Is $N=2$ Large?

We study $\theta$ dependence of the vacuum energy for the 4d SU(2) pure Yang-Mills theory by lattice numerical simulations. The response of topological excitations to the smearing procedure is investigated in detail, in order to extract topological information from smeared gauge configurations. We determine the first two coefficients in the $\theta$ expansion of the vacuum energy, the topological susceptibility $\chi$ and the first dimensionless coefficient $b_2$, in the continuum limit. We find consistency of the SU(2) results with the large $N$ scaling. By analytic continuing the number of colors, $N$, to non-integer values, we infer the phase diagram of the vacuum structure of SU(N) gauge theory as a function of $N$ and $\theta$. Based on the numerical results, we provide quantitative evidence that 4d SU(2) Yang-Mills theory at $\theta = \pi$ is gapped with spontaneous breaking of the CP symmetry.


I. INTRODUCTION
The θ term of the Yang-Mills theory determines how to weight different topological sectors in the path integral. Since the θ parameter is the coefficient of a total derivative term in the Lagrangian, the θ-dependences of observables can be explored only through non-perturbative methods.
The special value θ = π has been of particular interest. In the classic literature [1][2][3], spontaneous CP violation of the 4d SU(N) Yang-Mills theory at θ = π was demonstrated in the large N limit [4]. More recently, an anomaly matching argument involving generalized global symmetries [5] showed that the CP symmetry in the confining phase has to be broken even at finite N [6]. A similar conclusion was derived by studying restoration of the equivalence of local observables between SU(N) and SU(N)/Z N gauge theories in the infinite volume limit [7]. See, for example, Refs. [8][9][10] for other approaches.
While lattice numerical simulations are ideal tools to explore non-perturbative dynamics of gauge theories, direct simulations at θ = π has been challenging due to the notorious sign problem. 1 Nevertheless, lattice simulations have been successfully used to determine the first few coefficients in the θ expansion of the vacuum energy for finite N. On the one hand, below the critical temperature T c these coefficients turn out to be consistent with the large N scaling down to N = 3 [14][15][16], which indicates spontaneous CP violation and the discontinuity of the vacuum energy across θ = π. On the other hand, above T c the coefficients determined at N = 3 and 6 are found to be consistent with the dilute instanton gas approximation (DIGA) [17], which predicts continuous behavior for the vacuum energy across θ = π.
The CP N−1 model in two dimensions shares many non-perturbative properties with the four-dimensional SU(N) Yang-Mills theory [18,19], and hence provides useful insights into the latter. For N ≥ 3, the model is believed to show spontaneous CP violation at θ = π [20].
By contrast the case with N = 2 is believed to be special and argued to become gapless at θ = π with unbroken CP symmetry [21][22][23][24][25][26][27][28][29]. Motivated by similarities between the 4d Yang-Mills theory and the 2d CP N−1 model, it is natural to ask if the 4d SU(N) Yang-Mills theory at θ = π shows distinctive behavior for small values of N, such as N = 2. 2 In this work we explore the θ dependence of the vacuum energy of the 4d SU(2) pure Yang-Mills gauge theory. In sec. II, we perform lattice numerical calculations to determine the first two coefficients in the θ expansion of the vacuum energy. The response of topological excitations to the smearing procedure is investigated in detail, in order to efficiently extract physical information from lattice configurations. The coefficients determined for N = 2 are compared to those previously obtained for N ≥ 3, to see whether the result at N = 2 can be seen as a natural extrapolation of those for N ≥ 3. In sec. III, we begin with theoretical 1 Recent and related developments towards direct simulations are found, for example, in Refs. [11][12][13]. 2 The Z N subgroup of the flavor symmetry of the 2d CP N−1 model can be regarded as a counterpart of the 1-form Z N center symmetry of the 4d SU(N ) pure Yang-Mills theory.
arguments for different behaviors of 4d SU(N) theory, for large N and for small N as we analytically continue the values of N. We then interpret the numerical results of sec. II and provide quantitative evidence that the 4d SU(2) theory belongs to the "large N" class, and is gapped and has spontaneous breaking of CP symmetry at θ = π.

