Identification of Perturbative Ambiguity Canceled Against Bion

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[3, ormalon rwho h pertur- the of growth π π ai n general and basic a s e.Cnrr othe to Contrary ies. 2 2 KYUSHU-HET-210 htperturbation that n ~ ~( g rm PD,and (PFD), grams β 2 f h oeo the of role the ify ymndiagrams eynman 0 , g 2 ) , eeultimate ieve bion (1) where β0 is the one-loop coefficient of the beta function [e.g. β0 = 11N 3 for SU N quenched QCD]. One can see that the perturbative ambiguities reduce to nonp~ erturbative( ) factors. It is convenient to quantitatively characterize the perturbative ambiguities by singularities = ∞ k of the Borel transform B t k=0 ak k! t , the generating function of the perturbative coefficients. The PFD induces( ) the∑ singularities( ~ ) at t = 1, 2, . . . , while the renormalon at t = 1 β0, 2 β0, . . . . The PFD-type ambiguity is known to get canceled against the ambiguity ~ ~ 2 of the semi-classical calculation for the instanton–anti-instanton amplitude ∼ e− SI [5, 6], 2 2 where SI = 8π g is the one-instanton action. On the other hand, the cancellation of a renormalon ambiguity~ was shown only in a two dimensional non-linear sigma model [7]. In 2012, the conjecture was proposed [8–11] that renormalon ambiguities are canceled by the new semi-classical object called a bion [12], a pair of a fractional instanton and anti- fractional instanton. This conjecture first requires an S1 compactification of a spacetime as Rd Rd−1 S1 with a small S1-radius R such that RΛ ≪ 1, where Λ is a dynamical scale of → an asymptotically× free theory. A bion solution appears in this setup when twisted boundary conditions are imposed along the S1-direction (or equivalently when a non-trivial holonomy exists under the periodic boundary condition). The bion action is given by SB = SI N, where N is a parameter specifying the degree of freedom of dynamical variables. Accordingly,~ the 2 ambiguities in bion calculus are typically given by ∼ e− SI ~N . The corresponding perturbative ambiguities are specified by the Borel singularities at t = 1 N, 2 N, . . . . Due to similar N dependence to renormalon ambiguities, it was conjectured~ that~ the bion is responsible for the cancellation of renormalon ambiguities. Since then, active discussions on the conjecture have been initiated. However, recent studies have reported inconsistency of the conjecture. Two observations were made in the systems where bion ambiguities exist. (i) In the SU 2 and SU 3 gauge theories with adjoint fermions on R3 S1, renormalon ambiguities are( ) absent [13].( ) (ii) In the CP N−1 models on R S1, renormalon× ambiguities are specified by the Borel singularity at t = 3 2N [14, 15], which× conflicts with that of the bion ambiguity [16–19].2 The both cases indicate~( ) that the bion ambiguities do not correspond to the renormalon ambiguities. These observations give rise to a new question: what cancels bion ambiguities? If renormalon does not correspond to bions, there should be a different type of perturbative ambiguities which cancel bion ambiguities. Such a perturbative ambiguity has not been identified so far. This implies a lack of our understanding on the resurgence structure. In this Letter, we argue that bion ambiguities are canceled against PFD-type ambiguities. Although this possibility was mentioned in Ref. [13], so far, it has not been understood whether a seemingly different singularity (by the factor 1 N) can get compatible with the bion ambiguity. We clarify that it occurs by a non-trivial~ enhancement of the amplitudes of almost all Feynman diagrams as a consequence of the S1 compactification and twisted boundary conditions. As a result, an N times closer Borel singularity to the origin arises

2The renormalon analyses [14, 15] have been performed for NRΛ ≫ 1, in which the bion analyses are not always valid. An important point is that it was shown [14, 15] that the S1 compactification affects renormalon structure even in this so-called large-N volume independence domain. This is against an expectation of the conjecture. 2 consistent with the bion. Our findings are helpful in giving a unified understanding to a series of controversial discussions. We explain our setup. We consider Rd−1 S1 spacetime with the S1-radius R, and the × A ZN -twisted boundary conditions for an N-component field φ

