Vacuum energy of the supersymmetric C P N−1 model on R× S 1 in the 1/N expansion
Kosuke Ishikawa, Okuto Morikawa, Kazuya Shibata, Hiroshi Suzuki
aa r X i v : . [ h e p - t h ] M a y Preprint number: KYUSHU-HET-206
Vacuum energy of the supersymmetric C P N − model on R × S in the /N expansion Kosuke Ishikawa , Okuto Morikawa , Kazuya Shibata , and Hiroshi Suzuki , ∗ Department of Physics, Kyushu University 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan ∗ E-mail: [email protected] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
By employing the 1 /N expansion, we compute the vacuum energy E ( δǫ ) of thetwo-dimensional supersymmetric (SUSY) C P N − model on R × S with Z N twistedboundary conditions to the second order in a SUSY-breaking parameter δǫ . This quantitywas vigorously studied recently by Fujimori et al. using a semi-classical approximationbased on the bion, motivated by a possible semi-classical picture on the infrared renor-malon. In our calculation, we find that the parameter δǫ receives renormalization and,after this renormalization, the vacuum energy becomes ultraviolet finite. To the next-to-leading order of the 1 /N expansion, we find that the vacuum energy normalized bythe radius of the S , R , RE ( δǫ ) behaves as inverse powers of Λ R for Λ R small, whereΛ is the dynamical scale. Since Λ is related to the renormalized ’t Hooft coupling λ R as Λ ∼ e − π/λ R , to the order of the 1 /N expansion we work out, the vacuum energyis a purely non-perturbative quantity and has no well-defined weak coupling expansionin λ R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index B06, B16, B32, B34, B35 typeset using PTP
TEX.cls . Introduction
In this paper, by employing the 1 /N expansion (for a classical exposition, see Ref. [1]), wecompute the vacuum energy E ( δǫ ) of the two-dimensional (2D) supersymmetric (SUSY) C P N − model [2–4] on R × S with Z N twisted boundary conditions to the second order ina SUSY-breaking parameter δǫ . This quantity was vigorously studied recently by Fujimoriet al. [5] (see also Refs. [6–8]) using a semi-classical approximation based on the bion [9–14]. One of the motivations for their study was a possible semi-classical picture on theinfrared (IR) renormalon [15, 16] advocated in Refs. [17–20]. In these works, in the contextof the resurgence program (for a review, see Ref. [21] and the references cited therein), it isproposed that the ambiguity caused by the IR renormalon through the Borel resummation(for a review, see Ref. [22]) be cancelled by the ambiguity associated with the integration ofquasi-collective coordinates of the bion; this scenario is quite analogous to the Bogomolny–Zinn-Justin mechanism for the instanton–anti-instanton pair [23, 24].In Ref. [5], by using the Lefschetz thimble method [25–27], the integration over quasi-collective coordinates of the bion is explicitly carried out and it was found that the vacuumenergy E ( δǫ ) possesses the imaginary ambiguity which is of the same order as that caused bythe so-called u = 1 IR renormalon. On the other hand, for the four-dimensional SU ( N ) gaugetheory with the adjoint fermion (4D QCD(adj.)), for N = 2 and 3, it has been found [28]that when the spacetime is compactified as R × S , the logarithmic behavior of the vacuumpolarization of the gauge boson associated with the Cartan subalgebra (“photon”) disap-pears. Since the IR renormalon is attributed to such a logarithmic behavior, in Ref. [28] itis concluded that the circle compactification generally eliminates the IR renormalon. Thisappears inconsistent with the renormalon interpretation of the result in Ref. [5].The original motivation in a series of works [29–31] by a group including the presentauthors was to investigate the fate of the IR renormalon under the circle compactificationto understand the above inconsistency. For this, we employed the 1 /N expansion (i.e. thelarge- N limit), in which Λ R = const. as N → ∞ , (1.1)where Λ is a dynamical scale and R is the S radius. We expected that in this way theIR renormalon and the bion can be highlighted, because the beta function of the ’t Hooftcoupling and the bion action remain non-trivial in the large- N limit, Eq. (1.1), whereas othersources to the Borel singularity such as the instanton–anti-instanton pair are suppressed.This intention was not so successful, because the calculations in Refs. [29–31] show thatthe behavior of the IR renormalon rather depends on the system; in the 2D SUSY C P N − model, the compactification from R to R × S shifts the location of the Borel singularityassociated with the IR renormalon [29, 31]. In the 4D QCD(adj.), because of the twistedmomentum of the gauge boson associated with the root vectors (“W boson”), R × S iseffectively decompactified in the large- N limit [35–37] and the IR renormalon gives rise tothe same Borel singularity as the uncompactified R [30]. It appears that a unified pictureon the semi-classical understanding of the IR renormalon is still missing. Recent related works are Refs. [32–34]. In this analysis, we relied on the so-called large- β approximation [38–40].
2n the present paper, as announced in Ref. [29], in the 1 /N expansion with Eq. (1.1),we compute the vacuum energy E ( δE ) of the 2D SUSY C P N − model on R × S with Z N twisted boundary conditions to the second order in a SUSY-breaking parameter δǫ ; this isthe quantity computed in Ref. [5] by the bion calculus. First, we find that the parameter δǫ receives renormalization and, after this renormalization, the vacuum energy becomes ultravi-olet (UV) finite. To the next-to-leading order of the 1 /N expansion, we find that the vacuumenergy is IR finite, as should be the case for a physical quantity. Finally, we find that thevacuum energy normalized by the radius of the S , RE ( δǫ ) behaves as inverse powers of Λ R for Λ R small, as shown in Eqs. (3.52)–(3.57) and Figs. 2 and 3. Since Λ is related to therenormalized ’t Hooft coupling λ R as Λ ∼ e − π/λ R , to the order of the 1 /N expansion wework out, the vacuum energy is a purely non-perturbative quantity and has no well-definedweak coupling expansion in λ R . This implies that one cannot even define the perturbativeexpansion for this quantity computed in the 1 /N expansion and cannot even discuss therenormalon problem. Therefore, although our 1 /N calculation is robust, it does not giveany clue to the issue. We do not yet fully understand why the semi-classical calculation onthe basis of the bion cannot be observed in the 1 /N expansion. Nevertheless, we believethat it is worthwhile to report our 1 /N calculation for future consideration because ourcalculation itself is rather non-trivial.
