aa r X i v : . [ h e p - l a t ] O c t Is N = 2 Large?
Ryuichiro Kitano , , ∗ Norikazu Yamada , , † and Masahito Yamazaki ‡ High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan Graduate University for Advanced Studies (SOKENDAI), Tsukuba 305-0801, Japan Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa, Chiba 277-8583, Japan (Dated: October 20, 2020)
Abstract
We study θ dependence of the vacuum energy for the 4d SU(2) pure Yang-Mills theory by latticenumerical simulations. The response of topological excitations to the smearing procedure is investi-gated in detail, in order to extract topological information from smeared gauge configurations. Wedetermine the first two coefficients in the θ expansion of the vacuum energy, the topological sus-ceptibility χ and the first dimensionless coefficient b , in the continuum limit. We find consistencyof the SU(2) results with the large N scaling. By analytic continuing the number of colors, N , tonon-integer values, we infer the phase diagram of the vacuum structure of SU( N ) gauge theory asa function of N and θ . Based on the numerical results, we provide quantitative evidence that 4dSU(2) Yang-Mills theory at θ = π is gapped with spontaneous breaking of the CP symmetry. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] ontents I. Introduction II. Lattice Simulations
III. Discussion N versus Small N N N N N − Model 23B. Quantitative Analysis of Lattice Results 241. N = 2 242. N inst IV. Summary and Discussion Acknowledgments References I. INTRODUCTION
The θ term of the Yang-Mills theory determines how to weight different topological sectorsin the path integral. Since the θ parameter is the coefficient of a total derivative term in theLagrangian, the θ -dependences of observables can be explored only through non-perturbativemethods.The special value θ = π has been of particular interest. In the classic literature [1–3],spontaneous CP violation of the 4d SU( N ) Yang-Mills theory at θ = π was demonstrated2n the large N limit [4]. More recently, an anomaly matching argument involving general-ized global symmetries [5] showed that the CP symmetry in the confining phase has to bebroken even at finite N [6]. A similar conclusion was derived by studying restoration ofthe equivalence of local observables between SU( N ) and SU( N )/ Z N gauge theories in theinfinite volume limit [7]. See, for example, Refs. [8–10] for other approaches.While lattice numerical simulations are ideal tools to explore non-perturbative dynamicsof gauge theories, direct simulations at θ = π has been challenging due to the notorioussign problem. Nevertheless, lattice simulations have been successfully used to determinethe first few coefficients in the θ expansion of the vacuum energy for finite N . On the onehand, below the critical temperature T c these coefficients turn out to be consistent withthe large N scaling down to N = 3 [14–16], which indicates spontaneous CP violation andthe discontinuity of the vacuum energy across θ = π . On the other hand, above T c thecoefficients determined at N = 3 and 6 are found to be consistent with the dilute instantongas approximation (DIGA) [17], which predicts continuous behavior for the vacuum energyacross θ = π .The CP N − model in two dimensions shares many non-perturbative properties with thefour-dimensional SU( N ) Yang-Mills theory [18, 19], and hence provides useful insights intothe 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 gaplessat θ = π with unbroken CP symmetry [21–29]. Motivated by similarities between the 4dYang-Mills theory and the 2d CP N − model, it is natural to ask if the 4d SU( N ) Yang-Millstheory at θ = π shows distinctive behavior for small values of N , such as N = 2. In this work we explore the θ dependence of the vacuum energy of the 4d SU(2) pureYang-Mills gauge theory. In sec. II, we perform lattice numerical calculations to determinethe first two coefficients in the θ expansion of the vacuum energy. The response of topologicalexcitations to the smearing procedure is investigated in detail, in order to efficiently extractphysical information from lattice configurations. The coefficients determined for N = 2 arecompared to those previously obtained for N ≥
3, to see whether the result at N = 2 canbe seen as a natural extrapolation of those for N ≥
3. In sec. III, we begin with theoretical Recent and related developments towards direct simulations are found, for example, in Refs. [11–13]. The Z N subgroup of the flavor symmetry of the 2d CP N − model can be regarded as a counterpart of the1-form Z N center symmetry of the 4d SU( N ) pure Yang-Mills theory. N ) theory, for large N and for small N as weanalytically continue the values of N . We then interpret the numerical results of sec. II andprovide quantitative evidence that the 4d SU(2) theory belongs to the “large N ” class, andis gapped and has spontaneous breaking of CP symmetry at θ = π . II. LATTICE SIMULATIONS