II. LATTICE SIMULATIONS
The vacuum energy can be expanded around θ = 0 as where χ is the topological susceptibility, and b 2i (i = 1, 2, 3, · · · ) are dimensionless coefficients describing the deviation of the topological charge distribution from the Gaussian. These quantities can be determined from the lattice configurations generated at θ = 0 as where Q is the topological charge, whose precise definition is given in eqs. (10)- (14), and · · · θ=0 denotes an ensemble average over configurations generated at θ = 0. According to the large N analysis [1,3], these quantities can be expressed, as a function of N, as By contrast the dilute instanton gas approximation leads to E(θ) − E(0) = χ(1 − cos θ), and hence the coefficients, b DIGA 2 = −1/12, b DIGA 4 = 1/360, · · · , are completely determined.
We attempted calculating b 4 as well as χ and b 2 . We could obtain only a loose bound −0.1 < b 4 < 0.1 due to a large statistical uncertainty. In the following, we focus on the determinations of χ and b 2 .   [31]. The uncertainties of N Tc are below 1% and hence neglected in the following.

A. Lattice Setup
The SU(2) gauge action on the lattice is described as where β = 4/g 2 is the lattice gauge coupling, W P and W R are the 1 × 1 plaquette and the 1 × 2 rectangle averaged over four dimensional lattice sites, respectively, and c 0 and c 1 satisfying c 0 = 1 − 8c 1 are the improvement coefficients. We take the tree-level Symanzik improved action [30], which is realized by c 1 = −1/12. To investigate the continuum limit, three values of the lattice coupling (β=1.750, 1.850 and 1.975) are taken. The lattice size is N site = N 3 S × N T with N S = 16 and N T = 2 × N S . We also perform simulations with N S = 24 on our finest lattice to check finite volume effects. The lattice spacing at each β is taken from N Tc = 1/(a(β)T c ) obtained in Ref. [31], where T c is the critical temperature for the SU(2) pure Yang-Mills theory. The value N Tc is then transformed to (aT c ) 2 for later use. To have an intuition about how large our lattice is, we estimated L σ 1/2 str at each lattice, using T c / √ σ str = 0.7091(36) [32], where L denotes the physical length of the spatial direction, i.e. L = a N S , and σ str is the representative dynamical scale (the sting tension).
where Proj acts as the projection back to an SU(2) element. This procedure minimizes the action density. The parameter ρ is taken to be 0.2, which corresponds to α APE = 6ρ/(1 + 5 ρ) = 0.6 in Ref. [35].

C. Response to Smearing
As mentioned above, the smearing is introduced to remove short distance fluctuations, which distort physical topological excitations through local lumps with the size of the lattice spacing. The measurement of the topological charge is therefore reliable only after a suitable number of smearing steps. However, the smearing may also affect physical topological excitations. The previous dedicated studies revealed that the smearing induces pair-annihilation, "melting away" or "falling through the lattice" [37][38][39]. In Ref. [39], it was found that topological objects go through several characteristic phases during the cooling procedure. In the first phase, the size of topological objects grows with the cooling, and some of them eventually melt away and some pair-annihilate. Then, the second phase comes where only relatively slow shrinkage of the objects takes place and eventually they disappear after long enough cooling. Assuming that the similar phases show up in the procedure of APE smearing, we will in the following determine the boundary between the two phases.
In order to explore how the smearing changes topological properties, we first look at the smearing history of the topological charge Q as a function of the smearing steps n APE . Fig. 1 shows the history obtained at β = 1.750 and 1.975. At relatively small n APE , Q changes frequently, and most of the changes here are expected to be associated with the removal of short distance fluctuations. We deduce that this range of n APE corresponds to the first phase. The frequency of change in Q is somewhat reduced as n APE and β increase, but the change steadily continues. In this region, both the increase and the decrease of Q happen mostly by one unit, and a change takes O(10) steps to be completed. This range of n APE is identified with the second phase. In the following, we discuss quantitative differences between the two phases.