2. Two-dimensional SUSY C P N − model Our spacetime is R × S , and −∞ < x < ∞ denotes the coordinate of R and 0 ≤ y < πR the coordinate of S . The Euclidean action of the 2D SUSY C P N − model in terms of thehomogeneous coordinate variables [2–4] is, in the notation of Eq. (2.24) of Ref. [29], S = Z d x Nλ (cid:2) − f + ¯ σσ + ¯ z A ( − D µ D µ + f ) z A + ¯ χ A ( / D + ¯ σP + + σP − ) χ A + 2 ¯ χ A z A η + 2¯ η ¯ z A χ A (cid:3) − Z d x iθ π ǫ µν ∂ µ A ν . (2.1)Here, and in what follows, it is understood that repeated indices are summed over; the lowerGreek indices, µ , ν , . . . , take the value x or y and the uppercase Roman indices, A , B , . . . ,run from 1 to N . λ is the bare ’t Hooft coupling and θ is the theta parameter. Also, D µ z A ≡ ( ∂ µ + iA µ ) z A , / Dχ A ≡ γ µ ( ∂ µ + iA µ ) χ A ,P ± ≡ ± γ , γ ≡ − iγ x γ y , γ x ≡ ! , γ y ≡ − ii ! , (2.2)and ǫ xy = − ǫ yx = +1. In Appendix A, by taking a particular limit R → ∞ , we illustrate that the perturbative part ofthe vacuum energy contains IR divergences, although when including the non-perturbative part itbecomes IR finite. The theta parameter θ may be eliminated by the anomalous chiral rotation χ A → e iαγ χ A , ¯ χ A → ¯ χ A e iαγ , η → e − iαγ η , ¯ η → ¯ ηe − iαγ , and σ → e iα σ . A (we call them N -fields), we impose the Z N twisted boundaryconditions along S : z A ( x, y + 2 πR ) = e πim A R z A ( x, y ) ,χ A ( x, y + 2 πR ) = e πim A R χ A ( x, y ) , ¯ χ A ( x, y + 2 πR ) = e − πim A R ¯ χ A ( x, y ) , (2.3)where the twist angle m A in these expressions depends on the index A as m A ≡ AN R for A = 1, . . . , N − , m N ≡ . (2.4)These twisted boundary conditions allow the fractional instanton/anti-instanton, theconstituent of the bion.For the auxiliary fields, f , σ , ¯ σ , A µ , η , and ¯ η , on the other hand, we assume periodicboundary conditions along S .For the calculation below, however, it turns out that an alternative form of the action,obtained by f → f + ¯ σσ (2.5)from Eq. (2.1), that is, S = Z d x Nλ (cid:2) − f + ¯ z A ( − D µ D µ + f + ¯ σσ ) z A + ¯ χ A ( / D + ¯ σP + + σP − ) χ A + 2 ¯ χ A z A η + 2¯ η ¯ z A χ A (cid:3) − Z d x iθ π ǫ µν ∂ µ A ν (2.6)is more convenient. This is because renormalization with the action in Eq. (2.1) requires aninfinite shift of the field f in addition to the multiplicative renormalization of the ’t Hooftcoupling (Eq. (2.10) below), whereas the action in Eq. (2.6) does not require such a shift.This difference comes from the fact that ¯ σσ in Eq. (2.5) is a composite operator and UVdivergent. In fact, the action in Eq. (2.6) can be obtained by the dimensional reduction of amanifestly SUSY-invariant non-linear sigma model in four dimensions [41]; we thus expecta simpler UV-divergent structure. For this reason, we adopt the action in Eq. (2.6) in thepresent paper. /N expansion Now, since the action of Eq. (2.1) (i.e. Eq. (2.24) of Ref. [29]) and the action of Eq. (2.6) aresimply related by the change of variable in Eq. (2.5), we can borrow the results in Ref. [29]in the leading order of the 1 /N expansion. With the twisted boundary conditions of Eq. (2.3), as we will note in Sect. 3.1, the effective actionarising from the Gaussian integration over N -fields is not simply proportional to N but dependsnontrivially on N . Such a non-trivial dependence on N in the Gaussian determinant is, however,exponentially suppressed in the large- N limit of Eq. (1.1) [29] and can be neglected in calculationsin the 1 /N expansion. A µ = A µ + δA µ , f = f + δf, σ = σ + δσ, (2.7)where the subscript 0 indicates the value at the saddle point in the 1 /N expansion and δ denotes the fluctuation, in the leading order of the 1 /N expansion in Eq. (1.1) we have A µ = A y δ µy , f = 0 , ¯ σ σ = Λ , (2.8)where Λ is the dynamical scale Λ = µe − π/λ R (2.9)defined from the renormalized ’t Hooft coupling λ R in the “MS scheme,” λ = (cid:18) e γ E µ π (cid:19) ε λ R (cid:18) λ R π ε (cid:19) − . (2.10)Here, we have used dimensional regularization with the complex dimension D = 2 − ε ; µ isthe renormalization scale. In Eq. (2.8), the constant A y is not determined from the saddlepoint condition in the present supersymmetric theory and, for Z N -invariant quantities suchas the partition function and the vacuum energy considered below, it should be integratedover with the measure [29] Z d ( A y RN ) . (2.11)Next, we need the propagators among fluctuations of the auxiliary fields. To obtain these,we add the gauge-fixing term S gf = N π Z d x d x ′ ∂ µ δA µ ( x ) ∂ ν δA ν ( x ′ ) Z dp x π πR X p y e − ip ( x − x ′ ) L ( p ) (2.12)and a local counter term S local ≡ N π Z d x (cid:18) − (cid:19) [ δσ ( x ) − δ ¯ σ ( x )] (2.13)to the action in Eq. (2.6) [29]. Then, in the leading order of the 1 /N expansion, we have (cid:10) δA µ ( x ) δA ν ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) L ( p ) D ( p ) ( δ µν + 4 " Λ + ¯ p y p K ( p ) L ( p ) p µ p ν ( p ) ) , (cid:10) δA µ ( x ) δR ( x ′ ) (cid:11) = (cid:10) δR ( x ) δA µ ( x ′ ) (cid:11) = 0 , (cid:10) δA µ ( x ) δI ( x ′ ) (cid:11) = − (cid:10) δI ( x ) δA µ ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) L ( p ) D ( p ) 2Λ ¯ p µ p , (cid:10) δA µ ( x ) δf ( x ′ ) (cid:11) = (cid:10) δf ( x ) δA µ ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) K ( p ) D ( p ) − p µ ¯ p y p , (cid:10) δR ( x ) δR ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) L ( p ) D ( p ) Λ , (cid:10) δR ( x ) δI ( x ′ ) (cid:11) = − (cid:10) δI ( x ) δR ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) K ( p ) D ( p ) − ¯ p y p , δR ( x ) δf ( x ′ ) (cid:11) = (cid:10) δf ( x ) δR ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) L ( p ) D ( p ) ( − ) , (cid:10) δI ( x ) δI ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) L ( p ) D ( p ) Λ , (cid:10) δI ( x ) δf ( x ′ ) (cid:11) = − (cid:10) δf ( x ) δI ( x ′ ) (cid:11) = 0 , (cid:10) δf ( x ) δf ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) L ( p ) D ( p ) ( − p ) , (cid:10) η ( x )¯ η ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) ( i / p + 2¯ σ P + + 2 σ P − ) L ( p ) + 2 i /¯ p ¯ p y /p K ( p ) D ( p ) (cid:18) − (cid:19) , (2.14)where the Kaluza–Klein (KK) momentum along S , p y , takes discrete values p y = n/R with n ∈ Z . We have also introduced the notations¯ p µ ≡ ǫ νµ p ν (2.15)and δR ( x ) ≡
12 [¯ σ δσ ( x ) + σ δ ¯ σ ( x )] , δI ( x ) ≡ i [¯ σ δσ ( x ) − σ δ ¯ σ ( x )] . (2.16)From the above results, we also have (cid:10) δσ ( x ) δ ¯ σ ( x ′ ) (cid:11) = 4 πN Z dp x π πR X p y e ip ( x − x ′ ) D ( p ) (cid:20) L ( p ) + 4 i ¯ p y p K ( p ) (cid:21) . (2.17)Various functions used in the above expressions are defined by L ( p ) ≡ L ∞ ( p ) + ˆ L ( p ) , L ∞ ( p ) ≡ p p ( p + 4Λ ) ln p p + 4Λ + p p p p + 4Λ − p p ! , ˆ L ( p ) ≡ Z dx X m =0 e − iA y πRNm e ixp y πRNm × πRN | m | p Λ + x (1 − x ) p K ( p Λ + x (1 − x ) p πRN | m | ) , K ( p ) ≡ i Z dx X m =0 e − iA y πRNm e ixp y πRNm πRN mK ( p Λ + x (1 − x ) p πRN | m | ) , D ( p ) ≡ ( p + 4Λ ) L ( p ) + 4 ¯ p y p K ( p ) , (2.18)where K ν ( z ) denotes the modified Bessel function of the second kind. For later calculations,it is important to note the properties L ( p ) = L ( − p ) , K ( p ) = K ( − p ) . (2.19)These can be shown by the change of the Feynman parameter, x → − x , noting that p y ∈ Z /R . 6oing back to the action in Eq. S (2.6), with the saddle point values in Eq. (2.8), thepropagators of the N -fields in the leading order of the 1 /N expansion are given by (cid:10) z A ( x )¯ z B ( x ′ ) (cid:11) = δ AB λN Z dp x π πR X p y e ip x ( x − x ′ ) e i ( p y + m A )( y − y ′ ) × (cid:2) p x + ( p y + A y + m A ) + Λ (cid:3) − , (cid:10) χ A ( x ) ¯ χ B ( x ′ ) (cid:11) = δ AB λN Z dp x π πR X p y e ip x ( x − x ′ ) e i ( p y + m A )( y − y ′ ) × [ iγ x p x + iγ y ( p y + A y + m A ) + ¯ σ P + + σ P − ] − . (2.20)To obtain these, we noted the twisted boundary conditions of Eq. (2.3).