A imA2πR A φ x, xd 2πR = e φ x, xd . (2) ( + ) ( ) d−1 1 Here x denotes the coordinates x1,...,xd−1 of R , xd does that of S , and the twist angle ( ) mA is given by = = ⎧A NR for A 1, 2, . . . , N 1, mA ⎪ ~( ) − (3) ⎨0 for A = N. ⎪ ⎩ d~2 2 This setup allows us to find a bion solution with the bion action SB = 4π mAR g ( ) ~ with A < N, and the bion ambiguities ∼ e−SB appear. The corresponding perturbative ambiguities are specified by the Borel singularities at t = 1 mAR = A N = 1 N, 2 N,.... We explain the keys to understanding the enhancement~( of the) PFD-~ type~ ambiguity~ under the above setup. First, the S1 compactification reduces the dimension of a momentum integral from d to d 1. Then, perturbation theory tends to suffer from severe infrared (IR) divergences. This is parallel− to finite-temperature and lower-dimensional super-renormalizable field theories with massless particles [20, 21]. Secondly, the twist angles play the role of mass terms for the zero Kaluza–Klein (KK) momentum, and hence work as IR regulators. As a result, an amplitude of a Feynman diagram is given by an inverse power of the twist 2 k angle as g mAR at the kth order, as a signal of the IR divergence. (It diverges when m 0.)[ For~( small)]A, this behavior gives ∼ Ng2 k due to Eq. (3). Thus, if we have the A → PFD exhibiting the IR divergences in the above( sense,) the kth order term of the perturbation series behaves as ∼ k! Ng2 k rather than ∼ k! g2 k. This causes the Borel singularities at t = 1 N, 2 N, . . . , corresponding( ) to the bion ambiguities.( ) In the~ following,~ we study the CP N model on R S1 as a concrete model, where an explicit calculation of the bion ambiguities has been performed× [19].

2. Enhancement of perturbative ambiguity: the CP N−1 model

The CP N−1 model in the two-dimensional Euclidean spacetime is given by

= 1 2 A A A A S 2 d x ∂µz¯ ∂µz jµjµ f z¯ z 1 , (4) g S − + ( − ) in terms of homogeneous coordinates zA andz ¯A, where f is introduced as the Lagrange multiplier ﬁeld to impose the constraintz ¯AzA = 1, and

1 A A jµ ≡ z¯ ←→∂ µz , ←→∂ µ ≡ ∂µ ←Ð∂ µ. (5) 2i −

3 A We implement the ZN -twisted boundary conditions (2) for z along the x2-direction. Then the propagator of zA is obtained as

i(pµ+mAδµ2)(x−y)µ A B = 2 AB e z x z¯ y g δ 2 2 a ( ) ( )f p p1 p2 mA f0 o + ( + ) + ≡ g2δAB −1 x y , (6) ◻A ( − ) with the vacuum expectation value f0, which is determined by the tadpole condition that the linear term of the ﬂuctuation δf vanishes. Here and hereafter, we use the abbreviation,

dp1 1 ≡ . (7) p π πR o S 2 2 p2Q∈Z~R 2 = Note that, in the denominator of Eq. (6), mA f0 serves as a mass parameter for p2 0. We consider the partition function in perturbation+ theory,

= f zA z¯A e−S Z S D S D D 2 2 = f e(1~g ) ∫ d xf ′ f (8) S D Z [ ] where

′ f = f0 δf Z [ + ] δ δ ≡ exp g2 ⋅ ◻−1 ⋅ δz¯A A δzA × 1 2 − A A exp 2 d x jµjµ δfz¯ z . (9) g S Vz=0 In the following, we consider only the A = 1 flavor. As we shall see, this contribution is significant for enhancement of the amplitude. (Note that a flavor symmetry is broken by the twisted boundary conditions.) We study the δf 0 term in ′ f for simplicity. O Z We first give a review on the PFD in the non-compactified( ) case.[ ] We estimate the kth order perturbative contribution by equally assigning one to every possible diagram. For this purpose, it is suitable to consider the replacements ◻−1 1 and j zz¯ . This ap- A → µ → proximately corresponds to employing the zero-dimensional version of the model. Then ′ ≡ ∞ 2 k f k=0 Tk g is estimated as Z [ ] ∑ ( ) 2k 1 δ δ 1 2 k Tk ∼ zz¯ 2k ! δz¯ δz k! ( ) ( ) 2k ! = ( ) ∼ 4kΓ k + 1 2 . (10) k! ( ~ ) This factorial growth is interpreted as the source of the Borel singularities at t = 1, 2, . . . in the non-compactified spacetime, corresponding to the ambiguities in the instanton–anti- instanton calculus [5, 6]. 4 Now we explain how the enhancement in terms of N occurs under the S1 compactification and twisted boundary conditions. For this purpose, the above naive counting is not sufficient and we need to have a closer look at the structure of loop integrals. The kth order perturbative contribution to ′ f is given by Z [ ] ′ f th order Z ]Sk [ 2 1 δ δ k = g2 ⋅ ◻−1 ⋅ 2k ! δz¯A A δzA ( ) 2 k × 1 1 −1 2 A A 2 d x z¯ ←→∂ µz . (11) k! g 4 S The number of vertices, V , is identical to the perturbation order, V = k. We first consider connected Feynman diagrams, for which the following relations follow:

