. In this case, the value of \(\chi ^2/{{\mathrm {dof}}}\) validates this hypothesis
Obs. | Fit | \(b_0\) | \(b_1\) | \(b_2\) | \(\chi ^2/{{\mathrm {dof}}}\) |
---|---|---|---|---|---|
\(H_E(1)\) | L | 0.078(44) | 66.3(19) | 319(19) | 0.30 |
\(H_E(1)\) | \(\hbox {L}_4\) | 0.069(75) | 66.8(35) | 314(43) | 0.59 |
\(H_E(1)\) | C | 0.04(84) | 67(39) | 293(373) | 0.34 |
\(H_E(1)\) | L* | 0.0 | 70.22(60) | 276(12) | 0.71 |
\(H_E(1)\) | C* | 0.0 | 70.3(98) | 237(200) | 0.27 |
\(G_W(1)\) | L | \(-\,0.00029(34)\) | 0.548(13) | \(-\,0.69(12)\) | 0.02 |
\(G_W(1)\) | \(\hbox {L}_4\) | \(-\,0.00017(69)\) | 0.540(44) | \(-\,0.59(54)\) | \(<0.01\) |
\(G_W(1)\) | C | \(-\,0.00003(36)\) | 0.49(13) | \(-\,0.48(86)\) | \(<0.01\) |
\(G_W(1)\) | L* | 0.0 | 0.5376(58) | \(-\,0.619(76)\) | 0.25 |
\(G_W(1)\) | C* | 0.0 | 0.485(28) | \(-\,0.48(27)\) | \(<0.01\) |
\(G_W(1/2)\) | L | 0.000045(373) | 0.167(17) | \(-\,0.17(15)\) | \(<0.01\) |
\(G_W(1/2)\) | \(\hbox {L}_4\) | \(-\,0.000008(646)\) | 0.170(42) | \(-\,0.23(55)\) | \(<0.01\) |
\(G_W(1/2)\) | C | \(-\,0.00049(247)\) | 0.168(82) | \(-\,0.32(56)\) | \(<0.01\) |
\(G_W(1/2)\) | L* | 0.0 | 0.1686(70) | \(-\,0.188(90)\) | \(<0.01\) |
\(G_W(1/2)\) | C* | 0.0 | 0.152(19) | \(-\,0.22(18)\) | 0.02 |
\(G_W(9/4)\) | L | \(-\,0.00124(63)\) | 2.781(38) | \(-\,3.95(43)\) | 0.13 |
\(G_W(9/4)\) | \(\hbox {L}_4\) | \(-\,0.0010(10)\) | 2.760(74) | \(-\,3.6(10)\) | 0.15 |
\(G_W(9/4)\) | C | 0.0014(99) | 2.43(40) | \(-\,1.8(29)\) | 0.13 |
\(G_W(9/4)\) | L* | 0.0 | 2.711(14) | \(-\,3.23(24)\) | 1.39 |
\(G_W(9/4)\) | C* | 0.0 | 2.485(73) | \(-\,2.22(81)\) | 0.07 |
\(H_W(9/4)\) | L | \(-\,0.24(14)\) | 120(9) | \(-\,129(84)\) | 0.39 |
\(H_W(9/4)\) | C | \(-\,3.3(25)\) | 226(88) | \(-\,1063(618)\) | 0.26 |
\(H_W(9/4)\) | L* | 0.0 | 106(3) | \(-\,7(40)\) | 1.17 |
\(H_W(9/4)\) | C* | 0.0 | 112(17) | \(-\,290(186)\) | 0.99 |
Concerning the \(N\rightarrow \infty \) limit itself, the extrapolated value is within two standard deviations from zero in the worst case. Notice that at finite lattice spacing, the errors in the extrapolation are two orders of magnitude smaller than the value of \(H_E(1)\) at \(N=3\). To further validate factorization, an additional fit is performed for which \(b_0 =0\) is fixed, and only \(b_1\) and \(b_2\) are fitted to the data. This enforces factorization, so the value of \(\chi ^2/{{\mathrm {dof}}}\) from the fit can be used to asses the validity of the assumption (\(L^*,C^*\) in Table 3). To summarize, for \(H_E(1)\) we find excellent agreement with factorization in the continuum, and a deviation compatible with two standard deviations in the worst case at finite lattice spacing, still statistically consistent with factorization.
