Renormalon Structure in Compactified Spacetime

arXiv:1909.09579v3 [hep-th] 16 Dec 2019 oueIshikawa Kosuke compactified spacetime in structure Renormalon ioaaTakaura Hiromasa 1 25/2/2020 ∗ eateto hsc,Kuh nvriy74Mtoa Nish Motooka, 744 University Kyushu Physics, of Department -al [email protected] E-mail: ...... integer sntascae ihtetitdbudr odtosadtksth takes and conditions boundary twisted the with associated not is ntenncmatfidspacetime non-compactified the on hc r fe elzdi large in realized often are which spacetime compactified ytecmatfiain n i)telo oetmvral ln the along n variable is momentum it loop i.e. the independent, (ii) volume and is compactification, diagram the renormalon by a of integrand loop icecompactification circle u sidpneto eal fteqatte ne osdrto.A a As consideration. under quantities the the of details of independent is ujc ne 0,B2 B35 B32, B06, Index Subject ...... epitotta h oaino eomlnsnuaiisi hoyon theory in singularities renormalon of location the that out point We pae hr h eomlnabgiyof ambiguity renormalon the where -plane, C P 1 N n kt Morikawa Okuto , 1 − efidta h oe iglrt sgnrlysitdby shifted generally is singularity Borel the that find We . , ∗ 1 oe on model R R × R d d S → − 1 1 1 with R × auaShibata Kazuya , d S − 1 1 N Z × R wt ml radius small a (with N 1 d S hoiswt wse onaycniin:()a (i) conditions: boundary twisted with theories eageti ne h olwn assumptions, following the under this argue We . wse onaycniin ntelarge the in conditions boundary twisted 1 h euti eea o n dimension any for general is result The . O rpitnme:KYUSHU-HET-199 number: Preprint (Λ ...... k scagdto changed is ) -u uuk,8909,Japan 819-0395, Fukuoka, i-ku, 1 ioh Suzuki Hiroshi , ...... R Λ yee using typeset ≪ )cndffrfo that from differ can 1) O (Λ xml,w study we example, n values e − k − 1 1 1 and , / /R nteBorel the in 2 P S u othe to due ) tmodified ot T 1 n/R P direction circle- a N T E d limit. X.cls with and 1. Introduction In perturbation theory of quantum field theory, perturbative series are typically divergent due to the factorial growth of the perturbative coefficients. There are two sources of this growth. One is the rapid growth of the number of Feynman diagrams as n! at the nth ∼ order. The other typically originates from a single Feynman diagram whose amplitude grows factorially βnn!, and is related to the beta function of the theory, where β is the one- ∼ 0 0 loop coefficient of the beta function. The latter is known as the renormalon [1, 2]. Such divergence of perturbative series implies that the accuracy of perturbative predictions is limited. Through the so-called Borel procedure (which is used to obtain a finite result from divergent series), the first source induces an imaginary ambiguity of (e−2SI ) in terms of d/2 O the one-instanton action SI = (4π) N/(2λ) in d-dimensional spacetime, and the second is (e−2SI /(Nβ0)), called the renormalon ambiguity, where λ denotes the ’t Hooft coupling O (defined as λ = g2N from a conventional coupling g). It is believed that the perturbative ambiguities disappear after the nonperturbative contributions are added. It has been pointed out that the first kind of the ambiguity is canceled against the ambiguity associated with the instanton-anti-instanton calculation [3–6]. Here, the semiclassical configuration plays an important role, where the two-instanton action 2SI gives the same size of contribution as the first kind of perturbative ambiguity to the path integral. On the other hand, it is not really known how the renormalon ambiguity is cured. A clear exposition is only known in the O(N) non-linear sigma model on two-dimensional spacetime, where a nonperturbative condensate, which appears in the context of the operator product expansion, cancels the renormalon ambiguity [2, 7–11]. There are currently active attempts to seek the semiclassical object which cancels the renormalon ambiguity, expecting a scenario analogous to the cancellation mechanism of the first kind of perturbative ambiguity. Since the instanton action is not compatible with the renormalon ambiguity by the factor Nβ0, another configuration is needed. A recent idea is to find such a configuration by the S1 compactification of the spacetime as Rd Rd−1 S1 → × (see Ref. [12] and references therein). In order for the semiclassical calculation to be valid, the S1 radius R is taken to be small RΛ 1, where Λ denotes the dynamical scale. In ≪ some theories on the compactified spacetime, a semiclassical solution that may be able to cancel the renormalon ambiguity is found, called the bion. In this scenario the ambiguity associated with the bion calculation is expected to cancel the renormalon ambiguity in the theory on Rd−1 S1 first, and then smooth connection to the theory on Rd is assumed. × In Refs. [13–17], it was claimed that the bion ambiguity is consistent with the renormalon ambiguity. So far, however, there has been no explicit confirmation that the bion truly cancels the renormalon ambiguity. To examine the validity of the bion scenario, it is of great importance to clarify the renormalon structure on the compactified spacetime because, as mentioned, this scenario expects the cancellation of the ambiguities due to the renormalon and bion first on the compactified spacetime. The purpose of this paper is to present some insight on the renormalon structure of the theory on the compactified spacetime.

