Neutral bions in the ${\mathbb C}P^{N-1}$ model

We study classical configurations in the ${\mathbb C}P^{N-1}$ model on ${\mathbb R}^{1}\times S^{1}$ with twisted boundary conditions. We focus on specific configurations composed of multiple fractionalized-instantons, termed"neutral bions", which are identified as"perturbative infrared renormalons"by \"{U}nsal and his collaborators. For ${\mathbb Z}_N$ twisted boundary conditions, we consider an explicit ansatz corresponding to topologically trivial configurations containing one fractionalized instanton ($\nu=1/N$) and one fractionalized anti-instanton ($\nu=-1/N$) at large separations, and exhibit the attractive interaction between the instanton constituents and how they behave at shorter separations. We show that the bosonic interaction potential between the constituents as a function of both the separation and $N$ is consistent with the standard separated-instanton calculus even from short to large separations, which indicates that the ansatz enables us to study bions and the related physics for a wide range of separations. We also propose different bion ansatze in a certain non-${\mathbb Z}_{N}$ twisted boundary condition corresponding to the"split"vacuum for $N= 3$ and its extensions for $N\geq 3$. We find that the interaction potential has qualitatively the same asymptotic behavior and $N$-dependence as those of bions for ${\mathbb Z}_{N}$ twisted boundary conditions.

On the other side, neutral bions (zero topological charge and zero magnetic charge) can be identified as the infrared renormalon [8-15, 36, 37]. Here imaginary ambiguities arising in bion's amplitude and those arising in non-Borel-summable perturbative series cancel against each other, and it is expected that full semi-classical expansion including perturbative and non-perturbative sectors, which is called "resurgent" expansion [38], leads to unambiguous and self-consistent definition of field theories in the same manner as the Bogomolny-Zinn-Justin (BZJ) prescription in quantum mechanics [39][40][41]. However, it is not straightforward to verify these arguments in gauge theories directly, since it is difficult to find an explicit ansatz of bion configurations.
In order to reach deeper understanding on bions and the associated physics, it is of great importance to study examples in the low-dimensional models such as CP N −1 models [9,10], principal chiral models [12,15] and quantum mechanics [11,13,14]. In particular, the CP N −1 model in 1+1 dimensions has been studied for a long time as a toy model of the Yang-Mills theory in 3+1 dimensions [42], because of similarities between them such as dynamical mass gap, asymptotic freedom and the existence of instantons [43]. The CP N −1 model on R 1 × S 1 with twisted boundary conditions admits fractionalized instantons (domain wall-instantons) as configurations with the minimal topological charge [44,45] (see also Refs. [46]). In Ref. [9], generic arguments on bion configurations were given in the CP N −1 model on R 1 × S 1 with Z N twisted boundary conditions, which is a corresponding situation to U(1) N −1 center-symmetric phase in QCD(adj.), based on the independent instanton description taking account of interactions between far-separated fractionalized instantons and anti-instantons. According to the study, the renormalon ambiguity arising in non-Borelsummable perturbative series is compensated by the amplitude of neutral bions also in the CP N −1 model. This phenomenon, which is called "resurgence", works as follows [9]: The effective interaction action by bosonic exchange between one fractionalized instanton K i and one fractionalized anti-instanton K j is where τ stands for distance (divided by the compact scale L) between two fractionalized instantons. Vectors α i , α j are affine co-roots and α i · α j is an entry of the extended Cartan matrix. The total bion amplitude including the fermion zero-mode exchange contribution is mainly given by with V ij eff (τ ) = S int (τ ) + 2N f ξτ and S I being the instanton action. N f stands for fermion flavors. For neutral bion α i · α i > 0, semiclassical description of independent fractionalized instantons breaks down since the interaction is attractive and instantons are merged in the end. Here, the BZJ-prescription, replacing g 2 → −g 2 , works to extract meaningful information from this amplitude. The prescription turns the interaction (spuriously) into a repulsive one and the amplitude becomes well-defined as By the use of the analytic continuation in the g 2 complex plane, we can continue back to the original g 2 . For N f = 0 case, we then encounter the following imaginary ambiguity in the amplitude asB We can rephrase this situation as unstable negative modes of bions give rise to imaginary ambiguities of the amplitude. The imaginary ambiguity has the same magnitude with an opposite sign as the leading-order ambiguity (∼ ∓iπe −2S I /N ) arising from the non-Borel- Although this generic argument based on far-separated instantons is clear, it is