For the fits with factorization enforced by fixing \(b_0 = 0\), the values of \(\chi ^2/{{\mathrm {dof}}}\) are also excellent. These values, together with those of \(b_0\) reported in Table 3, give us confidence on the validity of factorization. Notice that the errors at large \(N\) are at least one order of magnitude smaller than the value at \({\mathrm {SU}}(3)\) itself. Concerning the finite \(N\) corrections, comparing the loops at different values of c, we observe that those at large c are characterized by large coefficients in front of the \(1/N^2\) and \(1/N^4\) correction terms.
5.2.1 Loop size dependence
Parameters of the large \(N\) extrapolation of \({\hat{W}}\) and \({\hat{G}}_W\) as a function of \(\xi \)
\(\xi \) | \(a_0\) | \(a_1\) | \(a_2\) | \(b_0\) | \(b_1\) | \(b_2\) |
---|---|---|---|---|---|---|
1.00 | 0.7760(7) | 0.36(3) | \(-0.09(24)\) | 0.00005(37) | 0.167(17) | \(-0.17(15)\) |
1.25 | 0.5956(7) | 0.68(4) | 0.40(34) | 0.0003(13) | 0.752(55) | \(-1.13(44)\) |
1.50 | 0.4087(5) | 1.16(5) | 1.34(48) | 0.0005(31) | 2.62(12) | \(-5.05(92)\) |
1.75 | 0.2512(4) | 1.78(7) | 3.16(71) | 0.0053(65) | 8.06(22) | \(-18.7(17)\) |
2.00 | 0.1390(3) | 2.48(13) | 6.7(12) | 0.030(13) | 26.12(50) | \(-81.0(45)\) |
We observe that in the case of the loops themselves, the coefficients of the \(1/N^2\) expansion do not change significantly with \(\xi \), while those of \({\hat{G}}_W\) grow rapidly for larger loops. In fact, they grow exponentially fast as shown in Fig. 9. At finite \(N\), larger loops are much further away from \(N\rightarrow \infty \) than smaller loops.
6 Conclusions
We have taken the large \(N\) limit of a few observable of \({\mathrm {SU}}(N)\) pure gauge theories numerically defining all dimensionfull quantities in units of \(t_0\). This means that we held \(t_0\), or equivalently the coupling \(\bar{\lambda }_{{\mathrm {GF}}}\), Eq. (2.5), at a low energy fixed in defining the approach to the limit. As explained in Sect. 3, the precise magnitude of \(1/N^2\) corrections do depend on this choice. For each quantity, the continuum limit was taken before the large \(N\) limit, but we have also investigated large \(N\) scaling at finite lattice spacing, defined by \(a^2/t_0=\) constant.
In both cases we find that finite \(N\) observables are very well and very precisely described by a leading order term and corrections \(\sim 1/N^2\) and \(\sim 1/N^4\). We recall for example Fig. 4 where the excellent precision, in particular at finite a, is visible. In the same way, factorization has been confirmed very precisely. Of course, a numerical computation cannot substitute a mathematical proof, but our results make it very implausible that anything goes wrong with the large \(N\) limit in general, or factorization in particular.
However, the magnitude of corrections to the large \(N\) limit is more complex. We found a strong dependence on the physical size of the observables. For example, we considered \(R\times R\) Wilson loops smoothed with a smoothing radius of size again \(\sqrt{8t}=R\). Table 2 shows the deviation, \(\eta \), of SU(3) from SU(\(\infty \)) of these smooth loops to increase from 3% at a loop-size of \(R=0.2 \, {\mathrm {fm}}\) to 10% at \(R=1 \, {\mathrm {fm}}\).