The vacuum energy can be expanded around θ = 0 as E ( θ ) − E (0) = χ θ (cid:0) b θ + b θ + · · · (cid:1) , (1)where χ is the topological susceptibility, and b i ( i = 1 , , , · · · ) are dimensionless coefficientsdescribing the deviation of the topological charge distribution from the Gaussian. Thesequantities can be determined from the lattice configurations generated at θ = 0 as χ = h Q i θ =0 V , (2) b = − h Q i θ =0 − h Q i θ =0 h Q i θ =0 , (3) b = h Q i θ =0 − h Q i θ =0 h Q i θ =0 + 30 h Q i θ =0 h Q i θ =0 , (4)where Q is the topological charge, whose precise definition is given in eqs. (10)-(14), and h· · · i θ =0 denotes an ensemble average over configurations generated at θ = 0. According tothe large N analysis [1, 3], these quantities can be expressed, as a function of N , as χ ( N ) = χ ( ∞ ) + O ( N − ) , (5) b i ( N ) = b (1)2 i N i + O (cid:18) N i +2 (cid:19) . (6)By contrast the dilute instanton gas approximation leads to E ( θ ) − E (0) = χ (1 − cos θ ),and hence the coefficients, b DIGA2 = − / b DIGA4 = 1 / · · · , are completely determined.We attempted calculating b as well as χ and b . We could obtain only a loose bound − . < b < . χ and b . 4 N S N T c ( a T c ) L σ / statistics1.750 16 4.65 0.0462 4.9 80,1001.850 16 6.50 0.0237 3.5 71,0401.975 16 9.50 0.0111 2.4 30,4901.975 24 9.50 0.0111 3.6 131,830TABLE I: The simulation parameters. T c denotes the critical temperature. N T c is determinedfrom Ref. [31]. The uncertainties of N T c are below 1% and hence neglected in the following. A. Lattice Setup
The SU(2) gauge action on the lattice is described as S g = 6 β N site { c (1 − W P ) + 2 c (1 − W R ) } , (7)where β = 4 /g is the lattice gauge coupling, W P and W R are the 1 × × c and c satisfying c = 1 − c are the improvement coefficients. We take the tree-level Symanzikimproved action [30], which is realized by c = − /
12. To investigate the continuum limit,three values of the lattice coupling ( β =1.750, 1.850 and 1.975) are taken. The lattice sizeis N site = N 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 T c = 1 / ( a ( β ) T c ) obtained in Ref. [31], where T c is the critical temperaturefor the SU(2) pure Yang-Mills theory. The value N T c is then transformed to ( aT c ) forlater use. To have an intuition about how large our lattice is, we estimated L σ / at eachlattice, using T c / √ σ str = 0 . L denotes the physical length of the spatialdirection, i.e. L = a N S , and σ str is the representative dynamical scale (the sting tension).Gauge configurations are generated by hybrid Monte Carlo method and are stored every 10trajectories. Simulation parameters including the lattice spacings, the lattice size and thenumber of configurations (denoted as statistics in the table) are summarized in Tab. I.5 . Smearing and Definition of Topological Charges on the Lattice Among several equivalent methods often used in the literature [33–35], we choose thecombination of the APE smearing [36] and the 5-loop improved operator [37] to calculatetopological charge on each configuration. Topological charges on the lattice are obscured byshort distance fluctuations, which we remove by introducing a smoothing technique. In theAPE smearing, new link variables U (new) µ are constructed from old ones U (old) µ as U (new) µ = Proj (cid:2) (1 − ρ ) U (old) µ ( x ) + ρ X µ ( x ) (cid:3) , (8) X µ ( x ) = X ν = µ h U (old) ν ( x ) U (old) µ ( x + ˆ ν ) U (old) ν † ( x + ˆ µ )+ U (old) ν † ( x − ˆ ν ) U (old) µ ( x − ˆ ν ) U (old) ν ( x − ˆ ν + ˆ µ ) i , (9)where Proj acts as the projection back to an SU(2) element. This procedure minimizesthe action density. The parameter ρ is taken to be 0.2, which corresponds to α APE =6 ρ/ (1 + 5 ρ ) = 0 . Q = X x q ( x ) , (10) q ( x ) = X i c i q m i ,n i ( x ) , (11) q m,n ( x ) = 132 π m n X µ,ν,ρ,σ ǫ µ,ν,ρ,σ Tr h ˆ F µ,ν ( x ; m, n ) ˆ F ρ,σ ( x ; m, n ) i , (12)ˆ F µ,ν ( x ; m, n ) = 18 Im (cid:2)(cid:8) oriented clover average of ( m × n ) plaquette (cid:9) + (cid:8) ( m ↔ n ) (cid:9)(cid:3) , (13)where ( m i , n i ) = (1 , , (2 , , (1 , , (1 , , (3 ,
3) for i = 1 , . . . , c = (19 − c ) / , c = (1 − c ) / , c = ( −
64 + 640 c ) / ,c = 1 / − c , c = 1 / . (14)This operator is free of O ( a ) and O ( a ) terms. The replacement above is done once forall link variables, for each step of the smearing. The smearing is carried out every 10trajectories, and the topological charge is measured after every smearing step.6 Q n APE N S =16, β =1.750 -20-15-10-5 0 5 10 15 20 0 100 200 300 400 500 600 700 800 Q n APE N S =24, β =1.975 FIG. 1: Smearing history of topological charge Q as a function of the number of smearing steps n APE for randomly chosen 200 configurations at β = 1 .