We studied the correlation between the topological charge Q and the value of the action S g . At the same time, we also investigated the direction of change of Q per one step of the smearing, by classifying each configuration at a given n APE into three classes: • "stable" if the change is small, i.e. |Q(n APE ) − Q(n APE − 1)| ≤ 0.05.
If not stable, • "decreasing" if Q(n APE ) is approaching zero, • "increasing" if Q(n APE ) is moving away from zero. where the Bogomolnyi bound, S g = 8 π 2 |Q|/g 2 is shown by dotted lines. The "stable", "decreasing" and "increasing" data points are shown in blue, red and green, respectively.
There is no qualitative difference in the same plot for other values of β. It is seen that points gradually accumulate on integer values of Q by "increasing" or "decreasing". The value of the action is never below the Bogomolnyi bound in each topological sector, as expected. This indicates that "increasing" data can not exist around the boundary because the smearing lowers the action value and only either instantons(s) or anti-instanton(s) can exist on the bound. It is also seen that the larger the value of |Q| is, faster the minimum of the action in the topological sector reaches the Bogomolnyi bound. Thus, the minimum value of S g in the Q = 0 sector arrives at the bound, i.e. S g = 0, last.
Instantons are known to saturate the Bogomolnyi bound. Therefore, the data points with nonzero Q around the bound are attributed to approximate instantons or anti-instantons, and the "decreasing" occurring around there are interpreted as (anti-) instantons "falling through the lattice". We expect that all the "increasing" and "decreasing" in the second phase are caused by "falling". In order to examine this expectation, we introduce the participation ratio defined by where q(x, n APE ) denotes the topological charge density q(x) in eq. (11) after n APE steps of smearing. The participation ratio takes a value between 1/N site and 1. The maximal value P (n APE ) = 1 is realized when q(x, n APE ) takes a flat distribution over the whole space-time.
On the other hand, the possible minimum value, 1/N site , is attained when the density forms a local peak, q(x, n APE ) = δ(x). Fig. 3 shows the smearing history of Q and ln P as a function of n APE for one particular configuration at β = 1.850. For n APE > ∼ 30, whenever Q changes, ln P shows a rapid increase after slow decrease over many smearing steps. This can be interpreted as that a local object in topological charge density gradually shrinks and suddenly disappears at some point with a change of Q. This is precisely what happens when the "falling through the lattice" occurs [39].
We can directly check this interpretation by studying the distribution of the topological charges. Fig. 4 shows the topological charge density, projected onto the z-t plane, of the same configuration as in Fig. 3. Between n APE = 50 and 60 and n APE = 450 and 470, Q increases by unity, at the same time a negative peak disappears. Between n APR = 100 and 200, a positive peak seems to be smeared but does not suddenly disappear. It seems that a complicated process such as a pair annihilation happens in the latter case.
From these observations, we conclude that the changes of Q occurring in the second phase are dominated by the "falling" of instantons or anti-instantons. We expect that the "falling" occurs also in the first phase, but it is overshadowed by changes originating from other reasons.
Instanton and anti-instantons will "fall" at an equal rate. In configurations with Q > 0, more instantons exist than anti-instantons and vice versa. Then, it is expected that the "decreasing" would happen more frequently than the "increasing" in the second phase. To see if this is the case, we calculate the ensemble average of The sign of ∆Q(n APE ) tells us which of "increasing" or "decreasing" happens when going from n APE to n APE + 1. show exponential fall with approximately a common exponent for n APE < ∼ 10, while they take almost constant negative values for n APE > ∼ 20. The result for β = 1.975 and N S = 16 (triangle-up) shows slightly different behaviors probably because of the small physical volume. At any rate, this plot clearly shows that the boundary separating the two phases is located n APE ∼ 20. In the following analysis, we only deal with the data for n APE ≥ 20, where the short distance fluctuations are gone.
