Multiloop amplitudes of light-cone gauge superstring field theory: odd spin structure contributions

Journal of High Energy Physics, Mar 2018

Nobuyuki Ishibashi, Koichi Murakami

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Multiloop amplitudes of light-cone gauge superstring field theory: odd spin structure contributions

HJE Multiloop amplitudes of light-cone gauge superstring Nobuyuki Ishibashi 0 1 3 Koichi Murakami 0 1 2 Theory, Superstrings and Heterotic Strings 0 Otanoshike-Nishi 2-32-1 , Kushiro, Hokkaido 084-0916 , Japan 1 Tsukuba , Ibaraki 305-8571 , Japan 2 National Institute of Technology, Kushiro College 3 Tomonaga Center for the History of the Universe, University of Tsukuba We study the odd spin structure contributions to the multiloop amplitudes of light-cone gauge superstring field theory. We show that they coincide with the amplitudes in the conformal gauge with two of the vertex operators chosen to be in the pictures different from the standard choice, namely (−1, −1) picture in the type II case and −1 picture in the heterotic case. We also show that the contact term divergences can be regularized in the same way as in the amplitudes for the even structures and we get the amplitudes which coincide with those obtained from the first-quantized approach. String Field Theory; BRST Quantization; Conformal Field Models in String - 1 Introduction 3 4 2.1 2.2 2.3 3.1 3.2 Odd spin structure Multiloop amplitudes Dimensional regularization Conclusions and discussions 2 The problems with odd spin structure A Operator valued coordinate B Correlation functions of free fermions C Dimensional regularization for odd spin structure C.1 Supersymmetric X± CFT C.2 A proof of equality of (3.14) and (3.15) which agree with those of the first-quantized theory. For bosonic strings, there are several proposals of such string field theories. For superstrings, because of the problems with the method to calculate multiloop amplitudes using the picture changing operators, the construction of a string field theory has been a difficult problem. Recently, Sen has constructed a gauge invariant formulation of the string field theory for closed superstrings [ 1–5 ], based on the formulation [6] of closed string field theory for bosonic strings with a nonpolynomial action. Light-cone gauge closed superstring field theory is a string field theory for superstrings which involves only three-string interaction terms. It can be proved formally that the Feynman amplitudes of the string field theory coincide with those of the first-quantized theory [7]. The proof was formal because there appear unphysical divergences which are called the contact term divergences [8–12]. In a previous paper [ 13 ], we have shown that these divergences can be dealt with by dimensional regularization. In the case of type II superstrings, for example, one formulates a light-cone gauge superstring field theory – 1 – in noncritical dimensions or the one whose worldsheet theory for transverse variables is a superconformal field theory with central charge c 6= 12 [14]. Although Lorentz invariance is broken by doing so, it does not cause so much trouble because the light-cone gauge theory is a completely gauge-fixed theory. In [ 13 ], we have shown that the multiloop amplitudes corresponding to the Riemann surfaces with even spin structure involving external lines in the (NS,NS) sector can be calculated using the dimensional regularization and the results coincide with those of the first-quantized approach. What we would like to do in this paper is to generalize these results to the case of the surfaces with odd spin structure. On the Riemann surfaces with odd spin structure, there exist zero modes of the fermionic variables on the worldsheet which make the manipulations HJEP03(218)6 of the amplitudes complicated. We will show that it is possible to deal with these zero modes and prove that the amplitudes are equal to those of the first-quantized method, when all the external lines are in the (NS,NS) sector, in the case of type II superstrings. It is straightforward to obtain similar results for heterotic strings. The organization of this paper is as follows. In section 2, we review the results in [ 13 ] and the problems with the odd spin structures. In section 3, we deal with the amplitudes for the odd spin structure and show that these also coincide with those from the firstquantized approach. Section 4 is devoted to discussions. In the appendices, we present details of the manipulations given in the main text. 2 Light-cone gauge superstring field theory In this section, we review the known results for the multiloop amplitudes of light-cone gauge superstring field theory and the problems with the odd spin structures. 