also worthwhile manifesting and studying an explicit solution or ansatz corresponding to bion configurations, which can be investigated from short to large separation. In Ref. [10], the authors found out non-Bogomol'nyi-Prasad-Sommerfield (BPS) solutions in the CP N −1 model on R 1 × S 1 with a Z N twisted boundary condition, and have shown that these solutions can be critical points, around which the resurgent semi-classical expansion is performed. The simplest non-BPS solution that they found is a four-instanton configuration composed of two fractionalized instantons (ν = 1/N) and two fractionalized anti-instantons (ν = −1/N) for N ≥ 3. (see also [48].) They used the Din-Zakrewski projection method [47] generating a tower of non-BPS solutions from a BPS solution. It is known that all possible classical solutions are exhausted by this method at least on R 2 and S 2 [47]. This result indicates that if a simple bion configuration containing one instanton and one anti-instanton in the CP N −1 model (N ≥ 2) exists, it may not be a solution of the equation of motion, but may be some classical configuration which can give significant contributions to path integrals. If it is true, one question arises what ansatz corresponds to such a bion. If such an ansatz exists, the other questions arise how the instanton constituents behave at short separations and whether it is consistent with the amplitude (1) obtained in the standard instanton calculus in a far-separated limit. In the present study, we consider and study an ansatz corresponding to bions beyond exact solutions. We also consider more general twisted boundary conditions similar to the "split phase" in QCD(adj.).
The purpose of our work is to study an explicit ansatz corresponding to topologically trivial bion configurations in the CP N −1 on R 1 × S 1 with several twisted boundary conditions, and show how the instanton constituents behave at an arbitrary separation. For the Z N twisted boundary condition, we consider a simple neutral-bion ansatz, which yields configuration involving one fractionalized instanton (ν = 1/N) and one fractionalized antiinstanton (ν = −1/N) in the well-separated limit. By studying separation dependence of the total action, we exhibit the attractive interaction between the instanton constituents and how they are merged in the end, which means that the configuration has a negative mode.
By looking into N-dependence of the interaction potential as a function of the separation in comparison with the result in the standard instanton calculus (1), we show that our ansatz is consistent with (1) even from short to large separations. Our ansatz can be used to study bions and related physics for a wide range of separations. For the non-Z N twisted boundary conditions with N = 3, which we term a "split" boundary condition, we find out a different fractionalized instanton-anti-instanton ansatz. We again show that the configuration has a negative mode. We extend the ansatz to general N ≥ 3 cases, and find that the interaction potential between the instantons has qualitatively the same properties as (1) up to some factors in the extended versions. This fact indicates universality of resurgence based on neutral bions for general boundary conditions.
In Sec. II we introduce CP N −1 models with some notations for calculations. In Sec. III we first introduce Z N twisted boundary conditions and discuss how fractionalized instantons emerge. We then propose a specific ansatz for neutral bions, and discuss the properties. In Sec. IV we consider non-Z N twisted boundary conditions, and discuss bion-like configurations for the cases. Section V is devoted to a summary. Then, the action and topological charge representing π 2 (CP N −1 ) ≃ Z of the CP N −1 model in Euclidean two dimensions are given by (see, e.g., Ref. [47]) respectively, where d 2 x ≡ dx 1 dx 2 and µ, ν = 1, 2. Here, we have defined the covariant The action S and topological charge Q can be expressed in terms of the projection operator P ≡ nn † = ωω † ω † ω and using the complex coordinate z ≡ x 1 + ix 2 , All through this paper, we focus the geometry R 1 × S 1 and configurations on it satisfying periodicity in the x 2 direction with compactification scale L. For all the configurations considered in the present paper, the action density and the topological charge density are reduced to be functions of x 1 : where we have defined the action density s(x 1 ) and the charge density q(x 1 ) depending only on x 1 . Here we redefine the action and topological charge as S/(2π) → S and Q/(2π) → Q for them to have multiples of 1/N after x 1 integration. In this paper, we omit the coupling 1/g 2 for simplicity. The with the Pauli matrices σ. Then, the action is In the present section, we propose a neutral bion ansatz for a Z N twisted boundary condition in the CP N −1 model on R 1 ×S 1 . Z N twisted boundary conditions in a compactified direction is expressed as [9,10] ω(x 1 , x 2 + L) = Ω ω(x 1 , x 2 ) , Ω = diag. 1, e 2πi/N , e 4πi/N , · · ·, e 2(N −1)πi/N .