Before closing this subsection, let us add one comment. In Ref. [40], the shape of topological objects in SU(3) gauge theory is examined, and low dimensional long range structures rather than local lumps are discovered. Note that the analysis presented above does not indicate anything about the shape because the smearing changes it. Clearly, it is interesting to perform a similar study in the SU(2) case because the analysis performed in Refs. [41,42] suggests that the structure could be more localized for SU(2) than for SU(3).

D. Results
The fit results are tabulated in Tab. II.   with N = 3, 4, 6 down to L σ str ∼ 2.5, and no finite volume effect is observed. Our lattice with β = 1.975 and N S = 16 corresponds to L σ str = 2.4 (see Tab. I), which is smaller than but close to 2.5 and hence finite volume effects, if any, should not be significant. Thus, 1.8 σ difference observed at β = 1.975 is considered as a statistical fluctuation, and we include both results in the following analysis.
Next we discuss the continuum limit. Fig. 10 shows the extrapolation of χ/T 4 c and b 2 to the continuum. The limit for both quantities is examined by applying two functional forms: where the errors are summed in quadrature.
No result is available in the continuum limit for N = 2. 3 Fig. 11 shows the summary plot for χ/σ 2 str and b 2 , including our results. In this plot, we use T c / √ σ str = 0.7091 (36) [32] to change the normalization to χ/σ 2 str . The solid lines shown in the plots are the linear fit performed in Ref. [16] using the data at N = 3, 4, 6.
The results of χ/σ str for SU(2) theory are slightly above than the solid line, but the deviation is accountable by the next leading order correction, which is of O(1/N 2 ) relative to the leading one. It is then natural to expect that the dynamics of SU(2) gauge theory is a smooth extrapolation of the large N dynamics to N = 2, and that nothing special happens in between.
The value of b 2 at N = 2 obtained in this work turns out to be consistent with the instanton prediction, b DIGA 2 = −1/12, within 1.7 σ. However, it is more consistent with the naive linear extrapolation from the N ≥ 3 data to N = 2. This observation gives further support to the above expectation, i.e. nothing special happens between N ≥ 3 and N = 2.
see Fig. 12. This in particular means that there are two different lowest-energy states for θ = π mapped each other under the CP symmetry, and hence we expect CP to be spontaneously broken.
One should notice that the vacuum energy takes a different functional form as expected from the semiclassical one-instanton calculation [49], which gives the free energy density as Here ρ is the size modulus of the instanton, g(µ) is the running gauge coupling constant at the energy scale µ, and b 1 := 11N/3 is the coefficient of the one-loop beta function.
One of the reasons for the discrepancy between the semi-classical instanton analysis and the large N analysis resides in the famous IR divergence of the instanton analysis: the integral (20) is divergent as ρ becomes large.

Small N
The situation can be different when N is small. Let us regard N as a real parameter.
Then the integral (20) has a UV divergence at ρ → 0, if the values of N is smaller than the threshold value N inst = 12/11 [50]. In this case, the integral (20) should be regularized in the UV by the cutoff scale M, so that the lower value of the integral for ρ is given by M −1 .
Note that the UV regularization is needed even though the pure Yang-Mills theory in itself is asymptotic free.
We thus expect that the one-instanton contribution to have the schematic expression where Λ is the dynamical scale of the theory. This is a different qualitative behavior as suggested by the large N analysis, (19). 4 There can be several questions to this narrative. First, in the analysis above for small N, we evaluated only the instanton corrections, and one might object that there can be many other contributions to the partition function. While this is certainly true, let us note that these non-instanton contributions give rise to contributions of O(Λ 4 ). Since the cutoff scale M is much larger than the dynamical scale Λ (M ≫ Λ), these non-instanton contributions are much smaller than the instanton contribution of (21). Since we do know that instanton contributes to the path integral, we are certain that there is contribution of the form (21), which we expect will dominate over other contributions. 5 Note that the UV cutoff M dependence will appear only as an overall divergence. This means that while the topological susceptibility depends on the UV cutoff, the coefficients however changes the asymptotic behavior as N → ∞. There is a mathematical theorem [54] which guarantees that two real-valued functions, with suitable asymptotic conditions at infinity and with the same values at all integers, coincide. Such considerations are actually implicit in the large N analysis, and makes it possible to discuss small non-integer values of N (even to N < 2). The threshold value N inst for the instanton calculus makes sense in this context. The situation is different for the special value of θ = π, which has the CP symmetry in the Lagrangian. In this case, we can define two phases by the presence or the absence of the CP symmetry; the CP symmetry is spontaneously broken for large N, while for smaller N the vacuum energy may be given by the cosine form and hence CP is preserved. In that case, there exists a critical value of N = N CP between the two phases.