3. Computation of the vacuum energy
Our objective in this paper is to compute the vacuum energy of the present system as a powerseries of the coefficient δǫ of a SUSY-breaking term—the quantity computed in Ref. [5]: E ( δǫ ) = E (0) + E (1) δǫ + E (2) δǫ + · · · . (3.1)Here, the supersymmetry breaking term introduced in Ref. [5] is δS ≡ Z d x δǫπR N X A =1 m A (cid:18) ¯ z A z A − N (cid:19) . (3.2)Note that this depends on the twist angles in Eq. (2.4). A quick way to incorporate the effectof Eq. (3.2) is to regard δS as a mass term of the z A -field, as S + δS = Z d x Nλ ¯ z A (cid:0) − ∂ µ ∂ µ + Λ + δ A (cid:1) z A + · · · , (3.3)where δ A ≡ λδǫπRN m A . (3.4)With this modification, the vacuum energy is given by − Z dx E ( δǫ ) = Z d x λ X A δ A − X A ln Det( − ∂ µ ∂ µ + Λ + δ A )+ (connected vacuum bubble diagrams) . (3.5)Here, the vacuum bubble diagrams, which start from two-loop order, are computed by usingthe modified z A -propagator (cid:10) z A ( x )¯ z B ( x ′ ) (cid:11) = δ AB λN Z dp x π πR X p y e ip x ( x − x ′ ) e i ( p y + m A )( y − y ′ ) (cid:2) p x + ( p y + A y + m A ) + Λ + δ A (cid:3) − (3.6)instead of the one in Eq. (2.20). Then, by expanding Eq. (3.5) with respect to δ A , we havethe series expansion in Eq. (3.1). In the following calculations, we set E (0) = 0 assuming that7he bare vacuum energy at δǫ = 0 is chosen so that the system is supersymmetric for δǫ = 0.This amounts to computing the difference E ( δǫ ) − E ( δǫ = 0).If all the N -fields obey the same boundary conditions along S , all z A (or χ A and ¯ χ A )contribute equally and the order of the loop expansion with the use of the auxiliary fieldsand the order of the 1 /N expansion would coincide [1]. With the twisted boundary condi-tions in Eq. (2.3), however, not all N -fields contribute equally. The SUSY-breaking termin Eq. (3.2) also treats each of N -fields differently. For these reasons, in the present systemthe order of the loop expansion and that of the 1 /N expansion do not necessarily coin-cide; we have to distinguish both expansions. For instance, although the one-loop Gaussiandeterminant in Eq. (3.5) gives rise to the contribution of O (1 /N ), it also contains terms ofsubleading orders, O (1 /N ) and O (1 /N ) (see Eq. (3.49), for instance). Let us start with the one-loop Gaussian determinant in Eq. (3.5). We first note that − X A ln Det( − ∂ µ ∂ µ + Λ + δ A )= − X A Z d x Z dp x π πR X p y ln (cid:2) p x + ( p y + m A + A y ) + Λ + δ A (cid:3) = − Z d x X A ∞ X n = −∞ Z d p (2 π ) e i ( p y − m A − A y )2 πRn ln( p + Λ + δ A ) , (3.7)where we have used the identity12 πR ∞ X n = −∞ F ( n/R ) = ∞ X n = −∞ Z dp y π e ip y πRn F ( p y ) . (3.8)Hence, subtracting the logarithm of the Gaussian determinant at δǫ = 0, we have − X A ln Det (cid:18) − ∂ µ ∂ µ + Λ + δ A − ∂ µ ∂ µ + Λ (cid:19) = − Z d x X A ∞ X n = −∞ Z d p (2 π ) e i ( p y − m A − A y )2 πRn ln (cid:18) p + Λ + δ A p + Λ (cid:19) . (3.9)In this expression, since the n = 0 terms are Fourier transforms, only the n = 0 term is UVdivergent. Under the dimensional regularization with D = 2 − ε , the momentum integrationyields − X A ln Det (cid:18) − ∂ µ ∂ µ + Λ + δ A − ∂ µ ∂ µ + Λ (cid:19) = − Z d x π (cid:20) ε − ln (cid:18) e γ E Λ π (cid:19)(cid:21) X A δ A − Z d x X A π (cid:20) δ A − (Λ + δ A ) ln (cid:18) δ A Λ (cid:19)(cid:21) − Z d x X A X n =0 e − i ( m A + A y )2 πRn a) (b) (c) (d) (e) (f)(g) (h) Fig. 1: Two-loop vacuum bubble diagrams that contribute to E ( δǫ ) | in Eq. (3.13). Thesolid line denotes the z A -propagator of Eq. (3.6). The wavy line denotes the δA µ -propagator,the dotted line the δf -propagator, the broken line the δσ -propagator, and the arrowed solidline the η -propagator in Eqs. (2.17) and (2.14). × π ( −
4) 12 πR | n | hp Λ + δ A K ( p Λ + δ A πR | n | ) − Λ K (Λ2 πR | n | ) i . (3.10)Since Eqs. (2.9) and (2.10) imply that14 π (cid:20) ε − ln (cid:18) e γ E Λ π (cid:19)(cid:21) = 1 λ , (3.11)we see that the first term on the right-hand side of Eq. (3.10) is precisely canceled by thefirst term on the right-hand side of Eq. (3.5).In this way, from Eq. (3.5) we have E ( δǫ ) | = 2 πR X A π (cid:20) δ A − (Λ + δ A ) ln (cid:18) δ A Λ (cid:19)(cid:21) + 2 πR X A X n =0 e − i ( m A + A y )2 πRn × π ( −
4) 12 πR | n | hp Λ + δ A K ( p Λ + δ A πR | n | ) − Λ K (Λ2 πR | n | ) i . (3.12) Next, we work out the vacuum bubble diagrams in the two-loop level; they are depictedin Fig. 1. By using the propagators in Eqs. (2.14), (2.17), (2.20), and (3.6), and interactionvertices in Eq. (2.6), from Eq. (3.5) we have E ( δǫ ) | = − πR πN X A ∞ X n = −∞ Z d p (2 π ) e i ( p y − A y − m A )2 πRn p + Λ + δ A × "Z dℓ x π πR X ℓ y p − ℓ ) + Λ + δ A