2.1 Light-cone gauge superstring field theory In the light-cone gauge string field theory, the string field is taken to be an element of the Hilbert space H of the transverse variables on the worldsheet and a function of |Φ (t, α)i t = x+ , α = 2p+ . – 2 – In this paper, we consider the string field theory for type II superstrings in 10 dimensional flat spacetime as an example. |Φ(t, α)i should be GSO even and satisfy the level-matching condition (L0 − L¯0) |Φ (t, α)i = 0 , where L0, L¯0 are the zero modes of the Virasoro generators of the worldsheet theory. (2.1) (2.2) (2.3) −∞ 4π hΦB (−α)| i∂t − −∞ 4π hΦF (−α)| i∂t − L0 + L¯0 − 1 |ΦF (α)i + 2 gs − 6 gs − 2 F X B1,B2,B3 X B1,F2,F3 Z Z 3 Y r=1 3 Y r=1 αrdαr 4π αrdαr 4π δ δ 3 3 r=1 r=1 α ! ! X αr hV3 |ΦB1 (α1) i |ΦB2 (α2)i |ΦB3 (α3)i X αr hV3 |ΦB1 (α1) i α2 2 |ΦF2 (α2)i α3 2 |ΦF3 (α3)i , which consists of the kinetic terms and the three-string interaction terms. PB and PF denote the sums over bosonic and fermionic string fields respectively. The three-string ∗ ∗ vertices hV3| are elements of H ⊗ H ⊗ H Here H∗denotes the dual space of H. ∗ whose definition can be found in [ 13, 15, 16 ]. It is straightforward to calculate the amplitudes by the old-fashioned perturbation theory starting from the action (2.4) and Wick rotate to Euclidean time. The propagator and the vertex are given by the worldsheets depicted in figure 1, where the left and right supercurrents TFLC, T¯LC of the transverse variables Xi, ψi, ψ¯i (i = 1, . . . , 8) are inserted at F the interaction points of the three-string vertices. Each term in the expansion corresponds to a light-cone gauge Feynman diagram for strings. A typical light-cone gauge Feynman diagram for strings is depicted in figure 2. A Wick rotated g-loop N -string diagram is conformally equivalent to an N punctured genus g Riemann surface Σ. A g-loop N -string amplitude is given as an integral over the moduli – 3 – AN (g) = (igs)2g−2+N C Z where R [dT ][αdθ][dα] denotes the integration over the moduli parameters and C is the combinatorial factor. The integrand F (g) is given as a path integral over the transverse variables Xi, ψi, ψ¯i on the light-cone diagram. A light-cone diagram consists of cylinders which correspond to propagators of the closed string. On each cylinder one can introduce a complex coordinate whose real part τ coincides with the Wick rotated light-cone time it and imaginary part σ ∼ σ + 2παr parametrizes the closed string at each time. The ρ’s on the cylinders are smoothly connected except at the interaction points and we get a complex coordinate ρ on Σ. The path integral on the light-cone diagram is defined by using the metric ρ = τ + iσ , ds2 = dρdρ¯ . (2.5) (2.7) (2.6) HJEP03(218)6 ρ is not a good coordinate around the interaction points and the punctures, and the metric (2.7) is not well-defined at these points. F (g) can be expressed in terms of correlation functions defined with a metric dsˆ2 = 2gˆzz¯dzdz¯ which is regular everywhere on the worldN sheet, as N spin structure N X p + r r=1 Z ! δ N X p − r r=1 ! e− 21 Γ[σ;gˆzz¯] dXidψidψ¯i gˆzz¯ e−SLC[Xi,ψi,ψ¯i] × 2g−2+N Y I=1 the anomaly factor e− 21 Γ[σ;gˆzz¯], where Here z is a complex coordinate of the Riemann surface and the coordinate ρ becomes a function ρ(z) of z (see e.g. [18–20]). SLC Xi, ψi, ψ¯i denotes the worldsheet action of the transverse variables and the path integral measure dXidψidψ¯i gˆzz¯ is defined with the metric dsˆ2 = 2gˆzz¯dzdz¯. Since the integrand was defined by using the metric (2.7), we need σ = ln ∂ρ∂¯ρ¯ − ln gˆzz¯ , Γ [σ; gˆzz¯] = − 4π 1 Z dz ∧ dz¯pgˆ gˆab∂aσ∂bσ + 2Rˆσ . (2.9) zI (I = 1, · · · , 2g − 2 + N ) denote the z-coordinates of the interaction points of the lightcone gauge Feynman diagram. VrLC denotes the vertex operator for the r-th external line inserted at z = Zr (r = 1, . . . , N ). The right hand side of (2.8) does not depend on the choice of gˆzz¯. – 4 – As was demonstrated in [ 13 ], if all the external lines are in the (NS,NS) sector and the spin structure for the left and right fermions are both even, the term in the sum in (2.8) can be recast into a conformal gauge expression: Z dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gˆzz¯ e−Stot × 6g−6+2N Y K=1 Here Stot denotes the worldsheet action for the variables Xμ, ψμ, ψ¯μ (μ = +, −, 1, . . . , 8), ghosts and superghosts, 1 4 1 4 X (z) = c∂ξ − eφTF + ∂bηe2φ + b 2∂ηe2φ + η∂e2φ (z) (2.11) is the picture changing operator (PCO), X¯ (z¯) is its antiholomorphic counterpart and TF denotes the supercurrent for ∂Xμ, ψμ. The contours CK and εK = ±1 are chosen so that the antighost insertions correspond to the moduli parameters for the light-cone amplitudes. The vertex operator Vr(−1,−1)(Zr, Z¯r) is defined as Vr(−1,−1)(Zr, Z¯r) ≡ cc¯e−φ−φ¯VrDDF(Zr, Z¯r) . VrDDF(Zr, Z¯r) is the supersymmetric DDF vertex operator given by ¯¯i1(r) ¯¯j1(r) VrDDF(Zr, Z¯r) = Ai−1n(r1) · · · A−n¯1 · · · B−j1s(1r) · · · B−s¯1 · · · e with the DDF operators Ai−(rn), B−j(sr) for the r-th string defined as −ipr+X−−i pr−− Npr+r X++ipirXi and A¯i−(rn), B¯−i(sr) are similarly given for the antiholomorphic sector. Here we use the notation Ai−(rn) = B−i(sr) = I I Zr 2πi dz dz iDX e pr i −i n+ XL+ (z) , DX + Zr 2πi ipr+∂X + 21 DX e pr i −i s+ XL+ (z) , k ∂ D ≡ ∂θ + θ ∂ ∂z l . Nr ≡ X nk + X sl = X n¯k¯ + X s¯¯, l ¯ k ¯ l X μ (z, z¯) = Xμ (z, z¯) + iθψμ (z) + iθ¯ψ¯μ (z¯) + θθ¯F μ , z = (z, θ) denotes the superspace coordinate on the worldsheet and XL+ denotes the leftmoving part of the superfield X . + 1 We take the vertex operators to satisfy