2 In Ref. [18], the renormalon ambiguity in the supersymmetric CP N−1 model on the circle- compactified spacetime R S1 with Z twisted boundary conditions was studied.1 In a × N systematic expansion in 1/N,2 it was found that the renormalon ambiguity of the photon condensate (and its gradient flow extension) changes from that in the non-compactified spacetime R2. In particular, the Borel singularity is shifted by 1/2 in the Borel u-plane − (whose definition is explained shortly) due to the compactification. In this paper, as a generalization of the previous work [18], we present a general mechanism to explain the shift of the Borel singularity, or renormalon ambiguity. Our argument proceeds under the assumption that a loop integrand of a renormalon diagram is not modified by the circle compactification, although we consider a sufficiently small radius ΛR 1. In other ≪ words, we assume that a loop integrand exhibits the so-called volume independence. This feature would be general in the large N limit with certain twisted boundary conditions [21– 28]. We also assume that the loop momentum variable of the renormalon diagram along the S1 direction is given by n/R with integer n, and is not associated with the twisted boundary conditions. These assumptions typically correspond to the situation explained in Fig. 1. Under these assumptions, we study a renormalon diagram in the compactified spacetime Rd−1 S1, not restricting the dimension d. It tells us that the Borel singularity is shifted × by 1/2 in the Borel u-plane due to the compactification, independently of the dimension − of spacetime or the details of the physical quantities under consideration. The origin of this shift can be easily and clearly understood by effective reduction of the dimension of the momentum integration, as shall be explained. We also treat, as an explicit example, the CP N−1 model on R S1 with the Z twisted boundary conditions in the large N limit, × N where we observe the shift of a Borel singularity for an observable defined from the gradient flow [29, 30]. Let us clarify the definitions adopted in this paper to study factorially divergent series. For a perturbative series, ∞ n β0λ λ dn , (1.1) (4π)d/2 n=0 X we define its Borel transform as ∞ d B(u)= n un , (1.2) n! n=0 X and correspondingly the Borel integral is given by d/2 ∞ (4π) d/2 du B(u)e−(4π) u/(β0λ) . (1.3) β 0 Z0 In our definition, a pole singularity of the Borel transform at u = u0 > 0 gives an ambi- d/2 guity in the Borel integral of order e−(4π) u0/(β0λ) = e−2SI u0/(Nβ0). Thus, our definition is −S convenient to grasp the ambiguity in terms of e I ; it is enough to focus on the value u0

1 Reference [19] is a pioneering work studying the renormalon in the compactiﬁed spacetime. The authors of Ref. [19] analyzed the renormalon in SU(N) QCD on R3 S1 with adjoint fermions for small N, in which the bion analysis has been carried out [13, 14].× On the other hand, Ref. [20] investigated the renormalon of the same system for large N. (The analysis of Ref. [20] utilizes the so-called large-β0 approximation, which is a conventional tool to analyze renormalon in non-Abelian gauge theory, combined with large N limit.) 2 The large N limit with RΛ 1 ﬁxed was considered. ≪ 3 k p

Fig. 1 Renormalon diagram. The typical situation we consider is that the ﬁeld corresponding to the wavy line satisﬁes the periodic boundary condition (and has the Kaluza–Klein

(KK) momentum pd = n/R), while the field making bubbles (solid line) satisfies the twisted boundary conditions (effectively corresponding to kd = n/(NR)) in large N theories. The twisted boundary conditions for the field making bubbles are responsible for the volume independence of the loop integrand f(p). and not necessary to pay attention to the dimension d. (This definition coincides with those in Refs. [18, 20].) As we mentioned, we show the shift of the Borel singularity by 1/2 in − the compactified spacetime compared to the non-compactified case. This paper is organized as follows. In Sect. 2 we explain the general mechanism of how the shift of the Borel singularity occurs with the circle compactification of spacetime. In Sect. 3, as an example we study the CP N−1 model on R S1 with the Z twisted bound- × N ary conditions in the large N limit. The effective action of auxiliary fields exhibit volume independence, and thus, our large N calculation essentially reduces to the one in Ref. [31]. We note that the above volume independence of this model has already been clarified in Ref. [27], although the clarification of the renormalon structure is the novel point in this paper. Section 4 is devoted to the conclusions.

2. Renormalon structure in compactiﬁed spacetime In asymptotically free theory on the non-compactiﬁed spacetime Rd, a typical form from which a renormalon ambiguity appears is given by

ddp F (p)λ(p2e−C ) , (2.1) (2π)d Z where ddp is typically a loop integral and C is a constant. We encounter Eq. (2.1) in analyzing the renormalon using the leading logarithmic approximation, the large-β approx- R 0 imation [32–34], and the large N approximation [2, 7–11]. Here, λ denotes the running coupling which satisﬁes the renormalization group equation

2 d 2 β0 2 2 µ λ(µ )= λ (µ ) with β0 > 0 , (2.2) dµ2 −(4π)d/2 whose solution is given by

d/2 2 (4π) 1 λ(µ )= 2 2 , (2.3) β0 log(µ /Λ ) 4 d/2 2 with a renormalization group invariant (dynamical) mass scale Λ2 = µ2e−(4π) /[β0λ(µ )]. When the asymptotic form of F (p) in the infrared (IR) region is given by3