750 and 1.975.
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 latticespacing. The measurement of the topological charge is therefore reliable only after a suitablenumber of smearing steps. However, the smearing may also affect physical topological excita-tions. The previous dedicated studies revealed that the smearing induces pair-annihilation,“melting away” or “falling through the lattice” [37–39]. In Ref. [39], it was found that topo-logical objects go through several characteristic phases during the cooling procedure. In thefirst phase, the size of topological objects grows with the cooling, and some of them eventu-ally melt away and some pair-annihilate. Then, the second phase comes where only relativelyslow shrinkage of the objects takes place and eventually they disappear after long enoughcooling. Assuming that the similar phases show up in the procedure of APE smearing, wewill in the following determine the boundary between the two phases.In order to explore how the smearing changes topological properties, we first look at thesmearing history of the topological charge Q as a function of the smearing steps n APE . Fig. 1shows the history obtained at β = 1 .
750 and 1.975. At relatively small n APE , Q changesfrequently, and most of the changes here are expected to be associated with the removalof short distance fluctuations. We deduce that this range of n APE corresponds to the firstphase. The frequency of change in Q is somewhat reduced as n APE and β increase, but thechange steadily continues. In this region, both the increase and the decrease of Q happenmostly by one unit, and a change takes O (10) steps to be completed. This range of n APE
7s identified with the second phase. In the following, we discuss quantitative differencesbetween 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 thesmearing, 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 − | ≤ . • “ decreasing ” if Q ( n APE ) is approaching zero, • “ increasing ” if Q ( n APE ) is moving away from zero.Fig. 2 shows the scatter plot for Q and S g at several values of n APE , obtained at β = 1 . S g = 8 π | Q | /g 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 pointsgradually accumulate on integer values of Q by “increasing” or “decreasing”. The value ofthe action is never below the Bogomolnyi bound in each topological sector, as expected. Thisindicates that “increasing” data can not exist around the boundary because the smearinglowers the action value and only either instantons(s) or anti-instanton(s) can exist on thebound. It is also seen that the larger the value of | Q | is, faster the minimum of the actionin the topological sector reaches the Bogomolnyi bound. Thus, the minimum value of S g inthe 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 withnonzero Q around the bound are attributed to approximate instantons or anti-instantons,and the “decreasing” occurring around there are interpreted as (anti-) instantons “fallingthrough the lattice”. We expect that all the “increasing” and “decreasing” in the secondphase are caused by “falling”. In order to examine this expectation, we introduce theparticipation ratio defined by P ( n APE ) := 1 N site X x q ( x, n APE ) ! X x q ( x, n APE ) , (15)8 S g ( n A P E = ) S g ( n A P E = )
400 800 S g ( n A P E = ) S g ( n A P E = ) S g ( n A P E = ) QBogomolnyi bound
FIG. 2: Correlation between the topological charge Q and the action S g at smearing steps n APE =1 , , , ,
800 (from top to bottom) at β = 1 . | Q | are “stable”, “decreasing”, and “increasing”. The dotted lines representthe Bogomolnyi bound. q ( x, n APE ) denotes the topological charge density q ( x ) in eq. (11) after n APE steps ofsmearing. 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 formsa local peak, q ( x, n APE ) = δ ( x ). Fig. 3 shows the smearing history of Q and ln P as afunction of n APE for one particular configuration at β = 1 . n APE > ∼
30, whenever Q changes, ln P shows a rapid increase after slow decrease over many smearing steps. Thiscan be interpreted as that a local object in topological charge density gradually shrinks andsuddenly disappears at some point with a change of Q . This is precisely what happens whenthe “falling through the lattice” occurs [39].We can directly check this interpretation by studying the distribution of the topologicalcharges. Fig. 4 shows the topological charge density, projected onto the z - t plane, of thesame 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 and200, a positive peak seems to be smeared but does not suddenly disappear. It seems that acomplicated process such as a pair annihilation happens in the latter case.From these observations, we conclude that the changes of Q occurring in the secondphase are dominated by the “falling” of instantons or anti-instantons. We expect that the -10-8-6-4-2 0 2 0 100 200 300 400 500 600 Q ( n A P E ) , l n P ( n A P E ) n APE