P = 2V, L = V + 1, (12) where P and L denote the numbers of propagators and loops, respectively. Noting these relations, an amplitude of a Feynman diagram is written as

(2k) + 2 k F pi,µ mAδµ2 2 ( ) V g 2k 2 2 , (13) ( ) p1,...,p +1 + + + o k i=1 qi,1 qi,2 mA f0 ∏ [ ( ) ] where V2 is the volume factor of the two-dimensional spacetime, pi denotes a loop momentum, and qi is the momentum of a propagator, which is given by a linear combination of (p1,..., (2k) pk+1). In the numerator, we have a 2k th-order homogeneous polynomial F , originating from the derivative in the interaction( term.) ∀ The enhancement of the amplitude is caused by the zero KK-modes where pi,2 = 0 for i, ∀ and hence, qi,2 = 0 for i. For this part, we have

2 k k+1 (2k) V2 g dpi,1 F pi,1, mA ( ) . (14) ( )k+1 2k 2 + 2 + 2πR S Mi=1 2π i=1 qi,1 m f0 ( ) ∏ [ A ] Note that each loop integral becomes a one-dimensional integral due to the S1 compactiﬁca- tion. Accordingly, this expression possesses an IR divergence in the massless limit m2 +f 0 A → with the degree of divergence k−1; the mass dimension of the integration measures is k+1 , ( ) whereas that of the integrand is −2k. Then, if f0 is negligible compared with mA=1 = 1 NR , we obtain ~( ) 2 k 2 k ∼ V2 1 g = V2 1 Ng 2 k−1 2 (15) R mAR 4π R N 4π ( ) 2 because mA works as an IR regulator. (See below for discussion on the size of f0 mA.) We note that this argument indicates that a general connected diagram has the contribution~ naturally speciﬁed by N kg2k rather than g2k.

5 Figure 1: Estimate of Ck~Tk.

We see that disconnected diagrams show weaker enhancement. As an example, let us consider a kth order disconnected diagram given as a product of two connected diagrams. Since each connected diagram satisﬁes Eq. (12), the amplitude is given by

2 2 k V2 1 Ng ∼ . (16) R2 N 2 4π from Eq. (15). (Note that the total number of vertices is fixed as k.) This is suppressed by the inverse power of N compared with the contribution of the connected diagram (15). We can show that connected diagrams, which have the strongest enhancement, increase factorially. To show this, it is sufficient to assign one to every connected diagram. One can use ∞ ∞ 2 k 2 k C = Ck g = ln Tk g , (17) kQ=0 ( ) kQ=0 ( ) since the partition function ′ is given by the exponential of the total sum of connected Z diagrams C (i.e. ′ = eC ). From the estimate of T in Eq. (10) and this formula, we Z k estimate Ck in Fig. 1. One sees that Ck grows factorially at the almost same rate as Tk. 2 k Thus, it is plausible that we have the perturbation series as k k! Ng and the Borel singularity at t = 1 N, corresponding to the bion ambiguity. ∑ ( ) We emphasize that~ both the bion ambiguity and the position of the Borel singularity are uniformly controlled by the twist angles. This indicates validity of the resurgence structure between the bion and enhanced perturbative contributions. 2 ≫ We note that the above enhancement is specific to the case of mA f0. In contrast, if