F (p) (p2)α , (2.4) ≃ the Borel singularity arises at d u = α + (2.5) 2 from perturbative expansion of Eq. (2.1), which gives the renormalon ambiguity of (Λ2α+d).4 O Suppose that, in asymptotically free theory on the compactified spacetime Rd−1 S1, we × have ∞ 1 dd−1p F (p,p = n/R)λ(p2e−C ) , (2.6) 2πR (2π)d−1 d n=−∞ X Z 1 as a perturbative contribution. Here pd denotes the KK momentum along S and is given d−1 by pd = n/R, whereas p denotes the (continuous) momentum on R . As in Eq. (2.6), we assume that the loop integrand/summand is not modified from the infinite-volume case (2.1). That is, we assume volume independence of the integrand/summand. We also assume the 1 5 discrete loop momentum along the S direction to be pd = n/R with integer n. In the next section, as an example where we have Eq. (2.6), we study the CP N−1 model on R S1 × with the ZN twisted boundary conditions. It will be shown that the large N limit and the ZN twisted boundary conditions play an essential role in realizing the volume independence of the loop integrand/summand [27]. We analyze the renormalon ambiguity involved in Eq. (2.6). To analyze the IR renormalon, it is sufficient to focus on the IR region by introducing a UV cutoff q to the momentum p2

3 We consider a ultraviolet (UV) convergent quantity and hence the behavior of F (p) in the UV region is not the same as that in the IR region. 4 This can be easily seen by repeating the subsequent argument but without compactification. 5 In Ref. [20], which studies SU(N) QCD with adjoint fermions on R3 S1, we encounter the case × that the discrete loop momentum is effectively given by pd = n/(NR) rather than pd = n/R. 6 We obtain Eq. (2.8) by changing the order of the momentum integration and the infinite sum. This is indeed justified, for instance, in the region u < [α + (d 1)/2]/2 (for real u). The result obtained in this region can be extended to the whole| u|-plane by analytic− continuation. 5 The Borel transform (2.8) is easily evaluated as 1 1 1 q2α+d−1−2u B(u) = (µ2eC )u . (2.10) (4π)(d−1)/2 Γ((d 1)/2) 2πR α + (d 1)/2 u − − − This possesses a simple pole at7 d 1 u = α + − > 0 . (2.11) 2 The Borel singularity is shifted by 1/2 compared to the infinite-volume case as shown − in Eq. (2.5). As a result, the renormalon ambiguity appears as d/2 ∞×e±iδ (4π) d/2 du B(u)e−(4π) u/[β0λ(µ)] β 0 Z0 1 α+ d−1 √4π 1 ( iπ) eC 2 Λ2α+d−1 , (2.12) ∼ ± β Γ((d 1)/2) 2πR 0 − where only the renormalon ambiguity is shown. Equations (2.11) and (2.12) are the main results of this argument. As one can see from Eq. (2.12), the renormalon ambiguity is independent of the artificial momentum cutoff q. In fact, this cutoff independence holds in a broader sense. Let us take the UV cutoff as (Λ )R−1

N−1 1 3. Renormalon of the CP model on R × S with ZN twisted boundary conditions As an example where the mechanism in Sect. 2 applies, we consider the CP N−1 model on R S1 with the Z twisted boundary conditions. The action of this model in terms of × N

7 We assume 2α + d 1 > 0 for IR finiteness of Eq. (2.6). 8 However, we restrict− the UV cutoff to be q Q, where Q denotes the typical scale of an observable (which is not explicitly considered here) such≪ that the expansion of F in the low energy region is justified. (F is generally a function of p and Q.) For instance, for the photon (or gluon) condensate defined in the gradient flow (as we consider in Sect. 3), the typical scale is Q2 = t−1, where t is the flow time. 6 the homogeneous coordinate zA (A = 1,...,N) obeying the constraintz ¯AzA = 1 is defined by N S = d2x ∂ z¯A∂ zA j j + S , (3.1) λ µ µ − µ µ top Z 0 with the current jµ, 1 j = z¯A∂ zA zA∂ z¯A . (3.2) µ 2i µ − µ The topological term is given by iθ S = d2x ǫ ∂ j , (3.3) top 2π µν µ ν Z where ǫ = ǫ = +1. Here and hereafter, summation over the repeated indices is always xy − yx understood. It is convenient to adopt the following action with auxiliary fields to carry out the large N expansion [35]: N iθ S′ = S + d2x (A + j )(A + j )+ f z¯AzA 1 d2x ǫ ∂ (A + j ) λ µ µ µ µ − − 2π µν µ ν ν Z 0 Z N iθ = d2x f +z ¯A( D D + f)zA d2x ǫ ∂ A , (3.4) λ − − µ µ − 2π µν µ ν Z 0 Z where Dµ = ∂µ + iAµ. To respect the U(1) gauge symmetry of the model, Aµ behaves as a gauge field under the transformation zA gzA with g U(1): → ∈ 1 A A g−1∂ g . (3.5) µ → µ − i µ 1 A We impose the following ZN twisted boundary conditions along the S direction for z :

zA(x,y + 2πR)= e2πimARzA(x,y) , (3.6) where (x,y) R S1 and ∈ × A m = for A =1, ..., N 1 , (3.7) A NR −

mN = 0 . (3.8)

The auxiliary ﬁelds, Aµ and f, satisfy the periodic boundary conditions. In what follows, we analyze the renormalon ambiguity in this model by using 1/N expansion. As we shall see, renormalon diagrams of this model possess the same integrands as the inﬁnite-volume case.