We expect that the two phases are different also in that whether the vacuum is gapped or gapless. This is because of the mixed anomaly between the Z N center symmetry and the CP symmetry [6], and the presence of the Z N symmetry can be regarded as a definition of the confinement. Possible phase diagrams are shown in Fig. 13, where the presence/absence of the CP symmetry is assumed to coinside with the gapped/gapless system. Of course, the presence of the mixed anomaly only shows that at least either the center symmetry or the CP symmetry should be broken, and allows for the possibility that both are broken.
Here, the existence of N CP is our assumption motivated by the phase structure of the CP N−1 model we discuss below. Once it is assumed, we need to discuss how the gapless phase extends to the θ = π region. In the figure, we show a possibility that the gapless theory is realized even at θ = 0 at some N. There are other possibilities that the line does not reach to θ = 0 axis, as well as the possibility that gapless theories are realized only on the θ = π line. Note that irrespective of the possible phase structures we define the critical value N CP by the presence/absence of the CP symmetry. Note that the value of N inst is not necessarily the same as the critical value N CP ; the former is defined purely for the semiclassical instanton computation applicable for generic values of θ, while the critical value N CP is the value separating the CP broken/preserved phases at the special value θ = π. It is not a priori clear if we expect general inequalities between the two values N inst and N CP . One may be tempted to imagine that N inst should always be smaller than N CP since the potential generated by the instanton is always smooth so that the spontaneous CP breaking does not happen. However, although it is certain that the contributions from the small instantons dominate the instanton density for N < N inst , one cannot exclude the possibility that non-trivial infrared physics still leads the spontaneous CP breaking and/or confinement at θ = π.
Let us next come to more quantitative aspects. In the instanton calculus, we obtained the threshold value of N inst = 12/11, which is smaller than N = 2. The estimation by the one-loop beta function is justified by the asymptotic freedom. Therefore, it is expected that the SU(2) gauge theory has a UV-independent value of the topological susceptibility. As we discussed already, however, one cannot conclude whether N CP < 2 holds or not only from this discussion. Below, we examine the lattice results of the θ-dependence of the theory and discuss whether N = 2 is small or large more carefully.

Comparison with the CP N−1 Model
It is useful to compare the 4d Yang-Mills theory with the celebrated CP N−1 model in two dimensions [18,19]. This theory has many similarities with the 4d SU(N) Yang-Mills theory, and could be of help in understanding the non-perturbative properties of the latter. 6 In the large N limit of the 2d CP N−1 model, there exists gap at any values of θ, and the vacuum energy is discontinuous at θ = π, where the CP symmetry is spontaneously broken. There seems to be a consensus that these properties takes over down to N = 3.
The situation is different from the N = 2 case, i.e. the CP 1 model, which is nothing but the O(3) spin model. This model is believed to be gapless and have continuous vacuum energy at θ = π [56].
When we consider N as a continuous parameter again, one expects that there will be a critical value of N between the two phases, which we denote by N CP . For N > N CP the theory has spontaneous CP breaking at θ = π, and we expect that the dependence on the parameter N is accounted by the large N scaling: we call this the "large N phase." By contrast for N < N CP we have an unbroken CP symmetry for θ = π, and the semiclassical instanton analysis applies: we call this the "small N phase." The computation similar to (20) gives the threshold value N inst = 2 for the CP N−1 model, consistent with the divergence of topological susceptibility for the the CP 1 -model. The most remarkable difference of the CP 1 model from other (N > 2) CP N−1 models is that the semi-classical calculation of the former leads to a UV divergence in the topological susceptibility. This is supported by lattice numerical calculations, unless a suitable counter term is added [50,[57][58][59][60][61][62][63][64] 7 .