Q(n
APE )ln P(n
APE ) FIG. 3: An example of Q ( n APE ) and ln P ( n APE ) as a function of n APE at β = 1 . APE = 50 0 4 8 12 16 20 24 28t 0 4 8 12 z -0.2-0.1 0 0.1 0.2 n APE = 60 0 4 8 12 16 20 24 28t 0 4 8 12 z -0.2-0.1 0 0.1 0.2n APE = 100 0 4 8 12 16 20 24 28t 0 4 8 12 z -0.2-0.1 0 0.1 0.2 n APE = 200 0 4 8 12 16 20 24 28t 0 4 8 12 z -0.2-0.1 0 0.1 0.2n APE = 450 0 4 8 12 16 20 24 28t 0 4 8 12 z -0.2-0.1 0 0.1 0.2 n APE = 470 0 4 8 12 16 20 24 28t 0 4 8 12 z -0.2-0.1 0 0.1 0.2 FIG. 4: Distribution of topological charge projected onto z - t plane at n APE = 50, 60, 100, 200,450, and 470. “falling” occurs also in the first phase, but it is overshadowed by changes originating fromother reasons.Instanton and anti-instantons will “fall” at an equal rate. In configurations with
Q > Q ( n APE ) = Q ( n APE + 1) − Q ( n APE ) for Q ( n APE ) > Q ( n APE ) − Q ( n APE + 1) for Q ( n APE ) < . (16)The sign of ∆ Q ( n APE ) tells us which of “increasing” or “decreasing” happens when goingfrom n APE to n APE + 1. Fig. 5 shows the n APE dependence of |h ∆ Q ( n APE ) i| , where thesymbols are filled when its original value is negative. The results from four ensembles | 〈 ∆ Q ( n A P E ) 〉 | n APE β =1.7501.8501.975N S =24, 1.975 FIG. 5: The decay rate of the topological charge. show exponential fall with approximately a common exponent for n APE < ∼
10, while theytake 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 physicalvolume. At any rate, this plot clearly shows that the boundary separating the two phasesis located n APE ∼
20. In the following analysis, we only deal with the data for n APE ≥ D. Results
Fig. 6 shows the Monte Carlo history of Q over a thousand configurations in four ensem-bles obtained at n APE = 800. It is seen that the fluctuation of Q is frequent enough, andthat the amplitude depends on β and N site . In the following analysis, all the measurementsare binned with the bin size of 100 configurations, and a single elimination jackknife methodis used to estimate statistical uncertainties.Fig. 7 shows the histogram of Q for four ensembles at n APE = 0, 20, 100. ApproximateGaussian shape is seen in all ensembles.Fig. 8 shows the topological susceptibility in lattice unit, a χ ( n APE ) = h Q i /N site , as afunction of n APE . A mild decrease is seen for n APE ≥
20 as expected from a negative constantobserved in Fig. 5. We determine topological susceptibility at each lattice by extrapolatingthe smeared data in the second phase to n APE → -15-10-5 0 5 10 15 10000 10500 11000 Q configuration number β =1.750 -15-10-5 0 5 10 15 10000 10500 11000 Q configuration number β =1.850-15-10-5 0 5 10 15 10000 10500 11000 Q configuration number β =1.975 -15-10-5 0 5 10 15 10000 10500 11000 Q configuration numberN S =24, β =1.975 FIG. 6: Monte Carlo history of Q at four ensembles. h i s t og r a m o f Q Q n
APE = 020100 0 0.2 0.4 0.6 0.8 -15 -10 -5 0 5 10 15 h i s t og r a m o f Q Q n
APE = 020100 0 0.5 1 1.5 2 2.5 -15 -10 -5 0 5 10 15 h i s t og r a m o f Q Q n
APE = 020100 0 0.2 0.4 0.6 0.8 -15 -10 -5 0 5 10 15 h i s t og r a m o f Q Q n
APE = 020100
FIG. 7: Histogram of Q for four ensembles at n APE = 0, 20, 100. place even in the first pahse. The data points in n APE ∈ [20 ,
40] are well described by alinear function, a χ ( n APE ) = a χ (0) + c n APE . (17)The fit results are tabulated in Tab. II.Fig. 9 shows n APE dependence of b . Since b is found to be constant for n APE ≥
20, weperform the constant fit to extract b at n APE = 0. The fit results are shown in Tab. II.The values of b obtained at β = 1 .
975 with two lattice volumes turns out to be consistentwith each other due to the large statistical uncertainty, while 1.8 σ difference is observed for χ . In Ref. [15, 16], these quantities are calculated with several different volumes for SU( N ) β N S a χ (0) × c × b (0) × . − . − . − . − . − . − . − . − β =1.750 0.6 0.8 1 1.2 a χ β =1.850 0.2 0.25 0 10 20 30 40n APE β =1.975N S =24, 1.975 FIG. 8: a χ for the four ensembles as a function of n APE . with N = 3, 4, 6 down to L σ str ∼ .