3.1. Volume independence of the eﬀective action

As pointed out in Ref. [27], the effective action for the auxiliary fields Seff [Aµ,f] exhibits volume independence due to the ZN twisted boundary conditions and the large N. We illustrate this point, aiming for a self-contained explanation. After integrating out zA, the effective action is obtained as

2 N Seff [Aµ,f]= d x f + Tr Ln( DµDµ + f) , (3.9) − λ0 − Z XA where the topological term should be treated separately. We first calculate the effective potential, which is obtained as S = V V (A ,f ); the fields with subscript 0 denote the eff 2 · eff µ0 0 7 constant values at the saddle point; V2 represents the volume of two-dimensional spacetime. Veff is explicitly given by

N dkx 1 2 2 Veﬀ (Aµ0,f0)= f0 + ln (kx + Ax0) + (ky + mA + Ay0) + f0 . −λ0 2π 2πR XA Z Xky (3.10) 9 Here, the KK momentum ky is discrete: n k = , n Z . (3.11) y R ∈ By using the formula

∞ ∞ 1 dky F (n/R)= eiky 2πRnF (k ) , (3.12) 2πR 2π y n=−∞ n=−∞ X X Z we can rewrite the infinite sum by the infinite sum of the integrals, where the momentum shift k k m A is allowed. Then, we obtain y → y − A − y0 ∞ N d2k V (A ,f )= f + e−i(mA+Ay0)2πRn eiky 2πRn ln(k2 + f ) . (3.13) eff µ0 0 −λ 0 (2π)2 0 0 n=−∞ XA X Z It is important to note that the sum over A yields

N−1 j N for n =0 mod N , e−imA2πRn = e−2πni/N = (3.14) (0 for n =0 mod N . XA Xj=0 6 ∞ Thus, in the sum n=−∞ in Eq. (3.13), only n such that n = Nm with integer m can contribute. Then, we have P ∞ N d2k V (A ,f )= f + N e−iAy02πRNm eiky 2πRNm ln(k2 + f ) , (3.15) eff µ0 0 −λ 0 (2π)2 0 0 m=−∞ X Z where the m = 0 term is the same contribution as the infinite-volume case, whereas the m = 0 terms are peculiar to the compactified spacetime. However, for m = 0 since we have 6 6 the oscillating factor eipy 2πRNm in the integrand, these integrals vanish in the large N limit where RN .10 Hence, we obtain the same effective potential as the infinite-volume →∞ case [31], N V (A ,f )= V (A ,f )= f log f /Λ2 1 . (3.16) eff µ0 0 eff,∞ µ0 0 −4π 0 0 − In Appendix A we present the explicit result of the m = 0 terms and one can give an explicit 6 proof that this contribution is negligible for large N in a parallel manner to Appendix B of Ref. [18]. (This contribution is exponentially suppressed as e−N .) ∼ In obtaining Eq. (3.16), we apply dimensional regularization to the m = 0 term in Eq. (3.15), where the dimension is set to be 2 d = 2 2ǫ, and accomplish the → −

9 As noted in footnote 10 below, the KK momentum of z fields effectively reduces to n/(NR) as seen from the subsequent calculations. 10 By applying the formula (3.12) to Eq. (3.15), one can see that the discrete momentum effectively reduces to ky = n/(NR) [27]. This is the situation explained in Fig. 1. 8 renormalization of the bare coupling in the MS scheme as ǫ −1 eγE µ2 λ(µ2) 1 λ = λ(µ2) 1+ . (3.17) 0 4π 4π ǫ The structure of the renormalization is not modified from the infinite-volume case, and the theory is indeed asymptotically free: d β µ2 λ(µ2)= 0 λ2(µ2) with β = 1 . (3.18) dµ2 −4π 0 2 The Λ scale used in Eq. (3.16) is defined as Λ2 = µ2e−4π/[β0λ(µ )]. From Eq. (3.16), the saddle point is given by 2 f0 = Λ , (3.19) as in the infinite-volume case. On the other hand, Ay0 is not determined and this moduli parameter should be integrated.11 Based on the same reasoning as above, thanks to the ZN twisted boundary condition and large N, the effective action for the fluctuation of the fields, Aµ = Aµ0 + δAµ and f = f0 + δf, reduces to the same form as the infinite-volume case. We show it to the quadratic order [31]: S [δA ,δf] eff µ |quadratic N dp 1 1 1 = x (p2δ p p ) δA(p)δA (p)δA ( p) δf (p)δf(p)δf( p) , 4π 2π 2πR 2 µν − µ ν L∞ µ ν − − 2L∞ − p Z Xy f f f f (3.22) where we define