N = 2
We have already seen that the semi-classical estimate of the topological susceptibility χ in 4d SU(N) gauge theory does not yield UV divergence even for the possible smallest value, N = 2. The continuum limit of lattice numerical calculations serves as an independent quantitative test of this expectation.
Although an extrapolation of numerical data is always subtle, and especially the continuum limit of quantities related to topological charge needs special care, our result as well as previous results in the literature demonstrate the finiteness of χ for SU(N) gauge theory, all the way to the value N = 2.
As shown in Fig. 11, the magnitude of |b 2 | obtained for N = 2 is slightly smaller than that of the instanton prediction b DIGA 2 = −1/12. The value of |b 4 | in Ref. [47] is much smaller than the instanton value. Both are rather consistent with the 1/N 2 and 1/N 4 scalings of the N ≥ 3 data as we discuss further below. This suggests the invalidity of the instanton description. One also expects that the vacuum energy has a cusp at θ = π due to small values of |b 2 | and |b 4 |. All these results suggest that N = 2 is "large" for the four-dimensional Yang-Mills theory.
In the literature there have been some attempts to analyze the vacuum of the 4d SU (2) theory at θ = π. For example, Ref. [66] analyzes the question for a suitable double-trace deformation of the 4d Yang-Mills theory via the semiclassical analysis and a twisted compactification, and obtained the results consistent with ours. It should be kept in mind, however, that any deformation of the theory, often needed for the semiclassical analysis, could potentially change the vacuum structure of the theory, let alone the precise values of N inst and N CP . It is also the case that for our discussion it is crucial to discuss the transition between small N and large N behaviors, as we will discuss a few paragraphs below.
Our result should be contrasted with the case of the 2d CP N−1 model, where N = 2 case is gapless and CP preserving at θ = π, as already mentioned before. This is an excellent demonstration of the quantitative differences between four-dimensional Yang-Mills theory and the two-dimensional CP N−1 model.
Notice that the relation between the 4d SU(N) Yang-Mills theory and the 2d CP N−1 model was further clarified in [67], which showed that the T 2 × S 1 compactification of the former with suitable 't Hooft magnetic flux gives rise to S 1 compactification of the two-dimensional sigma model whose target space has the topology of CP N−1 (see Refs. [9,10] for further checks via anomalies). A caution is needed, however, before any quantitative comparisons between the two. The two-dimensional model obtained from four-dimensional theory has a non-standard metric, and in addition there are special points (fixed points under the Weyl group action) in the CP N−1 where we encounter W-bosons of the four-dimensional theory [67]. Moreover for the analysis of [67] it was crucial to have a hierarchy of scales between the sizes of T 2 and S 1 , and any discussion of the standard flat space limit (where there is no such hierarchy) requires careful analytic continuation. These subtleties can easily affect quantitative discussions here.

N inst
Once we are settled with the case of N = 2, we can discuss even smaller values of N and ask how the theory approaches the N < N inst region.
We first pretend that N inst is unknown and try to determine its value by the lattice data by two methods. The first method uses topological susceptibility, which we expect to diverge at the value N = N inst . To determine this value we fit our N = 2 result together with those for N = 3, 4, 6 in Ref. [16] by an Ansatz where N inst is assumed to be a real number. Here the Ansatz is the simplest function of N 2 which has divergence at N = N inst and approaches to the large N value as N → ∞. 8 We then obtain which is shown as the dashed curve in Fig. 11 denoted by "ph fit" meaning phenomenological fit.