12 (2 p µ − ℓ µ )(2 p ν − ℓ ν ) L ( ℓ ) D ( ℓ ) ( δ µν + 4 " Λ + ¯ ℓ y ℓ K ( ℓ ) L ( ℓ ) ℓ µ ℓ ν ( ℓ ) ) (Fig. 1a) − L ( ℓ ) D ( ℓ ) ℓ (Fig. 1b) − p µ K ( ℓ ) D ( ℓ ) ¯ ℓ µ ¯ ℓ y ℓ (Fig. 1c)+ 2 L ( ℓ ) D ( ℓ ) Λ (Fig. 1d) − L ( ℓ ) D ( ℓ ) Λ ! (Fig. 1e)+ Z dℓ x π πR X ℓ y × − L ( ℓ ) D ( ℓ ) ( " Λ + ¯ ℓ y ℓ K ( ℓ ) L ( ℓ ) ℓ ) (Fig. 1f) − L ( ℓ ) D ( ℓ ) (Fig. 1g)+ 1( p − ℓ ) + Λ (cid:26) [ − ( p − ℓ ) · ℓ + 2Λ ] L ( ℓ ) D ( ℓ ) + 2 p µ ¯ ℓ µ ¯ ℓ y ℓ K ( ℓ ) D ( ℓ ) (cid:27)! (Fig. 1h) − (terms with δǫ = 0) , (3.13)where the contributions of each diagram in Fig. 1 are separately indicated by the equationnumbers. The total sum is E ( δǫ ) | = − πR πN X A ∞ X n = −∞ Z d p (2 π ) e i ( p y − A y − m A )2 πRn p + Λ + δ A × Z dℓ x π πR X ℓ y p − ℓ ) + Λ + δ A × (cid:26) L ( ℓ ) D ( ℓ ) (cid:20) p − p · ℓ − p · ℓℓ + 8Λ ( p · ℓ ) ( ℓ ) (cid:21) + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ (cid:20) − p · ℓℓ + 8 ( p · ℓ ) ( ℓ ) (cid:21) + K ( ℓ ) D ( ℓ ) ( − p · ¯ ℓ ¯ ℓ y ℓ (cid:27) + Z dℓ x π πR X ℓ y p − ℓ ) + Λ × (cid:26) L ( ℓ ) D ( ℓ ) (cid:20) − p + 4 p · ℓ + 8Λ p · ℓℓ − p ℓ − ℓ (cid:21) + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ (cid:20) − p · ℓℓ − p ℓ − ℓ (cid:21) + K ( ℓ ) D ( ℓ ) (8) p · ¯ ℓ ¯ ℓ y ℓ (cid:27)! (term with δǫ = 0) . (3.14)To examine the renormalizability of this expression, we first note that this can be written as E ( δǫ ) | = − πR πN X A n(cid:16) e δ A ∂ ξ e δ A ∂ η − (cid:17) I ( ξ, η ) + (cid:16) e δ A ∂ ξ − (cid:17) [ − I ( ξ,
0) + J ( ξ )] o(cid:12)(cid:12)(cid:12) ξ = η =0 , (3.15)where I ( ξ, η ) ≡ Z dℓ x π πR X ℓ y ∞ X n = −∞ Z d p (2 π ) e i ( p y − A y − m A )2 πRn p + Λ + ξ p − ℓ ) + Λ + η × (cid:26) L ( ℓ ) D ( ℓ ) (cid:20) p − p · ℓ − p · ℓℓ + 8Λ ( p · ℓ ) ( ℓ ) (cid:21) + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ (cid:20) − p · ℓℓ + 8 ( p · ℓ ) ( ℓ ) (cid:21) + K ( ℓ ) D ( ℓ ) ( − p · ¯ ℓ ¯ ℓ y ℓ (cid:27) , (3.16)and J ( ξ ) ≡ Z dℓ x π πR X ℓ y ∞ X n = −∞ Z d p (2 π ) e i ( p y − A y − m A )2 πRn p + Λ + ξ p − ℓ ) + Λ × " L ( ℓ ) D ( ℓ ) Λ + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ − p · ℓℓ − p ℓ − ℓ + 16 ( p · ℓ ) ( ℓ ) (cid:21) . (3.17)From Eq. (2.18), we see that, for | ℓ | → ∞ , ˆ L ( p ) and K ( p ) are exponentially small becauseof the Bessel functions, and thus L ( ℓ ) | ℓ |→∞ → ℓ ln( ℓ / Λ ) , D ( ℓ ) | ℓ |→∞ → ℓ L ( ℓ ) . (3.18)From these, we see that, in I ( ξ, η ) of Eq. (3.16), the integration over ℓ as well as the integra-tion over p are logarithmically UV divergent. In J ( ξ ) of Eq. (3.17), the integration over p islogarithmically UV divergent but the integration over ℓ is UV convergent. Assuming (say) thedimensional regularization, the change of integration variables ( p, ℓ ) → ( p − ℓ, − ℓ ) in I ( ξ, η ),Eq. (3.16), shows that I ( ξ, η ) = I ( η, ξ ) . (3.19)Now, in Eq. (3.15), using the identity e δ A ∂ ξ e δ A ∂ η − (cid:16) e δ A ∂ ξ − (cid:17) (cid:16) e δ A ∂ η − (cid:17) + e δ A ∂ ξ + e δ A ∂ η − E ( δǫ ) | = − πR πN X A h(cid:16) e δ A ∂ ξ − (cid:17) (cid:16) e δ A ∂ η − (cid:17) I ( ξ, η ) + (cid:16) e δ A ∂ ξ − (cid:17) J ( ξ ) i(cid:12)(cid:12)(cid:12) ξ = η =0 . (3.21)This shows that E ( δǫ ) | is UV finite provided that the parameter δ A is UV finite .That is, the operator e δ A ∂ ξ − J ( ξ ) increases the power of p + Λ in thedenominator in Eq. (3.17) and makes the p integration UV finite. Similarly, the opera-tor ( e δ A ∂ ξ − e δ A ∂ η −
1) acting on I ( ξ, η ) increases the power of ( p + Λ )[( p − ℓ ) + Λ ] inthe denominator of Eq. (3.16) and makes the integrations over p and ℓ UV convergent.11 .4. Renormalizability to the two-loop order
So far, we have observed that, from Eq. (3.12), E ( δǫ ) | = 2 πR X A π (cid:20) δ A − (Λ + δ A ) ln (cid:18) δ A Λ (cid:19)(cid:21) + 2 πR X A X n =0 e − i ( m A + A y )2 πRn × π ( −
4) 12 πR | n | hp Λ + δ A K ( p Λ + δ A πR | n | ) − Λ K (Λ2 πR | n | ) i , (3.22)and, from Eq. (3.21), E ( δǫ ) | = − πR πN X A h(cid:16) e δ A ∂ ξ − (cid:17) (cid:16) e δ A ∂ η − (cid:17) I ( ξ, η ) + (cid:16) e δ A ∂ ξ − (cid:17) J ( ξ ) i(cid:12)(cid:12)(cid:12) ξ = η =0 . (3.23)These representations show that the vacuum energy to the two-loop order is UV finite, ifthe parameter δ A defined in Eq. (3.4) is UV finite. This implies that the parameter δǫ mustreceive a non-trivial renormalization, as δ A = λδǫπRN m A is UV finite ⇒ δǫ = (cid:18) e γ E µ π (cid:19) − ε (cid:18) λ R π ε (cid:19) δǫ R , (3.24)so that λδǫ = λ R δǫ R is UV finite; here we have used Eq. (2.10).In terms of the renormalized parameters, the expansion of Eq. (3.22) with respect to δǫ yields E (1) δǫ (cid:12)(cid:12)(cid:12) = N Λ 1Λ
R λ R δǫ R πN RN X A m A X n =0 e − i ( m A + A y )2 πRn K (2 π Λ R | n | ) ,E (2) δǫ (cid:12)(cid:12)(cid:12) = N Λ 1(Λ R ) (cid:18) λ R δǫ R πN (cid:19) R N X A m A (cid:18) − (cid:19) × X n =0 e − i ( m A + A y )2 πRn π Λ R | n | K (2 π Λ R | n | ) . (3.25)For Eq. (3.23), we need to carry out momentum integrations in Eqs. (3.16) and (3.17).This is the subject of the next subsection. p -integration in E (1) δǫ | and E (2) δǫ | Let us next consider E (1) δǫ | , which is given by the O ( δ A ) term of Eq. (3.23). By usingthe formulas in Appendix B, p -integration in Eq. (3.17) yields E (1) δǫ (cid:12)(cid:12)(cid:12) = 2 πR N X A δ A Z dℓ x π πR X ℓ y " L ( ℓ ) D ( ℓ ) Λ + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ × Z dx X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn ( (2 πRn ) [ K ( z ) − K ( z )] 2 ℓ + (2 πRn ) K ( z )( − ℓ y ( ℓ ) + 2 πR | n | p x (1 − x ) ℓ + Λ K ( z ) (cid:20) ℓ + i πRn ℓ y ℓ ( − − x ) (cid:21)) , (3.26)where z ≡ p x (1 − x ) ℓ + Λ πR | n | . (3.27)Actually, the form of the integrand in the above expression depends on the choice of theFeynman parameter x . It can be changed by the change of variables x → − x and ℓ y → − ℓ y ,which keeps the integration region and the factor e ixℓ y πRn intact. It is convenient to fixthe form of the integrand I ( x, ℓ y ) by Z dx X ℓ y I ( x, ℓ y ) → Z dx X ℓ y