dpx 1 dpx 1 δA (x,y)= eipxx+ipyyδA (p) , δf(x,y)= eipxx+ipyyδf(p) , µ 2π 2πR µ 2π 2πR p p Z Xy Z Xy f f(3.23) and 2 2 2 2 2 δA 2 p + 4Λ p + 4Λ + p 4 ∞ (p)= log 2 , (3.24) L p2 p2 p2 + 4Λ2 p2 ! − p p p − p p p 2 2p 2 δf 2 p + 4Λ + p ∞(p)= log . (3.25) L p2(p2 + 4Λ2) p2 + 4Λ2 p2 ! p − p In Appendix A, we presentp the effective actionp including thep finite-volume contributions, which are omitted here. We again note that the finite-volume corrections are shown to be

11 The integration range is determined as follows. Noting that the theory is invariant under g U(1) satisfying the non-trivial boundary condition, ∈ g(x, y +2πR)= e2πi/N g(x, y) , (3.20) the shift of A induced by an element eiy/(RN) U(1), y ∈ A A 1/(RN) , (3.21) y → y − 1 reduces to an equivalent theory. Thus, the integral over 0 d(Ay0RN) should be considered. As long as the quantity to be integrated over this moduli parameter is independent of Ay0, this integral has no apparent eﬀect. R 9 exponentially suppressed e−N in the large N limit in a parallel manner to Appendix B ∼ of Ref. [18].

3.2. Renormalon To calculate the propagators of the auxiliary ﬁelds, we add the gauge-ﬁxing term,

N dp 1 1 S = x p p δA(p)δA (p)δA ( p) , (3.26) gf 4π 2π 2πR 2 µ νL∞ µ ν − p Z Xy f f to the eﬀective action Eq. (3.22). Then, the propagators read

4π 1 δAµ(p)δAν(q) = δµν 2 δA 2πδ(px + qx)2πRδpy +qy ,0 , (3.27) N p ∞ (p) D E L f f 4π 1 δf(p)δf(q) = δf 2πδ(px + qx)2πRδpy +qy ,0 . (3.28) − N ∞(p) D E L f f These are the leading-order results of the two-point functions in 1/N. Since they are obtained from the volume-independent eﬀective action, these results are of course volume independent. It is worth noting, however, that they do not contain renormalons.12 To see this, we consider δA δf 2 2 the expansion of 1/ (p) and 1/ ∞(p) in the high-energy region Λ /p 1 so that the L∞ L ≪ perturbative expansion in the asymptotically free theory works:

1 λ(p2e−2) Λ2 λ(p2e−2) 3λ2(p2e−2) = p2 + + (Λ4/p4) , (3.29) δA(p) 8π − p2 4π 16π2 O L∞ 2 2 2 2 2 1 2 λ(p ) Λ λ(p ) λ (p ) 4 4 δf = p + 2 2 + (Λ /p ) . (3.30) ∞(p) 8π p 4π − 16π O L

2 Since Λ2 = µ2e−4π/[β0λ(µ )] is zero in perturbative evaluation, these quantities are evaluated in perturbation theory (PT) as13

2 −2 2 1 2 λ(p e ) 1 2 λ(p ) δA = p , δf = p . (3.31) ∞ (p) PT 8π ∞(p) 8π L L PT

These results do not contain renormalon divergence; they are truncated at (λ). If one uses O a general renormalization scale µ in accordance with the concept of ﬁxed-order perturbation theory, the inﬁnite sums in λ(µ2) appear through Eq. (2.9) but they can be unambiguously resummed. Renormalons appear when the propagator containing the running coupling, as in Eq. (3.31), is involved in a loop integrand. As simple examples, let us consider the

12 Regarding the gauge field propagator, since it is gauge dependent, this result itself does not have physical meaning. 13 The coefficients of (Λ2/p2)k are regarded as Wilson coefficients, and they are calculated in perturbation theory as they are given by perturbative series in λ. 10 condensates, f(x)f(x) and F (x)F (x) : h i h µν µν i

4π dp 1 1 f(x)f(x) = Λ4 x , (3.32) N 2π 2πR δf h i − ∞(p) 2 2 Z py L p

Since the condensates are UV divergent, we introduce a UV cutoﬀ q to deﬁne them. The perturbative evaluations of these quantities are given by14

2 4π dpx 1 2 λ(p ) f(x)f(x) PT = p , (3.34) h i| − N 2π 2πR 8π p2

where we explicitly show that their integrands should be expanded in λ(µ).15 Now, we indeed encounter Eq. (2.6), assumed in the general argument in Sect. 2; the loop integrands are not modified from the infinite-volume case, but the integration measure is modified to that of the compactified spacetime. We note that since the auxiliary fields Aµ(x) and f(x) satisfy 1 the periodic boundary condition, the momentum along the S direction is given by py = n/R and has nothing to do with the twisted boundary conditions. Following the result in Sect. 2, the renormalon ambiguities are given by

1 1 f(x)f(x) = iπ Λ3 , (3.36) h i|renormalon ∓ N 2πR 2e3 1 F (x)F (x) = iπ Λ3 . (3.37) h µν µν i|renormalon ± N 2πR