the on-shell 1Although XL+ is not a well-defined quantity, it is used as a short-hand notation to express the vertex operator (2.13), which is well-defined. – 5 – (2.12) (Zr, Z¯r) , (2.13) (2.14) (2.15) 2 −2pr+pr− + pirpir + Nr = 1 2 VrDDF(Zr, Z¯r) turns out to be a weight 12 , 12 primary field made from Xμ, ψμ, ψ¯μ. Therefore Vr(−1,−1)(Zr, Z¯r) is an on-shell vertex operator in (−1, −1) picture. It is easy to see that the expression of the amplitude (2.5) given as an integral of (2.10) is BRST invariant. One way to derive the expression (2.10) is as follows [ 13 ]. Using a nilpotent fermionic charge, it is possible to show that the right hand side of (2.10) is equal to (2.16) (2.17) (2.18) HJEP03(218)6 dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gˆzz¯ In this form, the path integral factorizes into the contributions from Xμ, ψμ, ψ¯μ, ghosts and superghosts. Each of these contributions is calculated by taking gˆzz¯ to be the Arakelov metric gzAz¯ [21]. In the matter sector, integration over the longitudinal variables yields , Here ϑ[α] denotes the theta function with characteristic α and Ω is the period matrix. αL and αR denote the characteristics corresponding to the spin structures of the left- and right-moving fermions respectively. ZX± [gzz¯] and Zψ± [gzz¯] are respectively the partition functions of the free variables X± and ψ±, ψ¯± on the worldsheet endowed with the metric ds2 = 2gzz¯dzdz¯. zI(r) denotes the interaction point at which the r-th string interacts. The – 6 – contributions from the ghosts and superghosts are given as Substituting eqs. (2.18), (2.20), (2.21) into (2.17), we can easily see that (2.10) is equal to (2.8). 2.2 Dimensional regularization The amplitudes of superstring theory were calculated using the first-quantized formalism in [22] in which an expression using the PCO’s was given. The expression (2.10) is a special case of the one in [22], where the PCO’s are placed at the interaction points of the light-cone Feynman diagram.2 Unfortunately, the amplitude (2.5) given as an integral of (2.10) or (2.8) is not welldefined. (2.8) diverges when some of the interaction points collide, because TFLC(z) has the TFLC(zI )TFLC(zJ ) ∼ (zI − zJ )3 + · · · , 2 which makes the integral (2.5) ill-defined. This kind of divergence is called the contact Accordingly, the conformal gauge expression (2.10) suffers from the so-called spurious singularity. The holomorphic part of the correlation function of the superghost system has N r=1 2g−2+N Y I=1 N Y K=1 I dz dβdγdβ¯dγ¯ gzAz¯ e−Sβγ heφ (zI ) eφ¯ (z¯I )i Y he−φ (Zr) e−φ¯ Z¯r i A = Zψ± gzz¯ −1 e 12 Γ[σ;gzAz¯] Y e− Re N¯0r0r ∂2ρ (zI ) 2 . 2g−2+N Y I=1 *2g−2+N Y I=1 " ∼ ϑ[αL] − N r=1 eφ (zI ) Y e−φ (Zr) + X Z Zr ω + X Z zI r P0 I P0 QI,r E (zI , Zr) Z △ P0 ω !#−1 Qr σ2 (Zr) . × QI<J E (zI , zJ ) Qr<s E (Zr, Zs) QI σ2 (zI ) Here ω is the canonical basis of the holomorphic 1-forms, △ is the Riemann class, E(z, w) is the prime form of the surface and σ (z) is a holomorphic g2 form with no zeros or poles. The base point P0 is an arbitrary point on the surface.3 This correlation function diverges when 2Notice that in the light-cone setup, the positions of the PCO’s have the fixed coordinate in the coordinate patch on the surface and we do not need ∂ξ terms. 3For the mathematical background relevant for string perturbation theory, we refer the reader to [19]. – 7 – (2.20) (2.21) the spurious singularities. The first type of singularity corresponds to the contact term divergence mentioned above. The second type of singularity is due to existence of zero modes of γ. Singularities of this kind do not arise in our case. Since Zr (r = 1, . . . N ) and zI (I = 1, . . . , 2g − 2 + N ) are the poles and the zeros of the meromorphic one-form ∂ρ (z) dz respectively, PI zI − Pr Zr is a canonical divisor on the surface. Therefore we obtain Z △ P0 ω !#−1 = [ϑ[αL] (0)]−1 , (2.25) which is included in the factor Zψ± −1 in (2.21). [ϑ[αL] (0)]−1 may become singular at some points in the moduli space, but the [ϑ[αL] (0)]−1 cancels the factor ϑ[αL] (0) from the partition function Zψ± of ψ± and the whole amplitude is free from this type of singularity. Therefore, in order to make the amplitudes given in the previous subsection welldefined, we should deal with the contact term divergences. In our previous works, we employ the dimensional regularization to do so. Let us summarize the results: • One can formulate the light-cone gauge superstring field theory in d 6= 10 dimensional space time. The amplitudes are given in the form (2.5) with N spin structure N X p + r r=1 Z ! δ N X p − r r=1 ! e− d1−62 Γ[σ;gˆzz¯] ∂2ρ (zI ) 2 TFLC (zI ) T¯FLC (z¯I ) Y VrLC . (2.26) N r=1 Taking d to be large and negative, the factor e− d1−62 Γ[σ;gˆzz¯] tame the contact term divergences. • More generally we can regularize the divergences by taking the worldsheet superconformal field theory to be the one with central charge c 6= 12. One convenient choice of the worldsheet theory is the one in a linear dilaton background Φ = −iQX1, with – 8 – a real constant Q. The worldsheet action of X1 and its fermionic partners ψ1, ψ¯1 on a worldsheet with metric