In SU(N) gauge theories with adjoint quarks, this Z N twisted boundary condition corresponds to the vacuum with the gauge symmetry breaking SU(N) → U(1) N −1 , where Wilson-loop holonomy in the compactified direction is given by and where A 2 is the gauge field in the compactified direction. (See also [49][50][51][52] for topics related to Z N twisted boundary conditions.) We here omit permutation copies. We note that the gauge field defined in the CP N −1 model (6) also has the same Wilson-loop holonomy for Z N twisted boundary condition. Difference between (13) and (15) for N = 2 is just superficial and unphysical, since two different ansatz of ω(x) with an overall boundary condition factor e −iπ/2 result in the same projection field P(x) as we will show later. Fractionalized instantons (domain wall-instantons) carry the minimum topological charges in the CP N −1 model on R 1 × S 1 with a twisted boundary condition [44,45]. For simplicity, we begin with the CP 1 model and generalize the argument to the CP N −1 model subsequently. From next subsection we make all the dimensionful quantities and parameters dimensionless by using the compact scale L (L → 1) unless we have a special reason to recover it.

B. Fractionalized instantons
In this subsection, we illustrate fractionalized instantons in the CP 1 model satisfying a Z 2 twisted boundary condition (13) as on R 1 × S 1 with the unexplicit unit compactification scale L [53]. Here, we have used the relation (12) for the second equation.
Using the complex coordinate z = x 1 + ix 2 on R 1 × S 1 , fractionalized instanton solutions are given by with real constants λ and θ which are moduli. The configurations ω L and ω R are BPS which are holomorphic and depend on z, while their complex conjugate ω * L and ω * R are anti-BPS which are anti-holomorphic and depend onz only. Fig. 1 shows configurations in m(x) of these solutions. The configuration ω L (ω * L ) goes to n = (1, 0) (m = (0, 0, 1)) denoted by ⊙ at x 1 → −∞ and to n = (0, 1) (m = (0, 0, −1)) denoted by ⊗ at x 1 → +∞. The configuration and ω R (ω * R ) can be regarded as a domain wall and anti-domain wall, respectively. A domain wall at each constant x 2 slice corresponds to a path connecting the north pole m = (0, 0, +1) and the south pole m = (0, 0, −1) in the target space, as illustrated in Fig. 2(a). A U(1) modulus is localized on these domain walls characterizing which point on the equator in the target space a domain wall passes through [56]. This U(1) modulus is twisted along the domain wall to satisfy the boundary condition at x 2 = 0 and x 2 = 1. When one changes a constant x 2 slice from x 2 = 0 to x 2 = 1, a path in the target space changes with sweeping a half of the sphere as the target space, as shown in Fig.2(b) and (c). Therefore, these configurations give maps from the space R × S 1 to a half of the target space. BPS configurations ω L and ω R carry a half of the unit instanton corresponds to the configurations ω R and ω * L while (c) corresponds to the configurations ω L and ω * R .
The superposition ansatz such as ours has been studied long for Yang-Mills instantons and CP N −1 instantons on S 2 and R 2 (See [58] for example). On the other hand, for these theories, a multiple-type ansatz has been also investigated [59], in relation to the study on "zindons". However, due to the fixed twisted boundary condition, it is not straightforward to construct an ansatz for the present case, with keeping, non-triviality of configurations, finite energy and the boundary conditions. The twisted boundary condition strongly restricts patterns of ansatz. This is why we begin with the simple ansatz (20).
It is notable that the action density and topological charge density are independent of θ 1 . The operators P, ∂ z P, ∂zP, ∂ z P∂zP and ∂zP∂ z P have the following form as where a, b, c and d are some functions of z andz including λ 1 , λ 2 and θ 2 as parameters.