The second method uses the values of b 2 . Supposing that the semi-classical calculation becomes valid at N = N inst for SU(N) gauge theory, b 2 is expected to take b DIGA 2 = −1/12 at the same value of N. We again use the results for b 2 for N = 2, 3, 4, 6 to test this expectation. This time, by fitting the data to we obtain Substituting (23) and (25) into (24)  The two methods produce consistent estimates N inst ∼ 1.5, slightly larger than the semiclassical value 12/11. This numerology serves as a check of the overall picture, and moreover indicates that the large N scaling of b 2 holds well all the way until the value N ∼ N inst , where χ diverges. If we assume large N scaling for all the b 2n 's until N ∼ N inst , then certain derivatives of the free energy are necessarily discotinuous at θ = π, thus implying the breaking of the CP symmetry. This suggests the inequality N CP N inst . Further numerical studied are needed to make this inequality more precise.
Summarizing our discussion, the numerical data suggests the following shape of the vacuum energy density, as we change the value of N. At large N we have the quadratic form of the vacuum energy around θ = 0, while there is a cusp at θ = π. As we change N to smaller values, b 2 and b 4 grow while continue to obey the large N scaling to a good approximation.
The cusp of the vacuum energy at θ = π is gradually smoothened, however not completely; CP is still spontaneously broken. The transition of the large N picture to the instanton picture seems to be smooth as far as b 2n are concerned, and N = 2 is on the "large N" side. At N inst ∼ 1.5 the free energy approaches the cosine function with a diverging overall factor, χ. Once χ is diverging, it becomes difficult to infer the vacuum structure from the vacuum energy only, since the vacuum energy is masked by the large contributions of small instantons. In particular, it becomes invisible in practice whether or not there is a phase transition at θ = π.

IV. SUMMARY AND DISCUSSION
We performed lattice numerical simulations to explore the θ dependence of the vacuum energy in 4d SU(2) pure Yang-Mills theory, with special attention to the response of topological excitation to the smearing procedure. We discussed the method to extract the topological information from smeared configurations properly and estimated the first two coefficients in the θ expansion of the vacuum energy in the continuum limit, namely χ and b 2 . The value of χ turns out to be consistent with the previous results in the literature, while b 2 is determined for the first time.
We use these results to infer the phase structure of the 4d SU(N) theory as we change the values of N and θ. We highlighted the differences for "large N" and for "small N": we have the large N scaling for the former, while the vacuum energy is dominated by instantons in the latter. The differences between the two is most clear-cut for θ = π, where the CP symmetry is spontaneously broken for large N, while unbroken for small N.
We found that for N = 2 the topological susceptibility χ remains finite, and b 2 slightly deviates from the instanton predictions, while it is well fitted by the 1/N 2 extrapolation from the N ≥ 3 values. By further extrapolating N to small values by analytic continuation, we find that χ and b 2 reach the instanton predictions at N inst = 1.52 (2), and that the free energy will be dominated by instantons.
Our analysis gives strong quantitative evidence that the SU(2) theory, and hence all SU(N) theories for integer N, are in the large N category. While large N analysis is often regarded as an approximation applicable only to the large values of N, our results suggest that the large N analysis is more powerful, and can be useful for studying all possible values of N, even as small as N = 2. This is in contrast with the case of the 2d CP N−1 model, which is believed to be gapless for N = 2, while are gapped for N ≥ 3. It would be interesting to study for more general theories the applicability of large N analysis to smaller values of N.
In this work, we could not explore the question of precisely what topological object carries non-zero topological charges. In Ref. [40], it was pointed out that in SU(3) Yang-Mills theory the codimension-one objects ("sheets") are responsible for the non-zero topological charges, and from the similar study of CP N−1 model it was pointed out that the object becomes localized and becomes instantons as N → N CP . Thus, it is interesting to see what objects are responsible for the topological charges in the 4d SU(2) theory.
It is also interesting to see the θ dependence of the vacuum energy directly on the lattice, for finite real values of θ, especially near θ = π. This program has to overcome notoriously difficult problem, the sign problem. Since recent development in methodology is remarkable [11][12][13], such direct studies appear to be within reach in the near future.
Finally, it is interesting to ask if the analysis of this paper has any phenomenological considerations of the dynamical θ-angle, the axion [68][69][70][71]. For example, in the axionic inflationary models of Ref. [72] (see also Ref. [73]), the values of b 2n affect future observations of primordial gravitational waves from inflation [48].