dsˆ2 = 2gˆzz¯dzdz¯ becomes S X1, ψ1, ψ¯1; gˆzz¯ = dz ∧ dz¯pgˆ gˆab∂aX1∂bX1 − 2iQRˆX1 The amplitude is expressed in the form (2.26) with It was shown in [14] that (with the Feynman iε) by taking Q2 > 10, the amplitudes become finite. • We can define the amplitudes as analytic functions of Q2 and take the limit Q → 0 to obtain those in d = 10. In order to study the limit, it is useful to recast the expression (2.26) into the conformal gauge one [ 13 ] Z dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gzAz¯ e−Stot × 6g−6+2N Y K=1 which looks quite similar to the critical case (2.10).4 The crucial difference is that the worldsheet theory for the longitudinal variables X±, ψ±, ψ¯± is a superconformal field theory called the supersymmetric X± CFT, which has the central charge one can show that the amplitudes in the limit Q → 0 coincide with those given by the Sen-Witten prescription [23], if the latter exists. 2.3 The problems with odd spin structure The light-cone gauge amplitudes can be defined and calculated for odd spin structure, and we get the expression (2.5) with the integrand given by (2.8). The correlation functions 4Notice that the expression here is different from the one in [ 13 ] where the operators I zI(r) 2dπzi S (z, Zr) I z¯I(r) 2πi dz¯ S¯ z¯, Z¯r are inserted in place of e− iαQr2 X+ zˆ˜I(r) , z¯˜I(r) . The properties of operators of this kind with operator valued ˆ ˆ arguments z˜ˆI , z¯˜I are explained in appendix A. – 9 – of free fermions on higher genus Riemann surfaces are given in appendix B. Only the amplitudes with enough fermions from the vertex operators and TF insertions to soak up the zero modes are nonvanishing. However, we have a problem in rewriting the light-cone gauge expression (2.8) into the BRST invariant one (2.10), if we proceed as in the previous section. If α corresponds to an odd spin structure, (2.31) (2.32) (2.33) The correlation function (2.21) of the β, γ system diverges because it involves factors HJEP03(218)6 coming from Zψ± −1 on the right hand side of (2.21). On the other hand, the partition function (2.19) of the ψ± variables involves factors ϑ[α] (0) = 0 . (ϑ[α] (0))−1 , ϑ[α] (0) , 0 × ∞ − 3 which cancel the divergent contribution from the β, γ system. Therefore we need to make sense out of the combination to obtain the BRST invariant expression corresponding to the light-cone gauge amplitudes. 3 Odd spin structure The problem mentioned at the end of the previous section can be avoided by considering the amplitudes with insertions of ψ+, ψ− and δ(β), δ(γ). We would like to show that such insertions can be realized in a BRST invariant way, if we consider the conformal gauge amplitudes taking some of the vertex operators to have 0 or −2 picture, when all the external lines are in the (NS,NS) sector. 3.1 Multiloop amplitudes Let us consider the case where the spin structure αL for the left-moving fermions is odd and αR for the right-moving fermions is even. The case where αL is even and αR is odd or both of αL and αR are odd can be dealt with in the same way. We would like to show that the term in the sum in (2.8) corresponding to such a spin structure can be recast into a conformal gauge expression Z dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gzAz¯ e−Stot × 6g−6+2N Y K=1 I dz the combination 0 × ∞. (Zr, Z¯r) is the (−1, −1) picture vertex operator 2 1 V1(−2,−1) Z1, Z¯1 = − p+ cc¯e−2φe−φ¯ψ+V1DDF Z1, Z¯1 , V (0,−1) Z2, Z¯2 = 2 −cc¯e−φ¯ I dz Z2 2πi TF (z) + 1 4 c¯γe−φ¯ V2DDF Z2, Z¯2 , which satisfy XV1(−2,−1) Z1, Z¯1 = V (−1,−1)(Z1, Z¯1) , 1 XV2(−1,−1)(Z2, Z¯2) = V (0,−1) Z2, Z¯2 , 2 QBV1(−2,−1) Z1, Z¯1 = 0 , Since V DDF Z, Z¯ is expressed as (2.13), V2(0,−1) Z2, Z¯2 can be rewritten as where X is the picture changing operator (2.11) and QB denotes the BRST charge (C.12). (3.3) (3.4) (3.5) (3.6) V (0,−1) Z2, Z¯2 2 ¯¯i1(2) ¯¯j1(2) = −cc¯e−φ¯Ai−1n(21) · · · A−n¯1 · · · B−j1s(12) · · · B−s¯1 · · · + 1 4 1 1 = − 2 cc¯e−φ¯p2+ : ψ−V2DDF Z2, Z¯2 : + · · · , p − 2 − p N2 + 2 ψ+ − pi2ψi e −ip2+X−−i p2−− Np+2 X++ipi2Xi 2 where the ellipses in the last line denote the terms which do not involve ψ−. V (−2,−1) Z1, Z¯1 , V2(0,−1) Z2, Z¯2 can be considered to be the BRST invariant vertex 1 operators in (−2, −1) , (0, −1) pictures respectively. It is straightforward to define vertex operators V (−1,−2,), V (−1,0), or V (−2,−2), V (0,0) which can be used to express the amplitudes for the cases of the other spin structures mentioned above. Y K=1 I dz A poof of this fact can be found in appendix C.2. Therefore, in order to show that (3.1) is proportional to (3.2), we evaluate (3.7) and prove that it is proportional to (3.1). In (3.7), we can replace V2(0,−1) Z2, Z¯2 by 1 − 2 cc¯e−φ¯p2+ : ψ−V2DDF : Z2, Z¯2 , because only this term can soak up the zero mode of ψ−. After such a replacement, (3.7) factorizes into contributions from the ghosts, superghosts, longitudinal modes and the transverse modes. The parts of the transverse variables and ghosts are the same as those in (2.17) and we can use (2.20) to evaluate the latter. In this case, the correlation function of the longitudinal variables is modified from (2.18) into Z dX±dψ±dψ¯± gzAz¯ e−S± ψ+ (Z1) ψ− (Z2) Y VrDDF(Zr, Z¯r) ×hαL (Z1) hαL (Z2) Y αr 1 e− Re N¯0r0r VrLC(Zr, Z¯r) , and that of the superghosts is evaluated to be dβdγdβ¯dγ¯ gzAz¯ e−Sβγ e−2φ (Z1) Y heφ (zI ) eφ¯ (z¯I )i Y e−φ (Zr) Y e−φ¯ Z¯r N r=3 N r=1 It is possible to show that (3.2) is equal to dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gzAz¯ e−Stot N r=1 1 ! 