12 [ I ( x, ℓ y ) + I (1 − x, − ℓ y )] , (3.28)so that the form of the integrand is invariant under the above change of variables. Theparticular expression in Eq. (3.26) has been obtained in this way.Next, in Eq. (3.26) we use the identity K ν − ( z ) − K ν +1 ( z ) = − νz K ν ( z ) (3.29)with ν = 1. Then, by further using K ′ ( z ) = − K ( z ) (3.30)and ∂z∂x = 2 πR | n | p x (1 − x ) ℓ + Λ (1 − x ) ℓ , (3.31)which follows from Eq. (3.27), we have E (1) δǫ (cid:12)(cid:12)(cid:12) = 2 πR N X A δ A Z dℓ x π πR X ℓ y " L ( ℓ ) D ( ℓ ) Λ + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ × Z dx X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn × (cid:20) πRnℓ y K ( z ) − i ∂∂x K ( z ) (cid:21) πRn ( − ℓ y ( ℓ ) . (3.32)Finally, integration by parts with respect to x yields E (1) δǫ (cid:12)(cid:12)(cid:12) = 0 . (3.33) Recall that ℓ y ∈ Z /R . E (2) δǫ | , which is given by the O ( δ A ) terms in Eq. (3.23). First,the p -integration in the function J in Eq. (3.17) gives E (2) δǫ (cid:12)(cid:12)(cid:12) ( J )2-loop = − πR N X A δ A Z dℓ x π πR X ℓ y Z dx " L ( ℓ ) D ( ℓ ) Λ + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ × x (1 − x ) ℓ + Λ ] (cid:20) − x (1 − x )(3 − x + 10 x ) − (1 − x + 2 x ) Λ ℓ (cid:21) + 14 X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn × ( πR | n | p x (1 − x ) ℓ + Λ ! K ( z ) × (cid:20) − x (1 − x )(1 − x + 3 x ) − (1 − x + 2 x ) Λ ℓ (cid:21) + (2 πRn ) x (1 − x ) ℓ + Λ K ( z ) × (cid:20) − x + 2 x ) 1 ℓ + i πRn ℓ y ℓ ( − − x )(1 − x + 3 x ) (cid:21) + (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z ) × " (1 − x + 2 x ) 1 ℓ − − x + 2 x ) ℓ y ( ℓ ) . (3.34)On the other hand, from the function I in Eq. (3.16), E (2) δǫ (cid:12)(cid:12)(cid:12) ( I )2-loop = − πR N X A δ A Z dℓ x π πR X ℓ y Z dx × " L ( ℓ ) D ( ℓ ) ℓ x (1 − x ) ℓ + Λ ] ( − x (1 − x ) (cid:20) x (1 − x ) − (1 − x + 6 x ) Λ ℓ − ( ℓ ) (cid:21) + 14 X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn × ( πR | n | p x (1 − x ) ℓ + Λ ! K ( z )( − x (1 − x ) (cid:18) ℓ (cid:19) + (2 πRn ) x (1 − x ) ℓ + Λ K ( z ) × x (1 − x ) (cid:26) ℓ + 4 Λ ( ℓ ) − i πRn ℓ y ℓ (1 − x ) (cid:18) ℓ (cid:19)(cid:27)
14 (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z )( − x (1 − x ) " ℓ + 4Λ ℓ y ( ℓ ) + K ( ℓ ) D ( ℓ ) L ( ℓ ) ¯ ℓ y ℓ x (1 − x ) ℓ + Λ ] x (1 − x ) (cid:18) − x + 3 x + Λ ℓ (cid:19) + 14 X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn × ( πR | n | p x (1 − x ) ℓ + Λ ! K ( z )2 x (1 − x )(1 − x ) + (2 πRn ) x (1 − x ) ℓ + Λ K ( z )8 x (1 − x ) (cid:20) ℓ − i πRn ℓ y ℓ (1 − x ) (cid:21) + (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z )( − x (1 − x ) ℓ y ( ℓ ) )! + K ( ℓ ) D ( ℓ ) ¯ ℓ y ℓ X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn × (2 πRn ) x (1 − x ) ℓ + Λ K ( z ) i πRn ( − x (1 − x ) . (3.35)To obtain the expressions in Eqs. (3.34) and (3.35), we applied the procedure in Eq. (3.28).To further simplify the above expressions, we first note that all the terms linear in ℓ y areproportional to 1 − x , and thus to ∂z/∂x as in (3.31). Using this fact and the identity K ( z ) = − z (cid:20) z K ( z ) (cid:21) ′ , (3.36)we can carry out the integration by parts with respect to x in those terms linear in ℓ y . We thenuse the identity in Eq. (3.29) with ν = 2 to express K ( z ) in terms of K ( z ) and K ( z ). Theresulting expression contains the term K ( z ) x (1 − x )(1 − x ) , for which we use Eq. (3.31).We repeat the integration by parts as long as the factor 1 − x remains. In an intermediatestep, we use K ( z ) = − z [ zK ( z )] ′ . (3.37)Finally, we can carry out the x -integration in terms that do not contain the Bessel function. In this way, we have the following rather simple expression: E (2) δǫ (cid:12)(cid:12)(cid:12) = − πR N X A δ A Z dℓ x π πR X ℓ y We note that tanh − r ℓ ℓ + 4Λ ! = 14 p ℓ ( ℓ + 4Λ ) L ∞ ( ℓ ) . (3.38) " L ( ℓ ) D ( ℓ ) 4 − ℓ + 2Λ ) L ∞ ( ℓ ) ℓ ( ℓ + 4Λ )+ Z dx X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn × L ( ℓ ) D ( ℓ ) ( − (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z ) x (1 − x ) − (2 πRn ) x (1 − x ) ℓ + Λ K ( z ) x (1 − x )+ ℓ y ℓ " (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z ) x (1 − x )+ (2 πRn ) K ( z ) 2 ℓ − K ( ℓ ) D ( ℓ ) ¯ ℓ y ℓ (2 πRn ) x (1 − x ) ℓ + Λ K ( z ) i πRnx (1 − x ) ! . (3.39)This completes the p -integration in E (2) δǫ | .Let us examine whether the expression in Eq. (3.39) is IR finite or not. From the expressionsin Eq. (2.18) and L ∞ ( ℓ ) = 1Λ − ℓ Λ + O (( ℓ ) ) , (3.40)we see that the above ℓ x -integral for E (2) δǫ | is IR finite, as should be the case for anyphysical quantity.In what follows, we carry out the summation over the index A in Eqs. (3.25) and (3.39)and integrate the resulting expressions over the “vacuum moduli” A y as in Eq. (2.11).Then, we organize them according to the powers of 1 /N . Before doing these, however, itis helpful to further simplify Eq. (3.39) by noting that ˆ L ( p ) and K ( p ) in Eqs. (2.18) areexponentially suppressed for N → ∞ as . e − Λ RN because of the asymptotic behavior ofthe Bessel function, K ν ( z ) z →∞ ∼ p π/ (2 z ) e − z . Therefore, these functions can be neglected inthe power series expansion in 1 /N and we can set L ( ℓ ) → L ∞ ( ℓ ), K ( ℓ ) →