Then, it is obvious that both s( have no θ 1 dependence. It means that θ 1 corresponds to a bosonic zero mode, which does FIG. 3: Neutral bion. This is a composite of fractionalized instantons ω L and ω * R , where we have introduced a relative phase. The notation is the same as Fig. 1 s FIG. 4: Action density s(x 1 ) and topological charge density q(x 1 ) for the configuration of Eq. (20) for λ 1 = 1000, λ 2 = 1 and θ 2 = 0. The distance between the peaks of two fractionalized instantons is given by ∼ 4.3976, which is consistent with the separation (1/π) log(1000 2 ) obtained from Eq. (33).
not cost the configuration energy. On the other hand, the configuration depends on θ 2 . For now we assume θ 2 = 0, and will consider θ 2 = 0 cases later.
The total action and the net topological charge in the large-separation limit are given by respectively. We note that the topological charge is zero for any values of separation, and this configuration corresponds to a topologically trivial vacuum.
The corresponding configuration again has no θ 1 dependence. For λ 2 1 ≫ λ 2 , this configuration corresponds to a 1/N instanton (S = 1/N, Q = 1/N) and a 1/N anti-instanton (S = 1/N, Q = −1/N) at large separations. The total action and the net topological charge in this large-separation limit are given by respectively.
As x 1 varies from −∞ to ∞, the normalized complex vector n(x 1 ) takes the following three different values, which we denote as n 1 , n 2 , n 3 , for λ 2 1 > λ 2 . The above three domains are divided by two critical points corresponding to the locations of the two kinks. As shown in [9], the two affine co-roots α i and α j , which correspond to the two kinks (fractionalized instantons) in Fig. 4 are given by which satisfies In the present case, α i and α j are identical, which we define as α i = α j ≡ α. It is given by We note that α i · α j = α · α > 0 for this case.
The explicit form of the action density s(x 1 ) for general N is given by   (20). The action is independent of λ 2 for λ 2 1 /λ 2 fixed. The configuration is changed from S = 1 to S = 0, due to the attractive force.
We will from now look into the attractive interaction between the two fractionalized instantons. In order to understand precise separation dependence of action and interaction force, we need to know the exact separation between the two components of fractionalized instantons in our configuration. The positions τ 1 and τ 2 of fractionalized instantons and fractionalized anti-instantons in the x 1 -coordinate are given by the balance conditions [44,45,61,62], respectively. Then, the separation τ between them is given by For τ ≥ 0, τ can be interpreted as separation between the fractionalized-instanton components. The definition of separation depends on N for this configuration. As noted in the caption in Fig. 4, this definition of separation precisely describes the distance between the locations of two fractionalized instantons. Fig. 6 depicts the separation τ dependence of the total action S and the static force F = − dS dτ with λ 2 = 1 fixed for N = 2. It indicates that the total action monotonically decreases as τ gets smaller, and the interaction force is negative for wide τ range. It clearly shows that the fractionalized-instanton constituents exert an attractive force. To be precise, as will be shown later, the interaction force is exponentially suppressed for large separation τ ≫ 1 or the merged limit τ ≪ 0 (λ 2 1 /λ 2 ≪ 1). It indicates that our ansatz yields intermediate configurations between two (approximate) solutions, a two-separated fractionalized-instanton solution (S = 1, Q = 0) and a trivial perturbative vacuum (S = 0, Q = 0). Our analysis is easily generalized to λ 2 = 1, where we find that the total action is independent of λ 2 if λ 2 1 /λ 2 or τ are fixed. From this analysis, we see that the location of the center of mass λ 2 corresponds to a bosonic zero mode while λ 2 1 /λ 2 to a negative mode. The two constituents are getting closer and finally are merged by the attractive force, as shown in Fig. 7. The resultant configuration at τ = −∞ (λ 2 1 /λ 2 = 0) is given by for N = 2, and ω(τ = −∞) → 0, · · ·, 0, 1 + λ 2 e iθ 2 e 2π(z+z)/N , 0, 0, · · ·, 0 T .
for general N, with the quantum number The τ = (1/π) log λ 2 1 /λ 2 dependence of the total action S and the force F = − dS dτ with θ 2 = 0 for (20). For τ ≥ 0, we can interpret τ as separation between the instanton constituents.