5, and no finite volume effect is observed. Our latticewith β = 1 .
975 and N S = 16 corresponds to L σ str = 2 . σ difference observed at β = 1 .
975 is considered as a statistical fluctuation, and we includeboth results in the following analysis.Next we discuss the continuum limit. Fig. 10 shows the extrapolation of χ/T c and b tothe continuum. The limit for both quantities is examined by applying two functional forms:1. constant excluding the coarsest lattice2. linear in a using all latticesThese two are chosen because they yield smallest and largest values for χ/T c among otherreasonable choices. In either quantity, the constant fit is used to estimate the central value,and the difference between the two methods is taken as the systematic uncertainty in thefinal result. 15 β =1.750-0.1 0 0.1 b β =1.850-0.1 0 0.1 0 10 20 30 40n APE β =1.975N S =24, 1.975 FIG. 9: n APE dependence of b . The continuum limit of χ/T c turns out to strongly depend on the functional form, andas a result the error is dominated by the systematic uncertainty. On the contrary, thanks tothe constant behavior for b , the inclusion of the linear term into the functional form doesnot significantly alter the limit for the constant fit. The final results thus obtained are χT c = 0 . , χ / T c = 0 . , b = − . , (18)where the errors are summed in quadrature.In Refs. [14–16], the topological susceptibility χ is calculated in SU( N ) gauge theory withseveral values of N to study the large N behavior. In Refs. [14, 37, 43–45], χ is estimatedfor SU(2) gauge theory. As for b , the N dependence is studied for N ≥ N = 2. Fig. 11 shows the summary plotfor χ/σ and b , including our results. In this plot, we use T c / √ σ str = 0 . See, for an exploratory study, Ref. [46]. χ / T c (aT c ) constantlinear -0.2-0.1 0 0.1 0 0.01 0.02 0.03 0.04 0.05 b (aT c ) constantlinear FIG. 10: The continuum limit of χ/T c and b . The solid lines in both plots are the results from aconstant fit using only two finer lattices, and the dashed lines are those from a linear fit using alllattices. change the normalization to χ/σ . The solid lines shown in the plots are the linear fitperformed 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 thedeviation is accountable by the next leading order correction, which is of O (1 /N ) relativeto the leading one. It is then natural to expect that the dynamics of SU(2) gauge theory is asmooth extrapolation of the large N dynamics to N = 2, and that nothing special happensin between.The value of b at N = 2 obtained in this work turns out to be consistent with theinstanton prediction, b DIGA2 = − /
12, within 1.7 σ . However, it is more consistent with thenaive linear extrapolation from the N ≥ N = 2. This observation gives furthersupport to the above expectation, i.e. nothing special happens between N ≥ N = 2.Notice that, in Ref. [47] b = 6(2) × − is obtained in the continuum limit, which clearlydiffers from the value predicted from the instanton calculus, b DIGA4 = 1 / III. DISCUSSIONA. Large N versus Small N One of the motivations for present analysis of the 4d SU(2) pure Yang-Mills theory isto study the SU( N ) Yang-Mills theories for finite values of N . We expect that SU( N )Yang-Mills theories show qualitatively different behaviors, for large N and for small N .17 χ / σ LuciniDel DebbioBonatiprevious SU(2)this workph fit-0.1-0.05 0 0 0.1 0.2 0.3 b DIGADel DebbioBonatithis workph fit
FIG. 11: The N dependence of χ/σ and b . Each data point is slightly shifted horizontally tomake it easier to see. The horizontal dashed line in the b plot represents the dilute instanton gasapproximation (DIGA).
1. Large N In the large N limit, the values of the coefficients b i scales at O ( N − i ) (6), and hencebecomes smaller as N becomes large. This indicates that the vacuum energy, E ( θ, N ), willno longer be a 2 π -periodic function of θ . While this seems to be in tension with the 2 π -periodicity of θ in the Lagrangian, the apparent inconsistency is resolved by the possibility18 IG. 12: Schematic picture for the multi-valued vacuum energy for the large N pure Yang-Millstheory, reproduced from Ref. [48]. that the vacuum energy E ( θ, N ) is a multi-valued function of θ [1, 3]. Namely, there aremultiple branches ˜ E ( θ + 2 πn ) labeled by an integer n , where each ˜ E ( θ ) has a quadraticpotential near θ ∼ O (Λ ) as θ → ±∞ , with Λ being thedynamical scale. The correct minimal energy (over all the branches) is then E ( θ ) = min n ∈ Z ˜ E ( θ + 2 πn ) , (19)see Fig. 12. This in particular means that there are two different lowest-energy statesfor θ = π mapped each other under the CP symmetry, and hence we expect CP to bespontaneously broken.One should notice that the vacuum energy takes a different functional form as expectedfrom the semiclassical one-instanton calculation [49], which gives the free energy density as E ( θ ) ∼ Z ∞ dρρ ( µρ ) b exp (cid:20) − π g ( µ ) (cid:21) (1 − cos θ ) . (20)Here ρ is the size modulus of the instanton, g ( µ ) is the running gauge coupling constant atthe energy scale µ , and b := 11 N/ N analysis resides in the famous IR divergence of the instanton analysis: theintegral (20) is divergent as ρ becomes large.