When we increase the loop size R at fixed smoothing radius \(\sqrt{8t}=0.23 \, {\mathrm {fm}}\) from \(R=0.23 \, {\mathrm {fm}}\) to \(R=0.5 \, {\mathrm {fm}}\), the corrections \(\eta (1/9) \approx a_1/9\) (with \(a_1\) from Table 4 or Fig. 9) grow from 4% to more than 30%. The growth with R of the finite \(N\) corrections to factorization is even more dramatic as seen on the right panel in Fig. 9. These large corrections may also contribute to the fact that one has to go to very large \(N\) to approach the large \(N\) limit in the 1-point model [8, 57]. Of course, the dominating effect is expected to be that the color degrees of freedom provide the effective size of the system in that model.
One may also speculate that the growth of factorization violations with the loop size parameter \(\xi \) is so strong that it spoils the large \(N\) limit all together. We do not think that this is the case, but that indeed, it is important to take the limits in the right order: take the \(N\rightarrow \infty \) limit first and then the limit of large loop size. In order to investigate this issue further, one should probably first understand the large \(\xi \) limit at fixed \(N\), maybe just \(N= 3\). Here the relation to the effective string theory of Yang–Mills is likely to play a role [58, 59, 60, 61]. In a second step, one may then consider \(N\) large. Such a demanding programme is beyond the scope of our present work.
In summary, for the quantities studied explicitly, large \(N\) scaling is confirmed with high precision, but corrections to large distance observables can be substantial. One thus has to be careful when deriving quantitative information from large \(N\) considerations in gauge theories.
Footnotes
- 1.
In the lattice discretisation, one just needs to replace \(\int {\mathrm {d}}^3x \rightarrow a^3 \sum _{\vec {x}}.\)
- 2.
Numerically this is of interest because at not-so-small lattice spacing the first step can easily be investigated with a larger range in \(N\). Even more, as shown in the next section, the results at finite lattice spacing can be obtained with higher precision as only an interpolation to a common lattice spacing for all \(N\) is needed, and thus the statistical and systematic errors are greatly reduced when compared to the results of a continuum limit extrapolation.
Notes
Acknowledgements
We would like to thank M. Cè, L. Giusti and S. Schaefer for sharing part of the generation of the gauge configurations [2] and M. Koren for useful discussions. We are grateful to A. Gonzalez-Arroyo and U. Wolff for their valuable comments on our first manuscript. Our simulations were performed at the ZIB computer center with the computer resources granted by The North-German Supercomputing Alliance (HLRN). M.G.V acknowledges the support from the Research Training Group GRK1504/2 “Mass, Spectrum, Symmetry” founded by the German Research Foundation (DFG).
References
- 1.G. t Hooft, A planar diagram theory for strong interactions. Nucl. Phys. B 72, 461 (1974)ADSCrossRefGoogle Scholar
- 2.M. Cè, M. García Vera, L. Giusti, S. Schaefer, The topological susceptibility in the large-N limit of SU( N ) Yang–Mills theory. Phys. Lett. B 762, 232 (2016). arXiv:1607.05939 ADSCrossRefGoogle Scholar