The configuration is changed from S = 1 to S = 0, due to the attractive force. The configuration for τ 1 corresponds to neutral bions. which is identical to a trivial vacuum.
Here we discuss the characteristic size of neutral bions. As shown in [9], the size of "charged" bions N f > 0 is clearly determined since the instanton constituents are bound due to the balance of bosonic repulsive and fermionic attractive forces. This calculation can be extended to the case of N f = 0, and it gives the length scale of charged bions as τ * = N 2π log 8π g 2 N , which reads τ * = 0.5 ∼ 1.5 for N = O(1) with g 2 = 1. On the other hand, the calculation of the neutral bion size is not straightforward since its bosonic interaction is attractive and it requires analytic continuation of g 2 to negative values and its returning back to positive values. However, taking into account the fact that the actions of neutral and charged bions in the amplitudes are common except for the imaginary part, we speculate that neutral bions have a similar size or length scale to that of the charged bions τ * ∼ 1.
In the rest of the paper, we assume that the neutral bions arise from the separation scale τ 1. We thus regard our ansatz as the neutral bions only for the separation τ > 1. We will discuss whether this assumption is appropriate or not in terms of BZJ-prescription in Sec. V.
We now investigate interaction part of the action for this configuration, to compare our concrete ansatz to the far-separated instanton argument (1) in Ref. [9]. The interaction part of the action density is written as the action density s(x 1 ) minus the one fractionalizedinstanton density and one fractionalized-anti-instanton density s ν=1/N (x 1 ) + s ν=−1/N (x 1 ), The integrated interaction action is then given by In Fig. 8, we plot the logarithm of the total interaction action S int (N, τ ) as a function of τ for N = 2, 3, 4. For τ 1 region, log(−S int (N, τ )) can be well approximated by analytic lines, where ξ(N) is a slope and C(N) is a y-intercept. In Fig. 8 we simultaneously depict these analytic lines for the three cases. The slopes ξ of the approximate lines read ξ ∼ π for N = 2, ξ ∼ 2π/3 for N = 3 and ξ ∼ π/2 for N = 4, which indicates that the slope ξ can be generally expressed as Therefore we observe that the interaction action can be written as the following form for τ 1 region, This ξ is equivalent to the (dimensionless) lowest Kaluza-Klein spectrum Lm LKK , which is given as Lm LKK = |q i − q j | = 2π/N, where q i and q j are two nonzero components of Wilson-loop holonomy in (14)(15).
We next determine the N dependence of the y-intercept C(N). In Fig. 9 we plot exponential of the intercept exp[C(N)] as a function of N for N = 2, 3, 4, 5, 6, 7. We find out that this dependence is well approximated by exp[C(N)] ∼ 4/N, and depict it simultaneously in the figure. This result shows that the interaction action for τ 1 can be written as It means, for a wide range of separations τ 1, the interaction part of the action for our configuration is consistent with the neutral bion action (1) obtained from the far-separated instanton calculation up to 2π factor, which we introduced for convenience, as with α i · α j = α · α = 1 for our ansatz following the Lie algebra notation in [9].
We have shown that our ansatz (20) gives a configuration consistent to (1) except in the merged region τ < 1. It means that (20) is a good ansatz describing the neutral bion, and can be identified as an infrared renormalon since the imaginary part of its amplitude obtained through the BZJ-prescription (g 2 → −g 2 ) and analytic continuation cancels the notorious ambiguity arising in the Borel re-summation of the perturbative series. By use of the present ansatz, we can study properties of bions and the related physics, not only at large separation τ ≫ 1, but also at short separation τ 1.
We here discuss a physical meaning of our ansatz for the merged region τ < 1. The coincidence of the interactive actions for τ 1 (Fig. 8) implies that the neutral bion scale can be determined by such a coincidence point, or τ ∼ 1 for this case, which is consistent with the charged bion scale. On the other hand, for the region τ < 1, the configurations (see the right two columns in Fig. 7) are regarded as those around the perturbative vacuum rather than the bion saddle. It means that, in the semi-classical calculation, these configurations correspond to fluctuations around the perturbative saddle, but not around the bion saddle point. It is also notable that our ansatz connects these two different saddle points continuously by a  single parameter. We will briefly discuss how to classify the parameter regions into the two sectors in Sec. V.