2. 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 thethreshold value N inst = 12 /
11 [50]. In this case, the integral (20) should be regularized in19he UV by the cutoff scale M , so that the lower value of the integral for ρ is given by M − .Note that the UV regularization is needed even though the pure Yang-Mills theory in itselfis asymptotic free.We thus expect that the one-instanton contribution to have the schematic expression E ( θ, N ) ∼ M − N Λ N (1 − cos θ ) , (21)where Λ is the dynamical scale of the theory. This is a different qualitative behavior assuggested by the large N analysis, (19). 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 manyother contributions to the partition function. While this is certainly true, let us note thatthese non-instanton contributions give rise to contributions of O (Λ ). Since the cutoff scale M is much larger than the dynamical scale Λ ( M ≫ Λ), these non-instanton contributionsare much smaller than the instanton contribution of (21). Since we do know that instantoncontributes to the path integral, we are certain that there is contribution of the form (21),which we expect will dominate over other contributions. Note that the UV cutoff M dependence will appear only as an overall divergence. Thismeans that while the topological susceptibility depends on the UV cutoff, the coefficients b i do not depend on the UV cutoff. This is in fact an advantage of the definition of b i in(1).Another possible objection is that it does not make sense to consider non-integer valuesof N ; SU( N ) theory in the conventional thinking is defined only for integers N ≥
2. For ourpurposes, however, it is useful to promote N to be a real parameter and discuss the vacuumenergy E ( θ, N ) as a function of real values of N as well as θ . Mathematically, one mightworry that there are huge ambiguities in extending the functions E ( θ, N ) to non-integervalues. Indeed, when we multiply the vacuum energy by an expression F (sin πN ) for anyfunction F ( x ) with F ( x = 0) = 1, the integer values of E ( θ, N ) will be preserved. This While we discuss only pure Yang-Mills theory in this paper, similar issue arises for the SU(2) electroweakgauge group for the standard model, and we need to introduce the UV regulator for the small-size in-stantons. Interestingly, the size of the resulting integral could explain the smallness of the cosmologicalconstant [51–53]. There can be still multi-instanton corrections. These subleading corrections preserve the 2 π -periodicityand hence the CP symmetry, and does not play important roles in what follows. N → ∞ . There is a mathematical theorem[54] which guarantees that two real-valued functions, with suitable asymptotic conditions atinfinity and with the same values at all integers, coincide. Such considerations are actuallyimplicit in the large N analysis, and makes it possible to discuss small non-integer values of N (even to N < N inst for the instanton calculus makes sense in thiscontext.
3. Intermediate N We have seen that χ ∼ O ( N ) and b i ∼ O ( N − i ) for large N while χ depends onUV-cutoff and b i ∼ b DIGA2 i for small N .What happens at intermediate values of N ? For generic values of θ we do not necessarilyexpect a sharp transition between “large” and “small” N : there is no good order parameter.The situation is different for the special value of θ = π , which has the CP symmetry in theLagrangian. In this case, we can define two phases by the presence or the absence of the CPsymmetry; the CP symmetry is spontaneously broken for large N , while for smaller N thevacuum 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 gappedor gapless. This is because of the mixed anomaly between the Z N center symmetry and theCP symmetry [6], and the presence of the Z N symmetry can be regarded as a definition ofthe confinement. Possible phase diagrams are shown in Fig. 13, where the presence/absenceof the CP symmetry is assumed to coinside with the gapped/gapless system. Of course, thepresence of the mixed anomaly only shows that at least either the center symmetry or theCP 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 theCP N − model we discuss below. Once it is assumed, we need to discuss how the gaplessphase extends to the θ = π region. In the figure, we show a possibility that the gaplesstheory is realized even at θ = 0 at some N . There are other possibilities that the line doesnot reach to θ = 0 axis, as well as the possibility that gapless theories are realized only onthe θ = π line. Note that irrespective of the possible phase structures we define the criticalvalue N CP by the presence/absence of the CP symmetry.21 IG. 13: Possible phase structures of 4d SU( N ) pure Yang-Mills theory as a function of 1 /N and θ . At θ = π and N > N CP , there is a first-order phase transition. If such N CP exists, agapless theory should be realized for N < N CP at θ = π . Such a region may extends to θ = π as in the figures although it is totally unknown. The topological susceptibility, χ , diverges for N < N inst = 12 /
11 by the contributions from small instantons. In (a) and (b) we show possiblephase structures where N inst < N CP and N inst > N CP , respectively. The mixed anomaly in itselfallows for more complicated phase structures, and for example allows regions where both CP andcenter symmetries are broken. Note that the value of N inst is not necessarily the same as the critical value N CP ; theformer is defined purely for the semiclassical instanton computation applicable for genericvalues of θ , while the critical value N CP is the value separating the CP broken/preservedphases at the special value θ = π . It is not a priori clear if we expect general inequalitiesbetween the two values N inst and N CP . One may be tempted to imagine that N inst shouldalways be smaller than N CP since the potential generated by the instanton is always smoothso that the spontaneous CP breaking does not happen. However, although it is certain thatthe 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 spontaneousCP breaking and/or confinement at θ = π .Let us next come to more quantitative aspects. In the instanton calculus, we obtainedthe threshold value of N inst = 12 /
11, which is smaller than N = 2. The estimation by the22ne-loop beta function is justified by the asymptotic freedom. Therefore, it is expected thatthe SU(2) gauge theory has a UV-independent value of the topological susceptibility. As wediscussed already, however, one cannot conclude whether N CP < θ -dependence of the theory anddiscuss whether N = 2 is small or large more carefully.