6 2 ≪ mA f0, since f0 works as an IR regulator, we have

2 k V2 m R α g ∼ c A . 2 α ( 2 (k+)α−1)~2 (18) R αQ≥0 f0R 4π ( ) The enhancement does not occur in this case. 2 The size of mA f0, which is critical for the existence of the N times closer Borel singularity, is determined~ depending on the magnitude of NRΛ. To determine the vacuum expectation value f0, we consider the tadpole condition at the one-loop level. For NRΛ ≪ 1, 2 1 one finds a perturbative solution √f0R ∼ g NR − 4π in terms of the renormalized 2 2 ≫ (( ) )~( ) coupling g µ [22]. Hence mA f0 is satisfied within the perturbative expansion. Thus, the Borel singularity( ) at t = 1 N is likely to arise. It is worth noting that NRΛ ≪ 1 also ensures validity of bion analyses.~ This shows overall consistency. 2 On the other hand, for NRΛ ≫ 1, f0 is given by f0 = Λ as in the non-compactified case 2 = 2 ≪ due to the so-called large N volume independence [23]. Then since mA f0 1 NRΛ 1, we do not have the above enhancement or a Borel singularity at t =~1 N from~( the) PFD. In fact, since we have inverse powers of Λ in this case [24], genuine perturbative~ analysis cannot be considered reasonably. For NRΛ ≫ 1, the bion analyses are not always valid [8– 11, 16–19]. We give supplementary explanations in the case NRΛ ≪ 1, where we have the enhancement. First, we mention effects of other flavor contributions than A = 1. From the zero k−1 2 k k−1 2 k KK modes, we have k! mAR g 4π ∼ k! N A g 4π when we consider the A flavor alone. Here, A~(N corresponds) [ ~( to)] the position( ~ ) of[ the~( Borel)] singularity. Then, the closest singularity is given~ by A = 1, and so is the asymptotic behavior of the perturbative coefficients. This is the reason why we focused on this contribution. Secondly, we clarify difference in power counting of N from the non-compactified case. So far, we have not considered sums over flavor indices and limited the flavor to A = 1. Nevertheless, N dependence appears as a consequence of loop momentum integrations, where, in particular, the IR structure of loop integrands is essential. This is in contrast to the non-compactified case, where N dependence arises exclusively from the sums over flavor indices, and momentum integrations are irrelevant. When one takes into account sums over indices as well in the compactified case, one needs to pay attention to both the degree of IR divergence and structure of flavor indices. For instance, the N dependence of the diagram depicted in Fig. 2 is given by N 2k−1g2k at the kth order, provided that flavor indices for individual loops are independent. Here, we note that the loops outside a central loop do not possess IR divergences. Thirdly, we note that, in fact, the A = N − 1 sector also has the strongest enhancement. This is because p2 + mA can be −1 NR for p2 = −1 R. Thus, this part has the same order contribution as the A = 1 sector. ~( ) ~ Although we have considered the δf 0 term so far, we can repeat a similar analysis, O for instance, for the quadratic term in δf( of) the effective action for this field, which has a clearer physical meaning. Finally, we make some remarks. Adopting the homogeneous coordinates as in the present study, we actually have a difficulty in defining fixed order perturbation theory for the effective 7 ......

Figure 2: A series of Feynman diagrams whose amplitude at the kth order is of order N 2k−1g2k.

action. In integrating out the A = N flavor, the IR regulator is given by f0 since mA = 0. Namely, zN is subject to the periodic boundary condition and we partially have the same 4 2 situation as finite-temperature field theory. For NRΛ ≪ 1, f0 is given by f0 ∼ g 4πR and plays the role of a screening mass [20, 21]. It should be kept in the denominator~( ) of Eq. (14), otherwise calculations break down due to IR divergences. In this treatment, we have the term with α = 0 in Eq. (18), and the kth order perturbative contribution is partially 2 2 1 2 2 given by g k f0R (k− )~ = O g . It is irrelevant to the perturbation order k. Therefore, higher-loop diagrams~( ) can contribute( ) at the same order. This kind of problem is known as the Linde problem [25, 26]. Then, it is practically impossible to systematically obtain a series expansion in g2 for the effective action even at relatively low orders. If we instaed adopt inhomogeneous coordinates ϕa = za zN (a = 1, 2, . . . , N −1), perturbative calculations are free from the Linde problem because~ all twist angles are taken non-zero. With the use of the inhomogeneous coordinates we can also argue that perturbative coefficients behave as ∼ k! Ng2 k for the partition function in a similar manner. We have used the homogeneous coordinates( ) here for simpler illustration.

3. Conclusions

For nearly a decade, there have been vigorous discussions on the conjecture that the bion is responsible for the cancellation of renormalon ambiguities, and a unified interpretation has not been established. In this Letter, we argued convincingly that the bion cancels the perturbative ambiguity caused by the proliferation of Feynman diagrams. In particular, we demonstrated how a Borel singularity appears at t = 1 N upon the S1 compactification and twisted boundary conditions, which is located at t = ~1 in the non-compactified spacetime. Here, the IR structure of loop integrals is crucial, and the N times closer singularity is regarded as a signal of IR divergences in perturbation theory. This observation settles a recent controversy about the role of the bion and deepens our understanding on the resurgence structure.

Acknowledgments

The authors are grateful to H. Suzuki for encouragement and valuable comments. They also thank M. Ishii, E. Itou, and K. Yonekura for discussions. This work was supported by

8 JSPS Grants-in-Aid for Scientiﬁc Research numbers JP18J20935 (O.M.) and JP19K14711 (H.T.).

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