0, and D ( ℓ ) → ( p + 4Λ ) L ∞ ( ℓ ) in Eq. (3.39) to yield E (2) δǫ (cid:12)(cid:12)(cid:12) = − πR N X A δ A Z dℓ x π πR X ℓ y × " − ℓ + 2Λ ) L ∞ ( ℓ ) ℓ ( ℓ + 4Λ ) L ∞ ( ℓ )+ Z dx X n =0 e − i ( m A + A y )2 πRn e ixℓ y πRn ℓ + 4Λ ) L ∞ ( ℓ ) × ( − (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z ) x (1 − x ) − (2 πRn ) x (1 − x ) ℓ + Λ K ( z ) x (1 − x )16 ℓ y ℓ " (2 πR | n | ) p x (1 − x ) ℓ + Λ K ( z ) x (1 − x ) + (2 πRn ) K ( z ) 2 ℓ , (3.41)up to exponentially small terms. A and integration over A y We thus consider the sum over the index A and the integration over the vacuum moduli A y in Eq. (2.11). The summation over A can be carried out as X A e − im A πRn = N − X j =0 (cid:16) e − πni/N (cid:17) j = N ( , for n = 0 mod N , , for n = 0 mod N , (3.42) X A m A e − im A πRn = N R − N , for n = 0 mod N , N e − πni/N − , for n = 0 mod N , (3.43)and X A m A e − im A πRn = N R − N + 12 N , for n = 0 mod N , N e − πni/N − (cid:18) − N − e πni/N (cid:19) , for n = 0 mod N . (3.44)On the other hand, the integration over A y with the measure in Eq. (2.11) results in Z d ( A y RN ) e − iA y πRn = , for n = 0 , , for n = 0, n = 0 mod N ,iN πn (cid:0) e − πni/N − (cid:1) , for n = 0 mod N . (3.45)The combination of the above two operations therefore yields Z d ( A y RN ) X A m A e − i ( m A + A y )2 πRn = N R − N , for n = 0 , , for n = 0, n = 0 mod N ,iπn , for n = 0 mod N , (3.46)and Z d ( A y RN ) X A m A e − i ( m A + A y )2 πRn = N R − N + 12 N , for n = 0 , , for n = 0, n = 0 mod N , i πn (cid:18) − N (cid:19) + 32 N πn πn/N ) , for n = 0 mod N . (3.47)Using Eqs. (3.46) and (3.47) for Eq. (3.25), under the integration over A y , E (1) δǫ (cid:12)(cid:12)(cid:12) = N Λ 1Λ
R λ R δǫ R πN X n =0 mod N iπn K (2 π Λ R | n | ) = 0 , (3.48)17nd E (2) δǫ (cid:12)(cid:12)(cid:12) = N Λ 1(Λ R ) (cid:18) λ R δǫ R πN (cid:19) (cid:18) − (cid:19) − N + 12 N + 6 N X n> ,n =0 mod N Λ RK (2 π Λ Rn )tan( πn/N ) . (3.49)For the two-loop corrections, from Eq. (3.33), E (1) δǫ (cid:12)(cid:12)(cid:12) = 0 , (3.50)and for Eq. (3.40) we have E (2) δǫ (cid:12)(cid:12)(cid:12) = − π (cid:18) λ R δǫ R πRN (cid:19) Z dℓ x π πR X ℓ y × R (cid:18) − N + 12 N (cid:19) − ℓ + 2Λ ) L ∞ ( ℓ ) ℓ ( ℓ + 4Λ ) L ∞ ( ℓ )+ Z dx X n> ,n =0 mod N × (cid:20) N cos( xℓ y πRn )tan( πn/N ) − (cid:18) − N (cid:19) sin( xℓ y πRn ) (cid:21) ℓ + 4Λ ) L ∞ ( ℓ ) × ( − (2 πRn ) p x (1 − x ) ℓ + Λ K ( z ) x (1 − x ) − πRnx (1 − x ) ℓ + Λ K ( z ) x (1 − x )+ ℓ y ℓ " (2 πRn ) p x (1 − x ) ℓ + Λ K ( z ) x (1 − x ) + 2 πRnK ( z ) 2 ℓ , (3.51)up to exponentially small terms. Finally, we arrange the above results in powers of 1 /N . From Eqs. (3.48) and (3.50), we have E (1) δǫ = 0 · N + 0 · N − + O ( N − ) . (3.52)Thus, E (1) δǫ vanishes to the order we worked out.For E (2) δǫ , setting E (2) δǫ = E (2) δǫ (cid:12)(cid:12)(cid:12) O ( N − ) + E (2) δǫ (cid:12)(cid:12)(cid:12) O ( N − ) + O ( N − ) , (3.53)from Eq. (3.49), RE (2) δǫ (cid:12)(cid:12)(cid:12) O ( N − ) = N − ( λ R δǫ R ) (Λ R ) − F (Λ R ) , (3.54)18here F ( ξ ) ≡ − π [1 + c ( ξ )] , c ( ξ ) ≡ lim N →∞ N X n> ,n =0 mod N ξK (2 πξn )tan( πn/N ) . (3.55)From Eqs. (3.49) and (3.51), on the other hand, RE (2) δǫ (cid:12)(cid:12)(cid:12) O ( N − ) = N − ( λ R δǫ R ) (Λ R ) − G (Λ R ) , (3.56)where G ( ξ ) ≡ − π − ξ + lim N →∞ X n> ,n =0 mod N ξ K (2 πξn )tan( πn/N ) − N ξc ( ξ ) − π ξ Z ∞−∞ d ˜ ℓ x X ˜ ℓ y ∈ Z − ℓ + 2 ξ ) ˜ L ∞ (˜ ℓ, ξ )˜ ℓ (˜ ℓ + 4 ξ ) ˜ L ∞ (˜ ℓ, ξ )+ lim N →∞ Z dx X n> ,n =0 mod N × " N cos( x ˜ ℓ y πn )tan( πn/N ) − x ˜ ℓ y πn ) ℓ + 4 ξ ) ˜ L ∞ (˜ ℓ, ξ ) × ( − (2 πn ) q x (1 − x )˜ ℓ + ξ K ( z ) x (1 − x ) − πnx (1 − x )˜ ℓ + ξ K ( z ) x (1 − x )+ ˜ ℓ y ˜ ℓ (2 πn ) q x (1 − x )˜ ℓ + ξ K ( z ) x (1 − x ) + 2 πnK ( z ) 2˜ ℓ )! . (3.57)In this expression, we have defined˜ L ∞ (˜ ℓ, ξ ) ≡ q ˜ ℓ (˜ ℓ + 4 ξ ) ln q ˜ ℓ + 4 ξ + p ˜ ℓ q ˜ ℓ + 4 ξ − p ˜ ℓ (3.58)and z ≡ q x (1 − x )˜ ℓ + ξ π | n | . (3.59)We plot the function F (Λ R ) in Eq. (3.54) in Fig. 2 and the function G (Λ R ) in Eq. (3.56)in Fig. 3. These plots clearly show that, to the order of the 1 /N expansion we worked out,the vacuum energy is a well-defined finite quantity under the parameter renormalizationin Eqs. (2.10) and (3.24). Equations (3.52)–(3.57) and Figs. 2 and 3 are the main resultsof this paper. Since Figs. 2 and 3 show that the functions F (Λ R ) and G (Λ R ) remain finiteas Λ R →
0, Eqs. (3.54) and (3.56) [and Eq. (3.52)] show that the vacuum energy normal-ized by the radius of the S , RE ( δǫ ), behaves as inverse powers of Λ R for Λ R small, the19 .0 0.2 0.4 0.6 0.8 1.0 1.2 - - - - Λ R Fig. 2: The function F (Λ R ) from Eq. (3.54). Λ R Fig. 3: The function G (Λ R ) from Eq. (3.56). O ( N − ) term behaves as (Λ R ) − , and the O ( N − ) term behaves as (Λ R ) − . Since Λ isgiven by Eq. (2.9), this result implies that to the order of the 1 /N expansion we worked out,the vacuum energy is a purely non-perturbative quantity and it has no well-defined weakcoupling expansion in λ R .