- 3.A. Donini, P. Hernández, C. Pena, F. Romero-López, Nonleptonic kaon decays at large \(N_c\). Phys. Rev. D 94, 114511 (2016). arXiv:1607.03262 ADSCrossRefGoogle Scholar
- 4.T. DeGrand, Y. Liu, Lattice study of large \(N_c\) QCD. Phys. Rev. D 94, 034506 (2016). arXiv:1606.01277 ADSCrossRefGoogle Scholar
- 5.B. Lucini, M. Teper, U. Wenger, Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators. JHEP 06, 012 (2004). arXiv:hep-lat/0404008 ADSMathSciNetCrossRefGoogle Scholar
- 6.A. Gonzalez-Arroyo, M. Okawa, The string tension from smeared Wilson loops at large N. Phys. Lett. B 718, 1524 (2013). arXiv:1206.0049 ADSCrossRefGoogle Scholar
- 7.B. Lucini, A. Rago, E. Rinaldi, SU(\(N_c\)) gauge theories at deconfinement. Phys. Lett. B 712, 279 (2012). arXiv:1202.6684 ADSCrossRefGoogle Scholar
- 8.A. Gonzalez-Arroyo, M. Okawa, Testing volume independence of SU(N) pure gauge theories at large N. JHEP 12, 106 (2014). arXiv:1410.6405 ADSCrossRefGoogle Scholar
- 9.B. Lucini, M. Panero, SU(N) gauge theories at large N. Phys. Rep. 526, 93 (2013). arXiv:1210.4997 ADSMathSciNetCrossRefGoogle Scholar
- 10.M.J. Teper, SU(N) gauge theories in (2+1)-dimensions. Phys. Rev. D 59, 014512 (1999). arXiv:hep-lat/9804008 ADSCrossRefGoogle Scholar
- 11.B. Lucini, M. Teper, SU(N) gauge theories in (2+1)-dimensions: further results. Phys. Rev. D 66, 097502 (2002). arXiv:hep-lat/0206027 ADSCrossRefGoogle Scholar
- 12.B. Bringoltz, M. Teper, A precise calculation of the fundamental string tension in SU(N) gauge theories in 2+1 dimensions. Phys. Lett. B 645, 383 (2007). arXiv:hep-th/0611286 ADSMathSciNetCrossRefGoogle Scholar
- 13.J. Liddle, M. Teper, The deconfining phase transition in D=2+1 SU(N) gauge theories. arXiv:0803.2128
- 14.A. Athenodorou, M. Teper, SU(N) gauge theories in 2+1 dimensions: glueball spectra and k-string tensions. JHEP 02, 015 (2017). arXiv:1609.03873 ADSMathSciNetCrossRefGoogle Scholar
- 15.L.G. Yaffe, Large n limits as classical mechanics. Rev. Mod. Phys. 54, 407 (1982)ADSMathSciNetCrossRefGoogle Scholar
- 16.E. Witten, The 1/N expansion in atomic and particle physics. NATO Sci. Ser. B 59, 403 (1980)Google Scholar
- 17.S.R. Coleman, 1/N, in 17th International School of Subnuclear Physics: Pointlike Structures Inside and Outside Hadrons Erice, Italy, July 31–August 10, 1979 (1980), p. 0011. http://www-public.slac.stanford.edu/sciDoc/docMeta.aspx?slacPubNumber=SLAC-PUB-2484
- 18.P. Rossi, M. Campostrini, E. Vicari, The large N expansion of unitary matrix models. Phys. Rep. 302, 143 (1998). arXiv:hep-lat/9609003 ADSMathSciNetCrossRefGoogle Scholar
- 19.P.D. Vecchia, R. Musto, F. Nicodemi, R. Pettorino, P. Rossi, The transition from the lattice to the continuum: \(cp^{N - 1}\) models at large n. Nucl. Phys. B 235, 478 (1984)ADSCrossRefGoogle Scholar
- 20.M. Campostrini, P. Rossi, The 1/N expansion of two-dimensional spin models. Riv. Nuovo Cim. 16, 1 (1993)CrossRefGoogle Scholar
- 21.M. Campostrini, P. Rossi, The 1/N expansion of two-dimensional spin models. Nucl. Phys. Proc. Suppl. 34, 680 (1994). arXiv:hep-lat/9305007 ADSCrossRefGoogle Scholar
- 22.T. Eguchi, H. Kawai, Reduction of dynamical degrees of freedom in the large N gauge theory. Phys. Rev. Lett. 48, 1063 (1982)ADSCrossRefGoogle Scholar