We here make a comment on cases for θ 2 = 0. For 0 < θ 2 < π/2, the interaction force is qualitatively the same as the case for θ 2 = 0, or attractive. For π/2 ≤ θ 2 ≤ π, things change: The potential barrier emerges around λ 2 1 /λ 2 = 1 (τ = 0), and the height becomes infinite for θ 2 = π as shown in Fig. 10. Of course, this does not mean that the interaction is repulsive since θ 2 is also a dynamical field variable and should relax eventually to θ 2 = 0 in order to minimize the total action. The result indicates that θ 2 corresponds to a positive mode.

IV. BIONS WITH NON-Z N TWISTED BOUNDARY CONDITIONS
The configuration discussed in the previous section is specific to the Z N twisted boundary condition. In this section we consider different boundary conditions and bion-like configurations. We here begin with the CP 2 model and extend the result to the CP N −1 .
We first consider the following twisted boundary condition for CP 2 This boundary condition corresponds to the vacuum where we omit permutation copies. In gauge theory, this boundary condition is realized by special Wilson-loop holonomy in the exotic gauge-broken phase in SU(3) gauge theory with adjoint quarks [16,17], where the gauge symmetry is broken as SU(3) → SU(2) × U(1). In this vacuum two elements of the Wilson-loop holonomy have the same value, but the other has a different one. This phase is called "split phase", thus we term the above boundary condition a "split" twisted boundary condition. Although how a neutral bion works in the split vacuum has not yet been elucidated, it is worth investigating bion-like configurations in this vacuum.

A. Bions for the split twisted boundary condition
We first consider a configuration in CP 2 satisfying the split twisted boundary condition (44), This is an ansatz beyond the simple superposition ansatz. For λ 2 1 ≫ λ 2 , this configuration is composed of a BPS fractionalized instanton (S = 1/2, Q = 1/2) and a BPS fractionalized anti-instanton (S = 1/2, Q = −1/2) in Fig. 11. The total action and the net topological charge in a far-separated limit are given by We note the action density and topological charge density are independent of the parameters θ 1 and θ 2 . Fig. 12 depicts λ 2 1 /λ 2 dependence of the total action S for (46). The total action is independent of λ 2 with λ 2 1 /λ 2 fixed. For this ansatz, the normalized complex vector n(x 1 ) takes the following three different values as x 1 varies from −∞ to ∞, for λ 2 1 > λ 2 . For this case, the entry of extended Cartan matrix is again positive α i · α j > 0 The separation τ is given by which is obtained from the balance conditions [44,45,61,62], FIG. 11: Action density s(x 1 ) and topological charge density q(x 1 ) for the configuration in Eq. (46) for λ 1 = 1000 and λ 2 = 1. The distance between two fractionalized instantons is ∼ 4.3976, which is consistent with the (1/π) log(1000 2 ) in Eq. (49). The configuration is changed from S = 1 to S = 1/2, due to the attractive force.
with τ = τ 2 − τ 1 . For τ ≥ 0, τ stands for a separation between the fractionalized-instanton constituents. In Fig. 13 we depicts τ dependence of the total action S and the static force (46). The result indicates that the force is negative for −∞ < τ < ∞, and the fractionalized instanton constituents have an attractive force. As with the configuration in the previous section, the two fractionalized instantons are merged by the attractive force, and finally resulting in the configuration, From this analysis, we see that λ 2 , θ 1 and θ 2 correspond to bosonic zero modes while λ 2 1 /λ 2 to a negative mode.
13: The τ = (1/π) log λ 2 1 /λ 2 dependence of the total action S and the force F = − dS dτ for the configuration in Eq. (46). For τ ≥ 0, we can interpret τ as the separation between the instanton constituents. The configuration is changed from S = 1 to S = 1/2 with Q = 0 conserved, due to the attractive force. The configuration for τ 1 corresponds to neutral bions.