4. Comparison with the CP N − Model
It is useful to compare the 4d Yang-Mills theory with the celebrated CP N − model intwo dimensions [18, 19]. This theory has many similarities with the 4d SU( N ) Yang-Millstheory, and could be of help in understanding the non-perturbative properties of the latter. In the large N limit of the 2d CP N − model, there exists gap at any values of θ , andthe vacuum energy is discontinuous at θ = π , where the CP symmetry is spontaneouslybroken. 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 model, which is nothing but theO(3) spin model. This model is believed to be gapless and have continuous vacuum energyat θ = π [56].When we consider N as a continuous parameter again, one expects that there will be acritical value of N between the two phases, which we denote by N CP . For N > N CP thetheory has spontaneous CP breaking at θ = π , and we expect that the dependence on theparameter N is accounted by the large N scaling: we call this the “large N phase.” Bycontrast for N < N CP we have an unbroken CP symmetry for θ = π , and the semiclassicalinstanton 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 − model, consistent with the divergenceof topological susceptibility for the the CP -model.The most remarkable difference of the CP model from other ( N >
2) CP N − models isthat the semi-classical calculation of the former leads to a UV divergence in the topologicalsusceptibility. This is supported by lattice numerical calculations, unless a suitable counterterm is added [50, 57–64] . In 2d CP N − a confining linear potential appears even when instanton dominates the dynamics [55, 56]. See also Ref. [65], in which the divergence of the topological susceptibility is examined in detail. . Quantitative Analysis of Lattice Results N = 2 We have already seen that the semi-classical estimate of the topological susceptibility χ in4d 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 independentquantitative test of this expectation.Although an extrapolation of numerical data is always subtle, and especially the contin-uum limit of quantities related to topological charge needs special care, our result as wellas 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 | obtained for N = 2 is slightly smaller thanthat of the instanton prediction b DIGA2 = − /
12. The value of | b | in Ref. [47] is much smallerthan the instanton value. Both are rather consistent with the 1 /N and 1 /N scalings ofthe N ≥ θ = π due to smallvalues of | b | and | b | . All these results suggest that N = 2 is “large” for the four-dimensionalYang-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-tracedeformation of the 4d Yang-Mills theory via the semiclassical analysis and a twisted com-pactification, 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 transitionbetween 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 − model, where N = 2 caseis gapless and CP preserving at θ = π , as already mentioned before. This is an excellentdemonstration of the quantitative differences between four-dimensional Yang-Mills theoryand the two-dimensional CP N − model.Notice that the relation between the 4d SU( N ) Yang-Mills theory and the 2d CP N − model24as further clarified in [67], which showed that the T × S compactification of the formerwith suitable ’t Hooft magnetic flux gives rise to S compactification of the two-dimensionalsigma model whose target space has the topology of CP N − (see Refs. [9, 10] for furtherchecks via anomalies). A caution is needed, however, before any quantitative comparisonsbetween the two. The two-dimensional model obtained from four-dimensional theory has anon-standard metric, and in addition there are special points (fixed points under the Weylgroup action) in the CP N − 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 betweenthe sizes of T and S , and any discussion of the standard flat space limit (where there isno such hierarchy) requires careful analytic continuation. These subtleties can easily affectquantitative discussions here. N inst Once we are settled with the case of N = 2, we can discuss even smaller values of N andask 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 databy two methods. The first method uses topological susceptibility, which we expect to divergeat the value N = N inst . To determine this value we fit our N = 2 result together with thosefor N = 3, 4, 6 in Ref. [16] by an Ansatz χσ = χσ (cid:12)(cid:12)(cid:12)(cid:12) N = ∞ × N N − N , (22)where N inst is assumed to be a real number. Here the Ansatz is the simplest function of N which has divergence at N = N inst and approaches to the large N value as N → ∞ . Wethen obtain χσ (cid:12)(cid:12)(cid:12)(cid:12) N = ∞ = 0 . , N inst = 1 . χ / dof = 0 .