4. Conclusion and discussion
By employing the 1 /N expansion, we have computed the vacuum energy E ( δǫ ) of the 2DSUSY C P N − model on R × S with Z N twisted boundary conditions to the second orderin the SUSY-breaking parameter δǫ in Eq. (3.2). We found that the vacuum energy is purelynon-perturbative and, although it is a perfectly well-defined physical quantity in the 1 /N expansion, it has no sensible weak coupling expansion in λ R .Our original intention was to compare our result in the 1 /N expansion with the result bythe bion calculus in Ref. [5], because it appears that the calculation in Ref. [5] holds evenunder the limit in Eq. (1.1). 20ccording to Ref. [5], the contribution of a single bion to the vacuum energy in Eq. (3.1)is given by ( E (0) is set to be zero) RE (1) δǫ = − R N − X b =1 m b A b (Λ R ) Rm b N δǫ (4.1)and RE (2) δǫ = − R N − X b =1 m b A b (Λ R ) Rm b N (cid:20) − γ E − (cid:18) πRm b Nλ R (cid:19) ∓ πi (cid:21) δǫ , (4.2)where the last ∓ πi term is the imaginary ambiguity caused by the integration over quasi-collective coordinates of the bion. In these expressions, the index b corresponds to the“species” of the bion and the coefficient A b is given by using the twist angle m A in Eq. (2.4)as A b = (cid:20) Γ (1 − m b R ) Γ (1 + m b R ) (cid:21) N − Y a =1 ,a = b m a m a − m b Γ (1 + ( m a − m b ) R ) Γ (1 − ( m a − m b ) R ) Γ (1 − m a R ) Γ (1 + m a R )= ( − b +1 N b ( b !) . (4.3)Using this, the coefficient of the imaginary ambiguity in Eq. (4.2) is given by − R N − X b =1 m b A b (Λ R ) Rm b N = 2 N N − X b =1 ( − b b ( b !) (Λ RN ) b . (4.4)When N is fixed, in the weak coupling limit Λ R ≪ b = 1 term − R N dominates the sum in Eq. (4.4).Λ = µ e − π/λ R is the exponential of the action of the constituent of the minimum bion(the minimal fractional instanton–anti-instanton pair) and, at the same time, is consistentwith the order of the u = 1 IR renormalon ambiguity. On the other hand, in the large- N limit in Eq. (1.1), whether Eq. (4.4) possesses a sensible 1 /N expansion or not is not clear,because each term behaves as O ( N ), O ( N ), O ( N ), . . . ; we could not estimate the sum asa whole in the large- N limit.Thus, we cannot compare our result in the 1 /N expansion with the result in Ref. [5] by thebion calculus. We have no clear idea yet why this comparison is impossible. One phenomeno-logical observation from Eq. (4.4) is that it is a series in the combination Λ RN and thus theresult in Ref. [5] seems meaningful for Λ RN ≪ N limit in Eq. (1.1),with which Λ RN ≫ More thought seems to be necessary to clearly understand therelation between bions, the IR renormalon, and the 1 /N expansion. Acknowledgments
We are grateful to Akira Nakayama and Hiromasa Takaura for collaboration at various stagesof this work. We would also like to thank Toshiaki Fujimori, Tatsuhiro Misumi, NorisukeSakai, and Kazuya Yonekura for helpful discussions. This work was supported by JSPSGrants-in-Aid for Scientific Research numbers JP18J20935 (O.M.) and JP16H03982 (H.S.). We would like to thank Aleksey Cherman, Yuya Tanizaki, and Mithat ¨Unsal for providing uswith suggestive arguments on this point. ote added In this paper we considered the large- N limit specified by Eq. (1.1), with which N Λ R → ∞ .On the other hand, Ref. [42] discussed that the semi-classical picture such as that in Refs. [17–20] holds only for N Λ R ≪
1. This is natural because the characteristic mass scale with thetwisted boundary condition can be N Λ R instead of Λ R and in the weak coupling limitΛ →
0. In this paper, we also observed that the perturbative analyses cannot be availablereasonably for N Λ R ≫
1; our approximation is basically the expansion in 1 / ( N Λ R ) and itis impossible to read how the vacuum energy behaves as N Λ R → N result.In a recent paper [43], perturbation theory with the twisted boundary condition is carefullystudied for N Λ R → A. The perturbative part of the vacuum energy contains IR divergences
In the limit R → ∞ , the expression of the vacuum energy is considerably simplified because n = 0 terms in Eqs. (3.55) and (3.57) are exponentially suppressed in this limit. We have RE (2) δǫ R →∞ → − π ( λ R δǫ R ) (cid:26) N − (Λ R ) − + N − (cid:20) −
32 (Λ R ) − + G ∞ (cid:21) + O ( N − ) (cid:27) , (A1)where G ∞ ≡ πR Z dℓ x π πR X ℓ y − ℓ + 2Λ ) L ∞ ( ℓ ) ℓ ( ℓ + 4Λ ) L ∞ ( ℓ ) . (A2)Equation (A1) is a non-perturbative expression obtained to the next-to-leading order of the1 /N expansion. From Eq. (3.40), we see that the ℓ -integration in G ∞ is IR convergent.To extract the perturbative part from Eq. (A1), we expand G ∞ with respect to Λand neglect all terms with positive powers of Λ = µe − π/λ R . Noting the behavior L ∞ ∼ (2 /ℓ ) ln( ℓ / Λ ) from Eq. (2.18), we obtain the perturbative part as G ∞ ∼ πR Z dℓ x π πR X ℓ y ℓ ) (cid:20) ℓ / Λ ) − (cid:21) . (A3)The perturbative expansion with respect to λ R ( µ ) is then given by G ∞ ∼ πR Z dℓ x π πR X ℓ y ℓ ) " − ∞ X k =0 [ − ln( ℓ /µ )] k (cid:18) λ R π (cid:19) k +1 , (A4)where we have used ln( ℓ / Λ ) = ln( ℓ /µ ) + 4 πλ R ( µ ) . (A5)Equations (A3) and (A4) show that the perturbative part of G ∞ suffers from IR divergencesin the ℓ -integration, although the full G ∞ itself is IR finite. B. Integration formulas
