Renormalon structure in compactified spacetime

We point out that the location of renormalon singularities in theory on a circle-compactified spacetime $\mathbb{R}^{d-1} \times S^1$ (with a small radius $R \Lambda \ll 1$) can differ from that on the non-compactified spacetime $\mathbb{R}^d$. We argue this under the following assumptions, which are often realized in large $N$ theories with twisted boundary conditions: (i) a loop integrand of a renormalon diagram is volume independent, i.e. it is not modified by the compactification, and (ii) the loop momentum variable along the $S^1$ direction is not associated with the twisted boundary conditions and takes the values $n/R$ with integer $n$. We find that the Borel singularity is generally shifted by $-1/2$ in the Borel $u$-plane, where the renormalon ambiguity of $\mathcal{O}(\Lambda^k)$ is changed to $\mathcal{O}(\Lambda^{k-1}/R)$ due to the circle compactification $\mathbb{R}^d \to \mathbb{R}^{d-1} \times S^1$. The result is general for any dimension $d$ and is independent of details of the quantities under consideration. As an example, we study the $\mathbb{C} P^{N-1}$ model on $\mathbb{R} \times S^1$ with $\mathbb{Z}_N$ twisted boundary conditions in the large $N$ limit.


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 ∼ β n 0 n!, and is related to the beta function of the theory, where β 0 is the oneloop 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 O(e −2SI ) in terms of the one-instanton action S I = (4π) d/2 N/(2λ) in d-dimensional spacetime, and the second is O(e −2SI /(N β0) ), called the renormalon ambiguity, where λ denotes the 't Hooft coupling (defined as λ = g 2 N 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][4][5][6].Here, the semiclassical configuration plays an important role, where the two-instanton action 2S I 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][8][9][10][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 N β 0 , another configuration is needed.A recent idea is to find such a configuration by the S 1 compactification of the spacetime as R d → R d−1 × S 1 (see Ref. [12] and references therein).In order for the semiclassical calculation to be valid, the S 1 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 R d−1 × S 1 first, and then smooth connection to the theory on R d is assumed.In Refs.[13][14][15][16][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.In Ref. [18], the renormalon ambiguity in the supersymmetric CP N −1 model on the circlecompactified spacetime R × S 1 with Z N twisted boundary conditions was studied. 1 In a 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 R 2 .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][22][23][24][25][26][27][28].We also assume that the loop momentum variable of the renormalon diagram along the S 1 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 R d−1 × S 1 , 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 × S 1 with the Z N twisted boundary conditions in the large N limit, 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, we define its Borel transform as and correspondingly the Borel integral is given by In our definition, a pole singularity of the Borel transform at u = u 0 > 0 gives an ambiguity in the Borel integral of order e −(4π) d/2 u0/(β0λ) = e −2SI u0/(N β0) .Thus, our definition is convenient to grasp the ambiguity in terms of e −SI ; it is enough to focus on the value u 0 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 R 3 × S 1 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-β 0 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.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 × S 1 with the Z N twisted boundary 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.

Renormalon structure in compactified spacetime
In asymptotically free theory on the non-compactified spacetime R d , a typical form from which a renormalon ambiguity appears is given by where d d p 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 0 approximation [32][33][34], and the large N approximation [2,[7][8][9][10][11]. Here, λ denotes the running coupling which satisfies the renormalization group equation whose solution is given by with a renormalization group invariant (dynamical) mass scale When the asymptotic form of F (p) in the infrared (IR) region is given by3 the Borel singularity arises at from perturbative expansion of Eq. (2.1), which gives the renormalon ambiguity of O(Λ 2α+d ). 4uppose that, in asymptotically free theory on the compactified spacetime as a perturbative contribution.Here p d denotes the KK momentum along S 1 and is given by p d = n/R, whereas p denotes the (continuous) momentum on R d−1 .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 discrete loop momentum along the S 1 direction to be p d = n/R with integer n. 5 In the next section, as an example where we have Eq.(2.6), we study the CP N −1 model on R × S 1 with the Z N twisted boundary conditions.It will be shown that the large N limit and the Z N 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 p 2 < q 2 .We take the UV cutoff as Λ ≪ q ≪ R −1 .Then, due to p 2 = p 2 + (n/R) 2 , only the n = 0 term and the range 0 < p 2 < q 2 have to be considered: (2.7) We study the Borel transform [defined in Eq. (1. 2)] corresponding to Eq. (2.7), which is obtained as6 Here, we note that the perturbative expansion of Eq. (2.7) in terms of λ(µ 2 ) is obtained through (2.9) The Borel transform (2.8) is easily evaluated as This possesses a simple pole at 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 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 < q instead of Λ ≪ q ≪ R −1 . 8In this case, we obtain the Borel transform The first line is the same as Eq.(2.8).For the second line, since p 2 = p 2 + (n/R) 2 always has a non-zero positive value larger than 1/R 2 ≫ Λ 2 , the integrals never become singular for any u.Thus, we do not have additional singularities.
As we observed, p d = n/R with |n| ≥ 1 does not give any renormalon singularities.This is because the compactification radius 1/R ≫ Λ plays the role of an IR cutoff for this sector.Hence, only the lowest KK mode (with n = 0) can give the renormalon singularities and should be focused, where we have Eq.(2.7).This is nothing but Eq. (2.1) with the replacement d → d − 1 (apart from the overall factor 1/(2πR)).This replacement is the origin of the shift.Thus, the shift is simply understood as the reduction of the dimension of momentum integration-cf.Eqs.(2.5) and (2.11).

boundary conditions
As an example where the mechanism in Sect. 2 applies, we consider the CP N −1 model on R × S 1 with the Z N twisted boundary conditions.The action of this model in terms of the homogeneous coordinate z A (A = 1, . . ., N ) obeying the constraint zA z A = 1 is defined by with the current j µ , The topological term is given by where ǫ xy = −ǫ yx = +1.Here and hereafter, summation over the repeated indices is always understood.It is convenient to adopt the following action with auxiliary fields to carry out the large N expansion [35]: where D µ = ∂ µ + iA µ .To respect the U (1) gauge symmetry of the model, A µ behaves as a gauge field under the transformation z A → gz A with g ∈ U (1): We impose the following Z N twisted boundary conditions along the S 1 direction for z A : where (x, y) ∈ R × S 1 and m N = 0 .
The auxiliary fields, A µ 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.