2 P0 P0 + ω Z △ !∗#−1 Z Z2 !#−1 ω Z1 (3.7) (3.8) (3.9) QrN=3 E(Z2, Zr) QI E(zI , Z1) σ2(Z1) × hαL(Z1)hαL(Z2)ϑ[αR] (0)∗ . 1 1 QI<J E (zI , zJ ) Qr<s E (Zr, Zs) QI σ2 (zI ) e−12S Here hαL(z) defined in (B.10) is equal to the zero mode of spin 12 left-moving fermion with spin structure αL. The explicit form of S and its relation to e−Γ[σ;gzAz¯] can be found in [ 13 ]. In the manipulations in (3.10), we have used (2.24) and the following identities: E (Z2, Z1) = ϑ[αL] RZZ12 ω hαL(Z1)hαL(Z2) , QrN=3 E(Z2, Zr) QI E(zI , Z1) σ2(Z1) QI E(zI , Z2) QrN=3 E(Z1, Zr) · σ2(Z2) α1 = − α2 . (3.11) is proved by observing where C is a quantity independent of z. From this expression we can derive |∂ρ (z)|2 = C |σ (z)|4 QI E (z, zI ) 2 , Qr E (z, Zr) α1 = α2 = lim lim z→Z1, w→Z2 (w − Z2) ∂ρ (w) z→Z1, w→Z2 w − Z2 (z − Z1) ∂ρ (z) z − Z1 exp Z z w du ∂ ln |∂ρ (u)|2 = − QI E(zI , Z2) QrN=3 E(Z1, Zr) · σ2(Z2) QrN=3 E(Z2, Zr) QI E(zI , Z1) σ2(Z1) . Combining eqs. (2.20), (3.9) and (3.10), it is straightforward to show that (3.7) is proportional to (3.1). 3.2 Dimensional regularization The amplitudes given by the integral (2.5) with the integrand of the form (3.2) is not well-defined, because of the contact term divergences. In order to make them well-defined, we employ the dimensional regularization illustrated in subsection 2.2. The amplitudes are given in the form (2.5) with the integrand (2.26) with d = 10 − 8Q2. The light-cone gauge amplitudes are finite for Q2 > 10. As in the case of even spin structure, we can (3.10) HJEP03(218)6 (3.11) (3.12) (3.13) Here the worldsheet theory of the longitudinal variables are taken to be the supersymmetric X± CFT. We discuss the correlation functions of the supersymmetric X± CFT for odd spin structures in appendix C.1. As is shown in appendix C.2, this expression is equal to N r=3 N r=3 I Z dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gzAz¯ e−Stot × 6g−6+2N Y K=1 define the amplitudes as analytic functions of Q2 and take the limit Q → 0 to obtain those in d = 10. In order to study the limit, we recast the light-cone gauge expression into a conformal gauge one. The noncritical version of (3.2) is given as Z In (3.15), we can replace V2(0,−1) Z2, Z¯2 by (3.8) for the same reason as that in the critical case. With the replacement, the path integral (3.15) can factorize into those of matter, ghosts and superghosts. For the longitudinal variables, we get from (C.7) Z (3.16) e− iαQr2 X + Combining eqs. (2.20), (3.10) and (3.16), it is straightforward to show that (3.15) is proportional to the light-cone gauge expression in (2.26) with d = 10 − 8Q2. The expression (3.14), summed over spin structures and integrated over the moduli parameters, gives a BRST invariant expression of the amplitude. The operators X, X¯ , ˆ zˆ˜I(r) , z¯˜I(r) , V1(−2,−1), V2(0,−1), Vr(−1,−1) are all BRST invariant and the BRST variations of the antighost insertions yield total derivatives on moduli space. Since (3.14) coincides with (2.26), the amplitude is finite for Q2 > 10. We use the light-cone gauge expression of the amplitude for Q2 > 10 to define it as an analytic function of Q2, which is denoted by ALC Q2 . We would like to see what happens in the limit Q → 0. The conformal gauge expression (3.14) can be deformed to define the amplitudes following the Sen-Witten prescription [23, 24]. We can divide the moduli space into patches and put the PCO’s avoiding the spurious singularities as was explained in [23] and define the amplitude ASW Q2 . Moving the locations of the PCO’s, the amplitudes change by total derivative terms in moduli space. Taking Q2 big enough, these total derivative terms do not contribute to the amplitudes, because the infrared divergences are regularized. Therefore ASW Q2 coincides with ALC Q2 as an analytic function of Q2. Since ASW Q2 is free from the spurious singularities, it can be well-defined for Q2 < 10 and Q→0 lim ALC Q2 = ASW (0) , (3.17) if the right hand side is well-defined. 4 Conclusions and discussions In this paper, we have shown that the Feynman amplitudes of the light-cone gauge closed superstring field theory can be calculated using the dimensional regularization technique, for higher genus Riemann surfaces with odd spin structure, if the external lines are in the (NS,NS) sector. In order to deal with the fermion zero modes peculiar to odd spin structures, we need to change the pictures of the vertex operators in the conformal gauge expression. We obtain the amplitudes in noncritical dimensions which coincide with the ones defined by using the Sen-Witten prescription. The amplitudes in the critical dimensions correspond to the limit d → 10 or Q → 0, and the results coincide with those given by the Sen-Witten prescription. There are several things remain to be done. One is to check