As already noted in Sect. 2, the renormalon ambiguities are independent of the UV cutoff. These renormalon ambiguities are peculiar to the compactified spacetime, since they depend on R. To give an example of a UV-finite observable which possesses a renormalon ambiguity, we consider the gradient flow [29, 30]. The flow equation is given by

∂tBµ(t,x)= ∂νGνµ(t,x)+ α0∂µ∂νBν(t,x) , Bµ(t = 0,x)= Aµ(x) , (3.38) where G (t,x)= ∂ B (t,x) ∂ B (t,x) is the field strength of the flowed gauge field; α is µν µ ν − ν µ 0 a constant regarded as a gauge parameter; t is called the flow time, whose mass dimension

14 The perturbative results of the condensates are the same as the loop integrations of the perturbative expressions of the integrands. We also note that the perturbative expression is reliable only in the high-energy region. Using such a result in the low energy-region, which is not justified, is a cause of a renormalon ambiguity. 15 If the integrands are not expanded in λ(µ), the integrals are ill-defined since they contain the poles around p2 Λ2, which are related to the renormalon ambiguities. ∼ 11 is 2. The flowed gauge field is obtained as − Bµ(t,x)

dp 1 ′ p p 2 p p 2 = A + d2x′ x eip(x−x ) δ µ ν e−tp + µ ν e−α0tp δA (x′) . µ0 2π 2πR µν − p2 p2 µ p Z Z Xy (3.39) Using the flowed gauge field, we can construct an observable G (t,x)G (t,x) , h µν µν i 4π dp 1 2 2 G (t,x)G (t,x) = x e−2tp , (3.40) h µν µν i N 2π 2πR δA(p) p ∞ Z Xy L where the Gaussian damping factor makes this quantity UV finite. In perturbation theory, it is given by 2 −2 4π dpx 1 2 λ(p e ) −2tp2 Gµν (t,x)Gµν (t,x) PT = p e . (3.41) h i| N 2π 2πR 4π expansion in λ(µ) Z py X

To analyze the renormalon in this quantity, we introduce a UV cutoff Λ2 q2 t−1 as ≪ ≪ done in Sect. 2. Since the integrand has the same behavior as that of the photon conden- 2 sate (3.35) in the IR region due to e−2tp 1, we have the same renormalon ambiguity as ≃ F (x)F (x) : h µν µν i 2e3 1 G (t,x)G (t,x) = iπ Λ3 . (3.42) h µν µν i|renormalon ± N 2πR As seen from the above reasoning, it is fairly general that the leading renormalon ambiguity of the photon (or gluon) condensate defined by the gradient flow, which is a UV-finite observable, is the same as that of the photon (or gluon) condensate defined with the UV cutoff.

In Eq. (3.42), only the leading renormalon ambiguity of Gµν (t,x)Gµν (t,x) is shown. 2 h i By considering the expansion of e−2tp in tp2 at higher order, we obtain the renormalon ambiguity beyond this order of the form 1 1 1 G (t,x)G (t,x) = iπ c Λ3 + c t Λ5 + c t2 Λ7 + . . . , (3.43) h µν µν i|renormalon ± 0 R 1 R 2 R according to the argument in Sect. 2, where c0, c1, c2, . . . denote the constants. In the Borel u-plane, these renormalon ambiguities correspond to the singularities at u = 3/2, 5/2, 7/2, . . . . These positions are different from the infinite-volume case, where the singularities are located at u = 2, 3, 4, . . . . We finally note that the emergence of the renormalon ambiguities is indeed an artifact of perturbation theory. By seeing Eq. (3.40), which is not evaluated in perturbation theory, one can see that this quantity is unambiguous because any divergence is not found in this expression. It indicates that the renormalon ambiguities found above are cured after nonperturbative effects are properly added.

4. Conclusions In this paper, we have presented a general argument that the renormalon structure is sig- nificantly affected by the circle compactification of the spacetime as Rd Rd−1 S1 with → × small S1 radius RΛ 1. The assumptions of this argument are that (i) a loop integrand of ≪ 12 a renormalon diagram16 is not modified by the compactification, and that (ii) the discrete loop momentum along the S1 direction of the renormalon diagram is not associated with the twisted boundary condition of the system and is given by n/R with integer n. Under these assumptions, we showed that a shift of the renormalon singularity generally occurs due to the circle compactification Rd Rd−1 S1. In particular, the singularity is shifted by 1/2 → × − in the Borel u-plane regardless of the dimension of spacetime d and details of the quantities under consideration. This can be easily understood by the reduction of the dimension of the loop momentum integral, which stems from the fact that only the lowest KK mode can give renormalon singularities. As an example, we studied the CP N−1 model on R S1 with the Z twisted boundary × N conditions in the large N limit. In this model, the above properties (i) and (ii) are indeed realized. The shift by 1/2 in the Borel u-plane was explicitly shown by studying the photon − condensate which is defined by the gradient flow and is a UV-finite observable. As already mentioned, the previous work, Ref. [18], which studied the supersymmetric CP N−1 model on R S1, had provided examples where this mechanism applies in the large N approximation. × Finally, we emphasize that the volume independence of the effective action does not always indicate the volume independence of the renormalon structure, as we observed in the example of the CP N−1 model. In this model, the volume-independent effective action gave the two- point functions or propagators which do not contain renormalon ambiguity. The renormalon ambiguity arises when these propagators (determined from the effective action) are included as loop integrands. Such quantities do not show volume independence any more, and the renormalon structure is not kept intact.