- 23.J. Kiskis, R. Narayanan, H. Neuberger, Does the crossover from perturbative to nonperturbative physics in QCD become a phase transition at infinite N? Phys. Lett. B 574, 65 (2003). arXiv:hep-lat/0308033 ADSCrossRefGoogle Scholar
- 24.R. Narayanan, H. Neuberger, Large N reduction in continuum. Phys. Rev. Lett. 91, 081601 (2003). arXiv:hep-lat/0303023 ADSMathSciNetCrossRefGoogle Scholar
- 25.A. Gonzalez-Arroyo, M. Okawa, The twisted Eguchi–Kawai model: a reduced model for large N lattice gauge theory. Phys. Rev. D 27, 2397 (1983)ADSCrossRefGoogle Scholar
- 26.A. Gonzalez-Arroyo, M. Okawa, A twisted model for large \(N\) lattice gauge theory. Phys. Lett. B 120, 174 (1983)ADSCrossRefGoogle Scholar
- 27.A. Gonzalez-Arroyo, M. Okawa, Large \(N\) reduction with the twisted Eguchi–Kawai model. JHEP 07, 043 (2010). arXiv:1005.1981 ADSMathSciNetCrossRefGoogle Scholar
- 28.YuM Makeenko, A.A. Migdal, Exact equation for the loop average in multicolor QCD. Phys. Lett. B 88, 135 (1979)ADSCrossRefGoogle Scholar
- 29.G. Bhanot, U.M. Heller, H. Neuberger, The quenched Eguchi–Kawai model. Phys. Lett. B 113, 47–50 (1982)ADSCrossRefGoogle Scholar
- 30.G. Cossu, M. D’Elia, Finite size phase transitions in QCD with adjoint fermions. JHEP 07, 048 (2009). arXiv:0904.1353 ADSCrossRefGoogle Scholar
- 31.B. Bringoltz, S.R. Sharpe, Non-perturbative volume-reduction of large-N QCD with adjoint fermions. Phys. Rev. D 80, 065031 (2009). arXiv:0906.3538 ADSCrossRefGoogle Scholar
- 32.A. Chatterjee, D. Gangopadhyay, Factorization in large N limit of lattice gauge theories revisited. Pramana 21, 385 (1983)ADSCrossRefGoogle Scholar
- 33.Y. Makeenko, Large N gauge theories. arXiv:hep-th/0001047
- 34.S. Chatterjee, Rigorous solution of strongly coupled \(SO(N)\) lattice gauge theory in the large \(N\) limit. arXiv:1502.07719
- 35.J. Jafarov, Wilson loop expectations in \(SU(N)\) lattice gauge theory. arXiv:1610.03821
- 36.M. Lüscher, P. Weisz, Perturbative analysis of the gradient flow in non-abelian gauge theories. JHEP 02, 051 (2011). arXiv:1101.0963 ADSMathSciNetCrossRefGoogle Scholar
- 37.V.S. Dotsenko, S.N. Vergeles, Renormalizability of phase factors in the nonabelian gauge theory. Nucl. Phys. B 169, 527 (1980)ADSMathSciNetCrossRefGoogle Scholar
- 38.R.A. Brandt, F. Neri, M.-A. Sato, Renormalization of loop functions for all loops. Phys. Rev. D 24, 879 (1981)ADSCrossRefGoogle Scholar
- 39.H. Dorn, Renormalization of path ordered phase factors and related hadron operators in gauge field theories. Fortsch. Phys. 34, 11 (1986)ADSMathSciNetGoogle Scholar
- 40.M. Creutz, Monte Carlo Study of Renormalization in Lattice Gauge Theory. Phys. Rev. D 23, 1815 (1981)ADSCrossRefGoogle Scholar
- 41.R. Narayanan, H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators. JHEP 03, 064 (2006). arXiv:hep-th/0601210 ADSMathSciNetCrossRefGoogle Scholar
- 42.M. Lüscher, Properties and uses of the Wilson flow in lattice QCD. JHEP 08, 071 (2010). arXiv:1006.4518 ADSMathSciNetCrossRefGoogle Scholar
- 43.M. Okawa, A. Gonzalez-Arroyo, String tension from smearing and Wilson flow methods. PoS LATTICE 2014, 327 (2014). arXiv:1410.7862 Google Scholar