Volume independence of the effective action
As pointed out in Ref. [27], the effective action for the auxiliary fields S eff [A µ , f ] exhibits volume independence due to the Z N twisted boundary conditions and the large N .We illustrate this point, aiming for a self-contained explanation.After integrating out z A , the effective action is obtained as where the topological term should be treated separately.We first calculate the effective potential, which is obtained as S eff = V 2 • V eff (A µ0 , f 0 ); the fields with subscript 0 denote the constant values at the saddle point; V 2 represents the volume of two-dimensional spacetime.
V eff is explicitly given by (3.10) Here, the KK momentum k y is discrete: By using the formula we can rewrite the infinite sum by the infinite sum of the integrals, where the momentum shift k y → k y − m A − A y0 is allowed.Then, we obtain It is important to note that the sum over A yields 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 the oscillating factor e ipy2πRN m 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],

.16)
In Appendix A we present the explicit result of the m = 0 terms and one can give an explicit 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 renormalization of the bare coupling in the MS scheme as The structure of the renormalization is not modified from the infinite-volume case, and the theory is indeed asymptotically free: The Λ scale used in Eq. (3.16) is defined as Λ 2 = µ 2 e −4π/[β0λ(µ 2 )] .From Eq. (3.16), the saddle point is given by as in the infinite-volume case.On the other hand, A y0 is not determined and this moduli parameter should be integrated. 11 Based on the same reasoning as above, thanks to the Z N twisted boundary condition and large N , the effective action for the fluctuation of the fields, A µ = A µ0 + δA µ and f = f 0 + δf , reduces to the same form as the infinite-volume case.We show it to the quadratic order [31]: where we define and In Appendix A, we present the effective action including the 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, the shift of A y induced by an element e iy/(RN ) ∈ U (1), reduces to an equivalent theory.Thus, the integral over 1 0 d(A y0 RN ) should be considered.As long as the quantity to be integrated over this moduli parameter is independent of A y0 , this integral has no apparent effect.exponentially suppressed ∼ e −N in the large N limit in a parallel manner to Appendix B of Ref. [18].

Renormalon
To calculate the propagators of the auxiliary fields, we add the gauge-fixing term, to the effective action Eq.(3.22).Then, the propagators read 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. 12To see this, we consider the expansion of 1/L δA ∞ (p) and 1/L δf ∞ (p) in the high-energy region Λ 2 /p 2 ≪ 1 so that the perturbative expansion in the asymptotically free theory works: 1 Since Λ 2 = µ 2 e −4π/[β0λ(µ 2 )] is zero in perturbative evaluation, these quantities are evaluated in perturbation theory (PT) as 131 These results do not contain renormalon divergence; they are truncated at O(λ).If one uses a general renormalization scale µ in accordance with the concept of fixed-order perturbation theory, the infinite sums in λ(µ 2 ) 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 condensates, f (x)f (x) and F µν (x)F µν (x) : (3.33) Since the condensates are UV divergent, we introduce a UV cutoff q to define them.The perturbative evaluations of these quantities are given by14 where we explicitly show that their integrands should be expanded in λ(µ). 15Now, 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 A µ (x) and f (x) satisfy the periodic boundary condition, the momentum along the S 1 direction is given by p y = n/R and has nothing to do with the twisted boundary conditions.Following the result in Sect.2, the renormalon ambiguities are given by 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 where is the field strength of the flowed gauge field; α 0 is a constant regarded as a gauge parameter; t is called the flow time, whose mass dimension is −2.The flowed gauge field is obtained as (3.39) Using the flowed gauge field, we can construct an observable G µν (t, x)G µν (t, x) , where the Gaussian damping factor makes this quantity UV finite.In perturbation theory, it is given by

.41)
To analyze the renormalon in this quantity, we introduce a UV cutoff Λ 2 ≪ q 2 ≪ t −1 as done in Sect. 2. Since the integrand has the same behavior as that of the photon condensate (3.35) in the IR region due to e −2tp 2 ≃ 1, we have the same renormalon ambiguity as 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 G µν (t, x)G µν (t, x) is shown.By considering the expansion of e −2tp 2 in tp 2 at higher order, we obtain the renormalon ambiguity beyond this order of the form according to the argument in Sect.2, where c 0 , c 1 , c 2 , . . .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.

Conclusions
In this paper, we have presented a general argument that the renormalon structure is significantly affected by the circle compactification of the spacetime as R d → R d−1 × S 1 with small S 1 radius RΛ ≪ 1.The assumptions of this argument are that (i) a loop integrand of a renormalon diagram 16 is not modified by the compactification, and that (ii) the discrete loop momentum along the S 1 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 R d → R d−1 × S 1 .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 × S 1 with the Z N twisted boundary 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 × S 1 , 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 twopoint 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.

Fig. 1
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 p d = n/R), while the field making bubbles (solid line) satisfies the twisted boundary conditions (effectively corresponding to k d = n/(N R)) 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).