how the amplitudes obtained by our procedure are related to the standard results in more detail. In particular, we should study the conditionally convergent integrals which appear in the Feynman amplitudes of superstrings. We expect that our regularization makes the integrals well-defined but in a way different from those in [25, 26]. Another thing to be done is to generalize our results to the amplitudes with external lines in the Ramond sector. With the correlation functions involving spin fields given for example in [27], it will be straightforward to rewrite the light-cone gauge expression into the conformal gauge one. These problems are left to future work. Acknowledgments N.I. would like to thank Ashoke Sen for useful comments. He would also like to thank the organizers of “Recent Developments on Light Front” at Arnold Sommerfeld Center for Theoretical Physics and “SFT@HIT” at Holon, especially Ivo Sachs, Ted Erler, Sebastian Konopka and Michael Kroyter, for hospitality. This work was supported in part by Grantin-Aid for Scientific Research (C) (25400242) and (15K05063) from MEXT. HJEP03(218)6 Operator valued coordinate It is convenient to introduce the operator valued coordinate zˆI and its supersymmetric version z˜ˆI , in order to express the conformal gauge form of the amplitudes for noncritical dimensions. fluctuation as Let us first consider zˆI which is defined in the bosonic case. In the light-cone gauge setup, we consider the situation where the variable X+ (z, z¯) possesses an expectation value − 2i (ρ (z) + ρ¯(z¯)). Therefore we decompose it into the expectation value and the i X+ (z, z¯) = − 2 (ρ (z) + ρ¯(z¯)) + δX+ (z, z¯) . T (z)zˆI ∼ − z − zˆI where δ(n)zI is at the n-th order in the derivatives of δX+, in principle we can obtain δ(n)zI if ∂2ρ (zI ) 6= 0. Lower order examples are given by δ(1)zˆI = − ∂2ρ ∂δX+ (zI ) , 2i 4 δ(2)zˆI = − (∂2ρ)2 ∂δX+∂2δX+ (zI ) + 2∂3ρ (∂2ρ)3 ∂δX+ 2 (zI ) . In general δ(n)zI becomes a polynomial of the derivatives of δX+ at z = zI . Quantities of order n with n > N for some N > 0 do not contribute to the correlation functions we consider in this paper. z¯ˆI , which is the antiholomorphic counterpart of zˆI , can be obtained in the same way. The OPE of zˆI with the energy-momentum tensor T (z) comes from the contractions of ∂X− in T (z) with ∂kδX+ (zI ) in zˆI . Taking the OPE of (A.2) with T (z), we get T (z)zˆI ∂2X+ (zˆI ) + 1 (z − zˆI )2 ∂X+ (zˆI ) + 1 z − zˆI ∂2X+ (zˆI ) = 0 . Using (A.2) and the fact that ∂2X+ (zˆI ) = − 2i ∂2ρ (zI ) + · · · is invertible perturbatively, Roughly speaking, we define zˆI to be an operator valued coordinate which satisfies Substituting (A.1) into (A.2), we get ∂X+ (zˆI ) = 0 . i We take zˆI so as to coincide with zI when δX+ = 0. Assuming that zˆI is expanded in terms of the fluctuation δX+ as (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) Expanding zˆI as in (A.4), we can see that the right hand side of (A.7) involves poles of arbitrarily high order at z = zI . Only finite number of them are relevant in the correlation functions (2.29), (3.14) and (3.15). With the OPE (A.7) and its antiholomorphic version, we can show the OPE’s for any constant α. Therefore eiαX+ zˆI , z¯ˆI is a BRST invariant operator in the conformal gauge bosonic string theory in noncritical dimensions. ˆ zˆ˜I = z˜ˆI , θ˜I to be the operator valued supercoordinate which satisfies It is straightforward to define the operator valued supercoordinate z˜ˆI . We define T (z) eiαX+ zˆI , z¯ˆI ∼ regular , T¯ (z¯) eiαX+ zˆI , z¯ˆI ∼ regular , ∂X ∂DX + zˆ˜I + zˆ˜I = 0 , Notice that z˜ˆI is the operator version of the supercoordinate z˜I defined in [7, 28, 29] which is not superconformal, but it is sufficient for our purpose. Similarly to the bosonic case, we decompose X + (z, z¯) as X + (z, z¯) = − 2 i (ρs (z) + ρ¯s (z¯)) + δX + (z, z¯) , where ρs, ρ¯s are the supersymmetric version of ρ, ρ¯ whose explicit form is given in (C.5). Sα in that equation is the one in (B.5) or in (B.12) according to whether the spin structure of the fermions is even or odd. Using this decomposition, we get z˜ˆI , θ˜I as expansions around ˆ z˜I , θ˜I in terms of the fluctuation δX + as z˜ˆI = z˜I + X δ(n)zˆ˜I , θ˜I = θ˜I + X δ(n)θ˜ˆI , ˆ ∞ n=1 ∞ n=1 assuming ∂2ρ (z˜I ) 6= 0. For example, 2i δ(1)zˆ˜I = − ∂2ρ ∂δX + (z˜I ) , 2i δ(1)θ˜ˆI = − ∂2ρ ∂DδX + (z˜I ) . (A.8) (A.9) (A.10) (A.11) (A.12) We obtain the OPE’s From these OPE’s, we get operator which can be replaced by ˆ for any constant α. Therefore eiαX + zˆ˜I , z˜¯I is BRST invariant in the conformal gauge superstring theory in noncritical dimensions. Using the operator valued coordinate thus defined, we can define a BRST invariant T (z)z˜ˆI − T (z)θ˜ˆI θˆ˜I ∼ − T (z)θ˜ˆI ∼ − − ˆ ˜ θ − θI z − z˜I ˆ − 1 2 z − z˜I ˆ − 1 ˆ z − z˜I 1 ˆ z − z˜I θ − θI ˆ ˜ ˆ z − z˜I DX + 2 · 2∂2X + DX + 3 · ∂2X + ω − Xn Z yi ! ω − Xn Z yi ω! (A.13) HJEP03(218)6 (A.14) (A.15) (A.16) (B.1) on higher genus Riemann surfaces. αR for right can be given by [19, 31] Z hdψdψ¯dψ¯†dψ†i gˆzz¯ = det Im Ω R d2z√gˆ − 1 2 in evaluating (2.29), (3.14) and (3.15). This expression can be used instead of the complicated combination