Acknowledgments The authors are grateful to Akira Nakayama for fruitful discussions. They also thank Toshiaki Fujimori, Tatsuhiro Misumi, and Norisuke Sakai for discussions. This work was supported by JSPS Grants-in-Aid for Scientiﬁc Research numbers JP18J20935 (O.M.), JP16H03982 (H.S.), and JP19K14711 (H.T.).

A. Finite volume corrections We present the effective potential and effective action which contain the finite volume corrections. The effective potential is given by

Veff (Aµ0,f0)= Veff,∞(Aµ0,f0)+ Veff,finite(Aµ0,f0) , (A1) with the finite volume correction

N √f0 V (A ,f )= e−iAy02πRNm K ( f 2πRN m ) , (A2) eff,finite µ0 0 − π 2πRN m 1 0 | | m6=0 X | | p where Kν (z) denotes the modified Bessel function of the second kind.

16 Here, we mean by a renormalon diagram that its loop integrand possesses a one-loop running coupling. 13 The effective action including the finite volume effects is given by N dp 1 S = x eff 4π 2π 2πR p Z Xy 1 (p2δ p p ) δA(p)+ δA (p) δA (p)δA ( p) × 2 µν − µ ν L∞ Lfinite µ ν − h i 1 δf (p)+ δf (p) δf(p)δf( p)f f − 2 L∞ Lfinite − h p p i δ µ y mixf(p)fδA (p)δf( p)+ δf(p)δA ( p) (A3) − µy − p2 Lfinite µ − µ − h i with f f f f 1 δA −iAy02πRNm ixpy 2πRNm 2 finite(p)= dx e e (2x 1) L 0 − Z mX6=0 2πRN m 2 2 | | K1( Λ + x(1 x)p 2πRN m ) × Λ2 + x(1 x)p2 − | | − 1 p 2 p 0 dx e−iAy 2πRNmeixpy2πRNm(2x 1) − ipy 0 − Z mX6=0 2πRNmK ( Λ2 + x(1 x)p22πRN m ) , (A4) × 0 − | | 1 p δf −iAy02πRNm ixpy 2πRNm finite(p)= dx e e L 0 Z mX6=0 2πRN m 2 2 | | K1( Λ + x(1 x)p 2πRN m ) , (A5) × Λ2 + x(1 x)p2 − | | − p and p 1 mix (p)= i dx e−iAy0 2πRNmeixpy 2πRNm2πRNmK ( Λ2 + x(1 x)p22πRN m ) . Lfinite 0 − | | 0 m6=0 Z X p (A6)