The interaction part of the action takes the same form as that in the previous section S int (τ ) = 1 π dxs int (x 1 ). In Fig. 14, we plot the logarithm of the interaction action S int (τ ) as a function of τ for the present case (46). For τ 2 region, log(−S int (τ )) is approximated by log [−S int (τ )] ∼ −6 τ + 1.9657 , (τ 2) .
Therefore, the interaction action can be written as the following form for τ 2 region, This is qualitatively consistent with (1), while the coefficients are different from ξ(3) and C(3) in Z 3 twisted boundary conditions. Although how the coefficients are fixed needs to be uncovered, we at least argue that neutral bion-type configurations exist also for the split boundary condition with N = 3, which are responsible for the cancellation of infrared renormalon ambiguity.

B. Bions in extended split boundary conditions
As an extension of the split boundary condition, we consider the following ansatz of the In this case we no longer regard the boundary condition as the Wilson-loop holonomy in the split vacuum, rather one specific twisted boundary condition, with Ω = (1, 1, e 2πi/3 ). We will investigate the ansatz from pure-theoretical interest.
charge in this limit are given by respectively. We note that the topological charge is zero and conserved.
FIG. 17: The τ = (3/2π) log λ 2 1 /λ 2 dependence of the total action S and the force F = − dS dτ for the configuration in Eq. (56). For τ ≥ 0, we can interpret τ as the separation between the instanton constituents. The configuration is changed from S = 2/3 to S = 1/3 with Q = 0 conserved, due to the attractive force. The configuration for τ 1 corresponds to neutral bions.
The separation τ is given by which is obtained from the two balance conditions [44,45,61,62], with τ = τ 2 − τ 1 . In Fig. 17 we depict the τ dependence of the total action S and the static force F = − dS dτ for N = 3. The result clearly shows that the fractionalized instanton constituents have the attractive force. The two constituents are merged by the attractive force, as shown in Fig. 18. For N = 3, the configuration results in the following form at τ = −∞ (λ 2 1 /λ 2 = 0) with S = 1/3, Q = 0 .
For general N, the resultant configuration at τ = −∞ (λ 2 1 /λ 2 = 0) is given by We now investigate the interaction part of the action for this configuration. The interaction action is given by In Fig. 19, we plot the logarithm of the total interaction action S int (N, τ ) as a function of where ξ(N) is a slope and C(N) is an y-intercept. For this case, the slope is expressed as The interaction action can be written as the following form for large τ region, We next determine the N dependence of C(N). In Fig. 20   simultaneously in the figure. This result indicates that the interaction action (potential) for large-separation region can be written as which implies that the interaction part of the action at large τ region is expressed as This asymptotic form of the interaction potential is qualitatively consistent to (1). For our special boundary conditions in the present subsection, which can no longer be identified as Wilson-loop holonomy, it is not straightforward to understand the meaning of the values of ξ and e C . However, as with the case for the Z N twisted boundary condition, it is true that the prescription (g 2 → −g 2 ) and analytic continuation lead to the ambiguity in the imaginary part of the amplitude for this case too. It implies that the resurgence procedure based on neutral bions universally works for general boundary conditions and vacua in field theories.
By calculating the renormalon ambiguity in the Borel re-summation of the perturbative series for the present non-Z N boundary conditions, we can check if the two ambiguities are cancelled against each other. In the future work, we will investigate whether resurgence procedure based on bions or bion-like configuratons still works for non-Z N vacuum such as split phases and its extensions.

V. SUMMARY AND DISCUSSION
In this paper, we have revisited topologically trivial configurations in the CP N −1 model on R 1 × S 1 with twisted boundary conditions, to study properties of bions composed of multiple fractionalized-instantons. In the CP N −1 model with center-symmetric and noncenter-symmetric twisted boundary conditions, we have considered an explicit ansatz of a configuration containing one fractionalized instanton (ν = 1/N) and one fractionalized antiinstanton (ν = −1/N), which has an attractive force. We have shown that the separationdependence and N-dependence of the interaction potential of the ansatz agree with the results of the far-separated instanton calculus [9], even at small values of the separation.