In Sect. 3.5 we have used the following integration formulas (in practice, we are interestedin the cases ( α, β ) = (1 , , , Z d p (2 π ) e ip y πRn p − ℓ ) + Λ ] α p + Λ ) β p µ p µ p ν =0 = 1 Γ ( α ) Γ ( β ) Z dx x α − (1 − x ) β − × π Γ ( α + β − (cid:2) x (1 − x ) ℓ + Λ (cid:3) − α − β .Γ ( α + β − (cid:2) x (1 − x ) ℓ + Λ (cid:3) − α − β xℓ µ ,Γ ( α + β − (cid:2) x (1 − x ) ℓ + Λ (cid:3) − α − β x ℓ µ ℓ ν + Γ ( α + β − (cid:2) x (1 − x ) ℓ + Λ (cid:3) − α − β δ µν , n =0 = 1 Γ ( α ) Γ ( β ) Z dx x α − (1 − x ) β − × π − α − β e ixℓ y πRn (cid:18) πR | n | √ x (1 − x ) ℓ +Λ (cid:19) α + β − K α + β − ( z ) , (cid:18) πR | n | √ x (1 − x ) ℓ +Λ (cid:19) α + β − K α + β − ( z ) xℓ µ + (cid:18) πR | n | √ x (1 − x ) ℓ +Λ (cid:19) α + β − K α + β − ( z ) i πRnδ µy , (cid:18) πR | n | √ x (1 − x ) ℓ +Λ (cid:19) α + β − K α + β − ( z ) x ℓ µ ℓ ν + (cid:18) πR | n | √ x (1 − x ) ℓ +Λ (cid:19) α + β − K α + β − ( z ) × ( δ µν + ixℓ µ πRnδ νy + i πRnδ µy xℓ ν ) − (cid:18) πR | n | √ x (1 − x ) ℓ +Λ (cid:19) α + β − K α + β − ( z ) × (2 πRn ) δ µy δ νy , (B1)where z ≡ p x (1 − x ) ℓ + Λ πR | n | . (B2) References [1] S. Coleman, “Aspects of Symmetry : Selected Erice Lectures,” doi:10.1017/CBO9780511565045[2] E. Cremmer and J. Scherk, Phys. Lett. , 341 (1978). doi:10.1016/0370-2693(78)90672-X[3] A. D’Adda, P. Di Vecchia and M. L¨uscher, Nucl. Phys. B , 125 (1979). doi:10.1016/0550-3213(79)90083-X[4] E. Witten, Nucl. Phys. B , 285 (1979). doi:10.1016/0550-3213(79)90243-8[5] T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, JHEP , 190 (2019)doi:10.1007/JHEP02(2019)190 [arXiv:1810.03768 [hep-th]].[6] T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, Phys. Rev. D , no. 10, 105002 (2016)doi:10.1103/PhysRevD.94.105002 [arXiv:1607.04205 [hep-th]].[7] T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, Phys. Rev. D , no. 10, 105001 (2017)doi:10.1103/PhysRevD.95.105001 [arXiv:1702.00589 [hep-th]].[8] T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, PTEP , no. 8, 083B02 (2017)doi:10.1093/ptep/ptx101 [arXiv:1705.10483 [hep-th]].[9] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Phys. Rev. D , 025011 (2005)doi:10.1103/PhysRevD.72.025011 [hep-th/0412048].[10] M. Eto, T. Fujimori, Y. Isozumi, M. Nitta, K. Ohashi, K. Ohta and N. Sakai, Phys. Rev. D , 085008(2006) doi:10.1103/PhysRevD.73.085008 [hep-th/0601181].[11] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, J. Phys. A , R315 (2006) doi:10.1088/0305-4470/39/26/R01 [hep-th/0602170].[12] F. Bruckmann, Phys. Rev. Lett. , 051602 (2008) doi:10.1103/PhysRevLett.100.051602[arXiv:0707.0775 [hep-th]].[13] W. Brendel, F. Bruckmann, L. Janssen, A. Wipf and C. Wozar, Phys. Lett. B , 116 (2009)doi:10.1016/j.physletb.2009.04.055 [arXiv:0902.2328 [hep-th]].
14] F. Bruckmann and S. Lochner, Phys. Rev. D , no. 6, 065005 (2018) doi:10.1103/PhysRevD.98.065005[arXiv:1805.11313 [hep-th]].[15] G. ’t Hooft, Subnucl. Ser. , 943 (1979).[16] M. Beneke, Phys. Rept. , 1 (1999) doi:10.1016/S0370-1573(98)00130-6 [hep-ph/9807443].[17] P. Argyres and M. ¨Unsal, Phys. Rev. Lett. , 121601 (2012) doi:10.1103/PhysRevLett.109.121601[arXiv:1204.1661 [hep-th]].[18] P. C. Argyres and M. ¨Unsal, JHEP , 063 (2012) doi:10.1007/JHEP08(2012)063 [arXiv:1206.1890[hep-th]].[19] G. V. Dunne and M. ¨Unsal, JHEP , 170 (2012) doi:10.1007/JHEP11(2012)170 [arXiv:1210.2423[hep-th]].[20] G. V. Dunne and M. ¨Unsal, Phys. Rev. D , 025015 (2013) doi:10.1103/PhysRevD.87.025015[arXiv:1210.3646 [hep-th]].[21] G. V. Dunne and M. ¨Unsal, Ann. Rev. Nucl. Part. Sci. , 245 (2016) doi:10.1146/annurev-nucl-102115-044755 [arXiv:1601.03414 [hep-th]].[22] J. C. Le Guillou and J. Zinn-Justin, Amsterdam, Netherlands: North-Holland (1990) 580 p. (Currentphysics - sources and comments[23] E. B. Bogomolny, Phys. Lett. , 431 (1980). doi:10.1016/0370-2693(80)91014-X[24] J. Zinn-Justin, Nucl. Phys. B , 125 (1981). doi:10.1016/0550-3213(81)90197-8[25] E. Witten, AMS/IP Stud. Adv. Math. , 347 (2011) [arXiv:1001.2933 [hep-th]].[26] M. Cristoforetti et al. [AuroraScience Collaboration], Phys. Rev. D , 074506 (2012)doi:10.1103/PhysRevD.86.074506 [arXiv:1205.3996 [hep-lat]].[27] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano, JHEP , 147 (2013)doi:10.1007/JHEP10(2013)147 [arXiv:1309.4371 [hep-lat]].[28] M. M. Anber and T. Sulejmanpasic, JHEP , 139 (2015) doi:10.1007/JHEP01(2015)139[arXiv:1410.0121 [hep-th]].[29] K. Ishikawa, O. Morikawa, A. Nakayama, K. Shibata, H. Suzuki and H. Takaura, arXiv:1908.00373[hep-th].[30] M. Ashie, O. Morikawa, H. Suzuki, H. Takaura and K. Takeuchi, arXiv:1909.05489 [hep-th].[31] K. Ishikawa, O. Morikawa, K. Shibata, H. Suzuki and H. Takaura, arXiv:1909.09579 [hep-th].[32] M. Yamazaki and K. Yonekura, arXiv:1911.06327 [hep-th].[33] M. Mari˜no and T. Reis, arXiv:1912.06228 [hep-th].[34] A. Flachi, arXiv:1912.12376 [hep-th].[35] T. Eguchi and H. Kawai, Phys. Rev. Lett. , 1063 (1982). doi:10.1103/PhysRevLett.48.1063[36] D. J. Gross and Y. Kitazawa, Nucl. Phys. B , 440 (1982). doi:10.1016/0550-3213(82)90278-4[37] T. Sulejmanpasic, Phys. Rev. Lett. , no. 1, 011601 (2017) doi:10.1103/PhysRevLett.118.011601[arXiv:1610.04009 [hep-th]].[38] M. Beneke and V. M. Braun, Phys. Lett. B , 513 (1995) doi:10.1016/0370-2693(95)00184-M[hep-ph/9411229].[39] D. J. Broadhurst and A. L. Kataev, Phys. Lett. B , 179 (1993) doi:10.1016/0370-2693(93)90177-J[hep-ph/9308274].[40] P. Ball, M. Beneke and V. M. Braun, Nucl. Phys. B , 563 (1995) doi:10.1016/0550-3213(95)00392-6[hep-ph/9502300].[41] U. Lindstr¨om and M. Roˇcek, Nucl. Phys. B , 285 (1983). doi:10.1016/0550-3213(83)90638-7[42] M. ¨Unsal and L. G. Yaffe, Phys. Rev. D , 065035 (2008) doi:10.1103/PhysRevD.78.065035[arXiv:0803.0344 [hep-th]].[43] O. Morikawa and H. Takaura, [arXiv:2003.04759 [hep-th]]., 065035 (2008) doi:10.1103/PhysRevD.78.065035[arXiv:0803.0344 [hep-th]].[43] O. Morikawa and H. Takaura, [arXiv:2003.04759 [hep-th]].