We note that these results are consistent with the gauge invariance; the tensors before δAµ are transverse. f References [1] G. ’t Hooft, “Can We Make Sense Out of Quantum Chromodynamics?,” Subnucl. Ser. 15 (1979) 943. [2] M. Beneke, “Renormalons,” Phys. Rept. 317 (1999) 1–142, arXiv:hep-ph/9807443 [hep-ph]. [3] E. Brezin, J. C. Le Guillou, and J. Zinn-Justin, “Perturbation Theory at Large Order. 2. Role of the Vacuum Instability,” Phys. Rev. D15 (1977) 1558–1564. [4] L. N. Lipatov, “Divergence of the Perturbation Theory Series and the Quasiclassical Theory,” Sov. Phys. JETP 45 (1977) 216–223. [Zh. Eksp. Teor. Fiz.72,411(1977)]. [5] E. B. Bogomolny, “CALCULATION OF INSTANTON - ANTI-INSTANTON CONTRIBUTIONS IN QUANTUM MECHANICS,” Phys. Lett. 91B (1980) 431–435. [6] J. Zinn-Justin, “Multi - Instanton Contributions in Quantum Mechanics,” Nucl. Phys. B192 (1981) 125–140. [7] F. David, “Nonperturbative Effects and Infrared Renormalons Within the 1/N Expansion of the O(N) Nonlinear σ Model,” Nucl. Phys. B209 (1982) 433–460. [8] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Two-Dimensional Sigma Models: Modeling Nonperturbative Effects of Quantum Chromodynamics,” Phys. Rept. 116 (1984) 103. [Fiz. Elem. Chast. Atom. Yadra17,472(1986)]. 14 [9] F. David, “On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion,” Nucl. Phys. B234 (1984) 237–251. [10] V. A. Novikov, M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Wilson’s Operator Expansion: Can It Fail?,” Nucl. Phys. B249 (1985) 445–471. [Yad. Fiz.41,1063(1985)]. [11] M. Beneke, V. M. Braun, and N. Kivel, “The Operator product expansion, nonperturbative couplings and the Landau pole: Lessons from the O(N) sigma model,” Phys. Lett. B443 (1998) 308–316, arXiv:hep-ph/9809287 [hep-ph]. [12] G. V. Dunne and M. Unsal,¨ “What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles,” PoS LATTICE2015 (2016) 010, arXiv:1511.05977 [hep-lat]. [13] P. Argyres and M. Unsal,¨ “A semiclassical realization of infrared renormalons,” Phys. Rev. Lett. 109 (2012) 121601, arXiv:1204.1661 [hep-th]. [14] P. C. Argyres and M. Unsal,¨ “The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects,” JHEP 08 (2012) 063, arXiv:1206.1890 [hep-th]. [15] G. V. Dunne and M. Unsal,¨ “Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model,” JHEP 11 (2012) 170, arXiv:1210.2423 [hep-th]. [16] G. V. Dunne and M. Unsal,¨ “Continuity and Resurgence: towards a continuum definition of the CP(N-1) model,” Phys. Rev. D87 (2013) 025015, arXiv:1210.3646 [hep-th]. [17] T. Fujimori, S. Kamata, T. Misumi, M. Nitta, and N. Sakai, “Bion non-perturbative contributions − versus infrared renormalons in two-dimensional CP N 1 models,” JHEP 02 (2019) 190, arXiv:1810.03768 [hep-th]. [18] K. Ishikawa, O. Morikawa, A. Nakayama, K. Shibata, H. Suzuki, and H. Takaura, “Infrared − renormalon in the supersymmetric CP N 1 model on R × S1,” arXiv:1908.00373 [hep-th]. [19] M. M. Anber and T. Sulejmanpasic, “The renormalon diagram in gauge theories on R3 × S1,” JHEP 01 (2015) 139, arXiv:1410.0121 [hep-th]. [20] M. Ashie, O. Morikawa, H. Suzuki, H. Takaura, and K. Takeuchi, “Infrared renormalon in SU(N) QCD(adj.) on R3 × S1,” arXiv:1909.05489 [hep-th]. [21] T. Eguchi and H. Kawai, “Reduction of Dynamical Degrees of Freedom in the Large N Gauge Theory,” Phys. Rev. Lett. 48 (1982) 1063. [22] P. Kovtun, M. Unsal,¨ and L. G. Yaffe, “Volume independence in large N(c) QCD-like gauge theories,” JHEP 06 (2007) 019, arXiv:hep-th/0702021 [HEP-TH]. [23] M. Unsal¨ and L. G. Yaffe, “Center-stabilized Yang-Mills theory: Confinement and large N volume independence,” Phys. Rev. D78 (2008) 065035, arXiv:0803.0344 [hep-th]. [24] E. Poppitz and M. Unsal,¨ “Comments on large-N volume independence,” JHEP 01 (2010) 098, arXiv:0911.0358 [hep-th]. [25] M. Unsal¨ and L. G. Yaffe, “Large-N volume independence in conformal and confining gauge theories,” JHEP 08 (2010) 030, arXiv:1006.2101 [hep-th]. [26] A. Gonzalez-Arroyo and M. Okawa, “Large N reduction with the Twisted Eguchi-Kawai model,” JHEP 07 (2010) 043, arXiv:1005.1981 [hep-th]. [27] T. Sulejmanpasic, “Global Symmetries, Volume Independence, and Continuity in Quantum Field Theories,” Phys. Rev. Lett. 118 no. 1, (2017) 011601, arXiv:1610.04009 [hep-th]. [28] D. J. Gross and Y. Kitazawa, “A Quenched Momentum Prescription for Large N Theories,” Nucl. Phys. B206 (1982) 440–472. [29] M. Lüscher, “Properties and uses of the Wilson flow in lattice QCD,” JHEP 08 (2010) 071, arXiv:1006.4518 [hep-lat]. [Erratum: JHEP03,092(2014)]. [30] M. Lüscher and P. Weisz, “Perturbative analysis of the gradient flow in non-abelian gauge theories,” JHEP 02 (2011) 051, arXiv:1101.0963 [hep-th]. [31] A. D’Adda, M. Lüscher, and P. Di Vecchia, “A 1/n Expandable Series of Nonlinear Sigma Models with Instantons,” Nucl. Phys. B146 (1978) 63–76. [32] D. J. Broadhurst and A. L. Kataev, “Connections Between Deep Inelastic and Annihilation Processes at Next-to-Next-to-Leading Order and Beyond,” Phys. Lett. B315 (1993) 179–187, arXiv:hep-ph/9308274 [hep-ph]. N [33] P. Ball, M. Beneke, and V. M. Braun, “Resummation of (β0αs) Corrections in QCD: Techniques and Applications to the Tau Hadronic Width and the Heavy Quark Pole Mass,” Nucl. Phys. B452 (1995) 563–625, arXiv:hep-ph/9502300 [hep-ph]. [34] M. Beneke and V. M. Braun, “Naive Nonabelianization and Resummation of Fermion Bubble Chains,” Phys. Lett. B348 (1995) 513–520, arXiv:hep-ph/9411229 [hep-ph]. [35] S. Coleman, Aspects of Symmetry. Cambridge University Press, Cambridge, U.K., 1985.