In Sec. III, we have considered a simple neutral-bion ansatz for the Z N twisted boundary condition, which represents a molecule of one fractionalized instanton (ν = 1/N) and one fractionalized anti-instanton (ν = −1/N). From the separation dependence of the total action we show that the interaction between the instanton constituents are attractive, thus the configuration has a negative mode. The separation dependence and N-dependence of the interaction potential between the instanton constituents is compared with the result in the standard far-separated instanton calculus in Eq.
(1), we show that our ansatz is consistent with Eq. (1) even from short to large separations. This result indicates that our ansatz well describes the neutral bion related to renormalon ambiguity, which can be used from short (τ 1) to long (τ ≫ 1) separations.
In Sec. IV, we have proposed bion-like ansatze in non-Z N twisted boundary conditions including the one corresponding to the split vacuum in QCD(adj.) and its extensions for N ≥ 3. We have shown that the interaction between the constituents is again attractive. In this case, we have found that the separation and N dependences of the interaction potential at large separation is qualitatively consistent to the result for Z N twisted boundary conditions [9] up to a numerical coefficient. It implies that the bion resurgence procedure universally works for a wide range of boundary conditions and vacua in field theories.
Our ansatz in the CP N −1 model corresponding to bion configurations can be a good starting point for studying properties of bions and related physics explicitly. Indeed, by using our ansatz, we can study physics related to neutral bions, not only at large separations τ ≫ 1, but also at short separations τ 1, which cannot be reached by the far-separated instanton approach.
We here discuss how to determine which of the saddle points (perturbative or bion) configurations around τ ∼ 1 should be classified into. We so far have no systematic way to classify the parameters to the two associated regions by looking into our ansatz itself. On the other hand, from the viewpoint of the BZJ-prescription, the length scale where the imaginary ambiguity in the amplitude for our ansatz gets close to the ambiguity in the perturbative Borel-sum calculation should be regarded as the neutral bion scale. In our calculation, this scale is about τ ∼ 1 as shown in Fig. 8, which is consistent with the charged bion scale too.
Thus, for now, what we can do for this purpose is just to sort out the configurations by the separation of the instanton constituents based on the plausible bion size τ ∼ 1, as performed in this paper. Exact classification of the parameter regions for the two saddles should be pursued in the future study.
As a future work, we consider to study charged bion configurations, whose instanton constituents have a repulsive interaction in a bosonic sector and also have an attractive interaction due to the fermion zero mode exchange. Due to the balance between the attractive and repulsive interactions, the size of charged bions will be clearly determined and there will be no complicated problem on the size of bions for this case. In theÜnsal's argument [3,4], this configuration has great significance in weak-coupling-regime confinement via "bion condensation". While understanding of phase diagram in the L-m adj plane is required to elucidate its relation to confinement in pure Yang-Mills or QCD theories, it should be also worth investigating a concrete configuration contributing confinement in a toy model. Abelian gauge field. We will see that the Grassmanian sigma model admits charged bions in addition to neutral bions. The D-brane configurations in Ref. [45] will turn out to be very useful for analyzing this model.
One path to connect our results of bions in the CP N −1 model to QCD may be to consider a non-Abelian vortex [63][64][65] in Yang-Mills theory in the Higgs vacuum. U(N) Yang-Mills theory coupled with suitable number of Higgs matter fields in the fundamental representation admits a non-Abelian vortex, whose effective theory can be described by the CP N −1 model. In this case, the Yang-Mills instantons and monopoles become CP N −1 instantons and domain walls, respectively, when trapped inside a vortex [44,62,[66][67][68][69]. Therefore, when the vortex world-sheet is wrapped around S 1 with a Wilson-loop holonomy, bions (instanton-monopoles) in Yang-Mills theory can exist inside the vortex as the CP N −1 bions (instanton-domain walls). By taking a un-Higgsing limit, the vortex disappears, and therefore we expect that they remain as Yang-Mills bions.
The same relation holds between quark matter in high density QCD and the CP 2 model on a non-Abelian vortex [70] (see Ref. [71] for a review). This may give a hint to understand a quark-hadron duality between the confining phase at low density and the Higgs phase at high density, through a non-Abelian vortex [72].