05 (23) Our Ansatz is motivated by the analysis performed in Ref. [47], where the topological susceptibilities of2d CP N − model are calculated at several values of N and fitted to the function including 1 / ( N − N = 2 , , . . . , and all the points for N ≥ N scaling. b . Supposing that the semi-classical calculationbecomes valid at N = N inst for SU( N ) gauge theory, b is expected to take b DIGA2 = − / N . We again use the results for b for N = 2, 3, 4, 6 to test thisexpectation. This time, by fitting the data to b ( N ) = b (1)2 N , (24)we obtain b (1)2 = − . χ / dof = 0 . . (25)Substituting (23) and (25) into (24) yields b ( N inst ) = − . − /
12. Furthermore, assuming the functional form of b ( N ) = b (1)4 /N and using the result b = 6(2) × − at N = 2 [47], b ( N inst ) = 0 . b DIGA4 = 1 / N inst ∼ .
5, slightly larger than the semi-classical value 12 /
11. This numerology serves as a check of the overall picture, and moreoverindicates that the large N scaling of b holds well all the way until the value N ∼ N inst ,where χ diverges. If we assume large N scaling for all the b n ’s until N ∼ N inst , thencertain derivatives of the free energy are necessarily discotinuous at θ = π , thus implyingthe breaking of the CP symmetry. This suggests the inequality N CP . N inst . Furthernumerical studied are needed to make this inequality more precise.Summarizing our discussion, the numerical data suggests the following shape of the vac-uum energy density, as we change the value of N . At large N we have the quadratic form ofthe vacuum energy around θ = 0, while there is a cusp at θ = π . As we change N to smallervalues, b and b 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 instantonpicture seems to be smooth as far as b n are concerned, and N = 2 is on the “large N ”side. At N inst ∼ . χ . Once χ is diverging, it becomes difficult to infer the vacuum structure from thevacuum energy only, since the vacuum energy is masked by the large contributions of small26nstantons. In particular, it becomes invisible in practice whether or not there is a phasetransition at θ = π . IV. SUMMARY AND DISCUSSION
We performed lattice numerical simulations to explore the θ dependence of the vacuumenergy in 4d SU(2) pure Yang-Mills theory, with special attention to the response of topologi-cal excitation to the smearing procedure. We discussed the method to extract the topologicalinformation from smeared configurations properly and estimated the first two coefficientsin the θ expansion of the vacuum energy in the continuum limit, namely χ and b . Thevalue of χ turns out to be consistent with the previous results in the literature, while b isdetermined for the first time.We use these results to infer the phase structure of the 4d SU( N ) theory as we changethe values of N and θ . We highlighted the differences for “large N ” and for “small N ”: wehave the large N scaling for the former, while the vacuum energy is dominated by instantonsin the latter. The differences between the two is most clear-cut for θ = π , where the CPsymmetry 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 slightlydeviates from the instanton predictions, while it is well fitted by the 1 /N extrapolation fromthe N ≥ N to small values by analytic continuation, wefind that χ and b reach the instanton predictions at N inst = 1 . N ) theories for integer N , are in the large N category. While large N analysis is oftenregarded as an approximation applicable only to the large values of N , our results suggestthat the large N analysis is more powerful, and can be useful for studying all possible valuesof N , even as small as N = 2. This is in contrast with the case of the 2d CP N − model, whichis believed to be gapless for N = 2, while are gapped for N ≥
3. It would be interesting tostudy 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 carriesnon-zero topological charges. In Ref. [40], it was pointed out that in SU(3) Yang-Mills theorythe codimension-one objects (“sheets”) are responsible for the non-zero topological charges,27nd from the similar study of CP N − model it was pointed out that the object becomeslocalized and becomes instantons as N → N CP . Thus, it is interesting to see what objectsare 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 notoriouslydifficult problem, the sign problem. Since recent development in methodology is remark-able [11–13], 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 phenomenologicalconsiderations of the dynamical θ -angle, the axion [68–71]. For example, in the axionicinflationary models of Ref. [72] (see also Ref. [73]), the values of b n affect future observationsof primordial gravitational waves from inflation [48]. Acknowledgments
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