Hz(r) 2dπzi Hz¯(r) 2dπz¯i S (z, Zr) S¯ z¯, Z¯r in [ 13, 30 ], which has a similar effect. I I B Correlation functions of free fermions In this appendix, we review a few basic facts about the correlation functions of free fermions The correlation functions of a free Dirac fermion with spin structure αL for left and When both αL and αR correspond to even spin structures, using the formula ϑ ω − Xn Z yi ω − e! ϑ (e)n−1 Qi<j [E (xi, xj) E (yj, yi)] Qi,j E (xi, yj) = det   ϑ RPx0i ω − RPy0i ω − e  E (xi, yj)  , Sα (z, w) = E (z, w) ϑ [α] (0) 1 ϑ [α] Rw ω z is the Szego kernel. The expression (B.4) implies that the partition function is given by Zψ[gzz¯] 2 = 2 and the propagators of the fermions are given in [32] for the case = where gˆzz¯ eν = − Ωα′ + α′′ ν , it is straightforward to show that (B.1) can be transformed into e−Sψ†(x1)ψ¯†(x¯1) · · · ψ†(xn)ψ¯†(x¯n)ψ¯(y¯n)ψ(yn) · · · ψ¯(y¯1)ψ(y1) ϑ[αL] (0) ϑ[αR] (0)∗ det [SαL (xi, yj)] det [SαR (xi, yj)∗] , (B.4) (B.2) (B.3) ψ†(x)ψ(y) = SαL (x, y) , ψ¯†(x¯)ψ¯(y¯) = SαR (x, y)∗ . eν = − ΩαL′ + αL′′ ν − Z p q ων , When the spin structures are not even, we need to take care of the fermion zero modes. For example, let us consider the case where αL corresponds to an odd spin structure and αR corresponds to an even one. In this case, using the formula (B.2) for in the limit q → p, we get [32] ϑ[αL] ω − Xn Z yi ω! Qi<j [E(xi, xj)E(yj, yi)] Qi,j E(xi, yj)  = Z dψ0dψ0† det ψ0hαL(xi)ψ0hαL(yj) + † 1 E (xi, yj) Pν ∂νϑ [αL] Ryxji ω ων(p)  hαL(z) = sX ∂νϑ [αL] (0) ων(z) ν (B.10) gives the zero mode of the fermion. Substituting (B.9) into the right hand side of (B.4), gˆzz¯ e−Sψ†(x1)ψ¯†(x¯1) · · · ψ†(xn)ψ¯†(x¯n)ψ¯(y¯n)ψ(yn) · · · ψ¯(y¯1)ψ(y1) (B.11) − 21 Z dψ0dψ0† det [SαL (xi, yj)] ϑ[αR] (0)∗ det [SαR (xi, yj)∗] , where † SαL(x, y) = ψ0hαL(x)ψ0hαL(y) + 1 E (x, y) Pν ∂νϑ [αL] Ry ω ων(p) x . (B.12) SαL(x, y) can be identified with the propagator of the left-moving fermions and it involves the zero mode variables ψ0†, ψ0. ψ0† and ψ0 should be integrated over after all the contractions are performed. The other cases where αR corresponds to an odd spin structure can be dealt with in the same way. C Dimensional regularization for odd spin structure In this appendix, we explain the details of how dimensional regularization works in the case of odd spin structure. C.1 Supersymmetric X± CFT In order to get the expression of the amplitudes in the conformal gauge, we need to calculate the correlation functions of the supersymmetric X± CFT on the surface with odd spin structure. The action of the supersymmetric X± CFT is given in the form Ss±uper gˆzz¯, X ± = Sfree gˆzz¯, X ± + Γsuper gˆzz¯, 2iX + , (C.1) where Sfree [gˆzz¯, X ±] denotes the free action of X ±. When the spin structures are both even, the correlation functions of the supersymmetric X± CFT are evaluated as [ 13 ] Z δ X p + r r ! ZsXuper[gˆzz¯] 2 ×e− d−810 Γsuper[gˆzz¯, ρs+ρ¯s] Y e−pu− ρs+ρ¯s (wu, w¯ u) . 2 Regarding the second and third lines as a correlation function of of the free theory with the source term e− d−810 Γsuper[gˆzz¯, 2iX +] Y e−ipu−X + (wu, w¯ u) N r=1 Y e−ipr+X − (Zr, Z¯ r) , N r=1 N M u=1 × e− d−810 Γsuper[gˆzz¯, 2iX +] Y e−ipu−X + (wu, w¯ u) HJEP03(218)6 (C.2) (C.3) (C.4) (C.5) (C.6) we can calculate it by replacing the X + (z, z¯) by its expectation value − 2i (ρs (z) + ρ¯s (z¯)) and derive the fourth line. Here ρs, ρ¯s are the supersymmetric version of ρ, ρ¯ and expressed as ρs (z) = ρ (z) − θ X αrΘrSαL (z, Zr) , ρ¯s(z¯) = ρ¯ (z¯) − θ¯X αrΘ¯ rSαR z¯, Z¯r , r r ZsXuper[gˆzz¯] 2 = ZX± [gˆzz¯] Zψ± [gˆzz¯] . ZsXuper[gˆzz¯] 2 is described by using ZX± [gˆzz¯] and Zψ± [gˆzz¯] in (2.19) as where SαL and SαR are taken to be the Szego kernel (B.5). The partition function The explicit form of e− d−810 Γsuper[gˆzz¯, ρs+ρ¯s] can be found in [ 13 ]. In the case where αL corresponds to an odd spin structure, we can proceed in the same way. Since the correlation functions of the free fermions are given in (B.11) as an integral over the zero modes ψ0±, we obtain Z gˆzz¯ u × ! N r=1 ! u e−Ss±uper[gˆzz¯] Y e−ipr+X − (Zr, Z¯ r) Y e−ipu−X + (wu, w¯ u) M u=1 X p + r ZX± [gˆzz¯] where ρs, ρ¯s in this formula are (C.5) with SαL given in (B.12) and SαR taken to be the Szego kernel (B.5). With the correlation function (C.7), it is straightforward to check the following prop HJEP03(218)6 erties of the energy-momentum tensor: • T X ± (z) is regular at z = zI . • T X ± (z) satisfies θ − Θr T X ± (z) e−ipr+X −−ipr−X + (Zr, Z¯ r) ∼ (z − Zr)2 −pr+pr− e−ipr+X −−ipr−X + (Zr, Z¯ r) + + 1 1 z − Zr 2 z − Zr De−ipr+X −−ipr−X + (Zr, Z¯ r) θ − Θr ∂e−ipr+X −−ipr−X + (Zr, Z¯ r) . (C.8) • The OPE between X −’s is given by DX − (z) DX − z′ ∼ − θ − θ z − z′ z − z′ − − + 1 2 (∂X +)2 + ∂DX + − (∂X +)3 − ∂2X + 2 (∂X +)3 + + ∂2DX 2 (∂X +)3 + 4∂DX +DX + (∂X +)4 5∂2X +DX 2 (∂X +)4 + 2∂2DX +DX + (∂X +)4 3∂2X +∂DX + 2 (∂X +)4 z ′ z ′ ∂2X + 2 DX + (∂X +)5 − ∂2DX +∂DX +DX (∂X +)5 + ! z ′ # 8∂2X +∂DX +DX + (∂X +)5 + ∂3X +DX 2 (∂X +)4 z ′ (C.9) and we can deduce that the energy momentum tensor T X ± (z) satisfies the OPE T X ± (z) T X ± z′ 12 − d ∼ 4 (z − z′)3 + (zθ−−zθ′′)2 23 T X ± z′ + 1 1 z − z′ 2 DT X ± z′ + θ − θ′ z − z′ ∂T X ± which corresponds to the super Virasoro algebra with the central charge cˆ = 12−d. It follows that combined with the transverse variables Xi (z, ¯z), the total central charge of the system becomes cˆ = 10. This implies that with the ghost superfields B (z) and C (z) defined as B(z) = β(z) + θb(z) , C(z) = c(z) + θγ(z) , it is possible to construct a nilpotent BRST charge QB = I dz 2πi −C T X ± 1 DX i∂X i + C∂C − 4 1 (DC)2 B . (C.10) z′ , (C.11) (C.12) These properties can be proved in the same way as in the even spin structure case, because we need only the behaviors of the fermion propagators around the singularities to do so. In the same way as in the even spin structure case [ 13 ], we can derive from (C.7), Z dX +dX − gˆzz¯ e−Ss±uper[gˆzz¯] Y e−ipr+X−(Zr, Z¯r) Y e−ips−X+(ws, w¯s) M s=1 N r=1 ! × ψ+ (u1) · · · ψ+ (un) ψ− (v1) · · · ψ− (vm) × ψ¯+ (u˜1) · · · ψ¯+ (u˜n) ψ¯− (v˜1) · · · ψ¯− (v˜m) ! − δ s X p + r ZX± [gˆzz¯] Y e−ps− 21 (ρ+ρ¯)(ws, w¯s)e− d−1610 Γ[σ;gˆzz¯] s × Z dψ+dψ−dψ¯+dψ¯− gˆzz¯ e π1 R d2z(ψ−∂¯ψ++ψ¯−∂ψ¯+)−Sint × ψ+ (u1) · · · ψ+ (un) ψ− (v1) · · · ψ− (vm) × ψ¯+ (u˜1) · · · ψ¯+ (u˜n) ψ¯− (v˜1) · · · ψ¯− (v˜m) , (C.13) where Sint = d − 10 " 8 − X 2 ∂ψ+ψ+ r αr ∂2ρ (zI(r)) + X I # +c.c. . ( 5 ∂4ρ 3 (∂2ρ)3 − 3 ∂3ρ 2 ! (∂2ρ)4 (∂2ρ)3 + 4∂3ρ ∂2ψ+ψ+ + 3 (∂2ρ)4 (zI ) (C.14) The path integral over ψ±, ψ¯± can be computed by treating Sint perturbatively. Since Sint involves only ψ+, the perturbation series terminates at a finite order. A proof of equality of (3.14) and (3.15) In this appendix, we show that (3.14) is equal to (3.15). In the case Q = 0, this equality implies that (3.2) is equal to (3.7). Proving this can be done by using a fermionic charge5 Qˆ′ ≡ I dz b 2πi − 4∂ρ iXL+ − 2 1 ρ (z) + ψ+ (z) , β 2∂ρ and its antiholomorphic counterpart Q¯ˆ′. Here iXL+ − 12 ρ (z) is defined as iXL+ − 2 1 ρ (z) ≡ Z z w0 dz′ i∂X+ 1 z′ , with a generic point w0 on the surface. iXL+ − 2 1 ρ (z) thus defined is single valued on the surface in the correlation functions we consider here because − 2i ρ coincides with the expectation value of XL+ in the presence of the sources e−ipr+X−. In order to use Qˆ′, we need to rewrite the ghost part of the correlation function. Inserting (C.15) (C.16) dz b w0 2πi ∂ρ (z) ∂ρc (w0) 2 (C.17) into (3.2) and deforming the contours of the antighost insertions, (3.2) is transformed into Z dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gzAz¯ e−Stot I αj ∂ρ bzz + I dz¯ αj ∂¯ρ¯bz¯z¯ ! I dz βj ∂ρ bzz + I dz¯ βj ∂¯ρ¯bz¯z¯ !# I dz b Y hVr(−1,−1)(Zr, Z¯r)i . (C.18) N r=3 Here αj and βj are chosen so that they form a canonical basis of the first homology group of the Riemann surface. Using Qˆ′, the operators inserted at z = zI can be expressed as I dz b = − + I ( dz b dz b dz b 4 zI 2πi ∂ρ (z) I r r r dw A (w) eφ (zI ) Y′ e− iαQr2 X + zˆ˜I(r), z¯˜I(r) ˆ ) dw ∂ρψ− (w) eφ (zI ) Y′ e− iαQr2 X + zˆ˜I(r), z¯˜I(r) . (C.19) ˆ Here the prime in Y ′ means that the product is taken over those r which satisfy zI(r) = zI , and A (w) = −i∂X+∂ργ (w) − 2∂ (∂ρc) ψ− (w) − 4 i d − 10 " 5 ∂2X+ 2 ∂3X+ ! 4 (∂X+)3 − 2 (∂X;)2 (−2∂ργ) − (∂X+)2 ∂ (−2∂ργ) + ∂2 (−2∂ργ) ∂X+ − (−2∂ργ) ∂ψ+∂2ψ+ 2 (∂X+)3 2∂2X+ (w) . Substituting (C.19) into (C.18) and using the commutators (C.20) (C.21) nQˆ′, c (w0)o = 0 , nQˆ′, b (z)o = 0 , hQˆ′, Vr(p,−1)i = 0 Qˆ′, I dz b dz b = 0 , zI 2πi w − zI (z) I dw ∂ρψ− (w) eφ (zI ) Y′ e− iαQr2 X + zˆ˜I(r), z¯˜I(r) ˆ ) = 0 , (C.22) we can show that (C.18) is equal to Z dXμdψμdψ¯μdbd¯bdcdc¯dβdβ¯dγdγ¯ gzAz¯ e−Stot ×∂ρc (w0) ∂¯ρ¯c¯(w¯0) g " I × Y j=1 × Y I × Y I dz b I zI 2πi ∂ρ I dz αj ∂ρ bzz + I dz¯ αj ∂¯ρ¯bz¯z¯ ! I dz βj ∂ρ bzz + I dz¯ βj ∂¯ρ¯bz¯z¯ !# (z) −eφTFLC (zI ) + 1 I 4 zI 2πi w − zI dw ∂ρψ− (w) eφ (zI ) The correlation functions of ψ± which appear in (C.23) can be calculated using (C.13) treating Sint perturbatively. Since ∂ρ (zI ) = 0, HzI 2πi w−zI dw ∂ρψ−(w) vanishes unless ψ− (w) be N r=3 the ψ− (w) in H dw ∂ρwψ−−z(Iw) , the contour integral over w vanishes. Therefore only the terms which involve contraction of the ψ+ (Z1) in V V (0,−1) Z2, Z¯2 survive. The ψ− (w) in HzI 2πi 2 dw ∂ρψ−(w) should be contracted with ∂nψ+ (zI ) 1 (−2,−1) Z1, Z¯1 with the ψ− (Z2) in w−zI involved in e− iαQr2 X + ˆ zˆ˜I(r) , z¯˜I(r) or Sint but doing so induces another contraction I dw 1 zJ 2πi w − zJ ∂ρψ− (w) ∂nψ+ (zI ) , (C.24) with zJ 6= zI , because e− iαQr2 X + ˆ zˆ˜I(r) , z¯˜I(r) conclude that HzI 2πi w−zI dw ∂ρψ−(w) in (C.23) does not contribute to the path integral. We can do the same thing for the antiholomorphic part and prove that the X (zI ) , X¯ (z¯I ) which appear in (C.18) can be replaced by −eφT LC (zI ) , −eφ¯T¯LC (z¯I ) for all I. By deforming the F F contours of the antighost insertions back, we can see that (3.14) is equal to (3.15). and Sint are Grassmann even. Hence we Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. JHEP 06 (2015) 022 [arXiv:1411.7478] [INSPIRE]. [3] A. 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Nobuyuki Ishibashi, Koichi Murakami. Multiloop amplitudes of light-cone gauge superstring field theory: odd spin structure contributions, Journal of High Energy Physics, 2018, 63, DOI: 10.1007/JHEP03(2018)063