- 44.R. Lohmayer, H. Neuberger, Rectangular Wilson loops at large N. JHEP 08, 102 (2012). arXiv:1206.4015 ADSMathSciNetCrossRefGoogle Scholar
- 45.R. Lohmayer, H. Neuberger, Non-analyticity in scale in the planar limit of QCD. Phys. Rev. Lett. 108, 061602 (2012). arXiv:1109.6683 ADSCrossRefGoogle Scholar
- 46.B. Alles, A. Feo, H. Panagopoulos, The three loop Beta function in SU(N) lattice gauge theories. Nucl. Phys. B 491, 498 (1997). arXiv:hep-lat/9609025 ADSCrossRefGoogle Scholar
- 47.R.F. Dashen, D.J. Gross, The relationship between lattice and continuum definitions of the gauge theory coupling. Phys. Rev. D 23, 2340 (1981)ADSCrossRefGoogle Scholar
- 48.G. ’t Hooft, Large N, in Phenomenology of large N(c) QCD. Proceedings, Tempe, USA, January 9–11, 2002 (2002), pp. 3–18. arXiv:hep-th/0204069
- 49.M. Lüscher, S. Schaefer, Lattice QCD without topology barriers. JHEP 07, 036 (2011). arXiv:1105.4749 ADSCrossRefGoogle Scholar
- 50.N. Cabibbo, E. Marinari, A new method for updating SU(N) matrices in computer simulations of gauge theories. Phys. Lett. B 119, 387 (1982)ADSCrossRefGoogle Scholar
- 51.M. Cè, C. Consonni, G.P. Engel, L. Giusti, Non-gaussianities in the topological charge distribution of the SU(3) Yang–Mills theory. Phys. Rev. D 92, 074502 (2015). arXiv:1506.06052 ADSCrossRefGoogle Scholar
- 52.R. Sommer, A new way to set the energy scale in lattice gauge theories and its applications to the static force and alpha-s in SU(2) Yang–Mills theory. Nucl. Phys. B 411, 839 (1994). arXiv:hep-lat/9310022 ADSCrossRefGoogle Scholar
- 53.M. García Vera, S. Schaefer, Multilevel algorithm for flow observables in gauge theories. Phys. Rev. D 93, 074502 (2016). arXiv:1601.07155 ADSCrossRefGoogle Scholar
- 54.M.G. Vera, R. Sommer, Large \(N\) scaling and factorization in \({{\rm SU}}(N)\) Yang–Mills theory. EPJ Web Conf. 175, 11018 (2018). arXiv:1710.06057 CrossRefGoogle Scholar
- 55.M. Cè, M. García Vera, L. Giusti, S. Schaefer, The large \({N}\) limit of the topological susceptibility of Yang–Mills gauge theory. PoS LATTICE 2016, 350 (2016). arXiv:1610.08797 zbMATHGoogle Scholar
- 56.ALPHA collaboration, M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Sint, R. Sommer, Slow running of the gradient flow coupling from 200 MeV to 4 GeV in \(N_{{\rm f}}=3\) QCD. Phys. Rev. D 95, 014507 (2017). arXiv:1607.06423
- 57.A. González-Arroyo, M. Okawa, Large N meson masses from a matrix model. Phys. Lett. B 755, 132 (2016). arXiv:1510.05428 ADSCrossRefGoogle Scholar
- 58.Y. Nambu, QCD and the string model. Phys. Lett. B 80, 372–376 (1979)ADSCrossRefGoogle Scholar
- 59.M. Luscher, K. Symanzik, P. Weisz, Anomalies of the free loop wave equation in the WKB approximation. Nucl. Phys. B 173, 365 (1980)ADSMathSciNetCrossRefGoogle Scholar
- 60.M. Luscher, P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation. JHEP 07, 014 (2004). arXiv:hep-th/0406205 ADSMathSciNetCrossRefGoogle Scholar
- 61.O. Aharony, N. Klinghoffer, Corrections to Nambu–Goto energy levels from the effective string action. JHEP 12, 058 (2010). arXiv:1008.2648 ADSMathSciNetCrossRefGoogle Scholar
Copyright information
Open AccessThis 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}