Multiloop amplitudes of light-cone gauge NSR string field theory in noncritical dimensions

Journal of High Energy Physics, Jan 2017

Feynman amplitudes of light-cone gauge superstring field theory are ill-defined because of various divergences. In a previous paper, one of the authors showed that taking the worldsheet theory to be the one in a linear dilaton background Φ = −iQX 1 with Feynman iε (ε > 0) and Q 2 > 10 yields finite amplitudes. In this paper, we apply this worldsheet theory to dimensional regularization of the light-cone gauge NSR superstring field theory. We concentrate on the amplitudes for even spin structure with external lines in the (NS,NS) sector. We show that the multiloop amplitudes are indeed regularized in our scheme and that they coincide with the results in the first-quantized formalism through the analytic continuation Q → 0.

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Multiloop amplitudes of light-cone gauge NSR string field theory in noncritical dimensions

Received: December Multiloop amplitudes of light-cone gauge NSR string Nobuyuki Ishibashi 0 1 2 4 5 Koichi Murakami 0 1 2 3 5 0 Otanoshike-Nishi 2-32-1 , Kushiro, Hokkaido 084-0916 , Japan 1 Tsukuba , Ibaraki 305-8571 , Japan 2 Faculty of Pure and Applied Sciences, University of Tsukuba 3 National Institute of Technology, Kushiro College 4 Center for Integrated Research in Fundamental Science and Engineering (CiRfSE) 5 Open Access , c The Authors Feynman amplitudes of light-cone gauge superstring eld theory are ill-de ned because of various divergences. In a previous paper, one of the authors showed that taking the worldsheet theory to be the one in a linear dilaton background String Field Theory; BRST Quantization; Conformal Field Models in String - Feynman i" (" > 0) and Q2 > 10 yields nite amplitudes. In this paper, we apply this iQX1 with the analytic continuation Q ! 0. Contents 1 Introduction 2 Superstring eld theory in linear dilaton background 2.1 2.2 Linear dilaton background dilaton background 3 BRST invariant form of the amplitudes The prescription 4.2 Q ! 0 limit of the light-cone gauge amplitudes 5 Conclusions and discussions B Supersymmetric X CFT C.1 A formula for the superghosts C.2 A formula for the reparametrization ghosts D A proof of (3.9) Introduction 14], using the light-cone gauge closed NSR superstring eld theory, we have studied the to the worldsheet theory. theory in a linear dilaton background iQX1 is considered so that the central charge of light-cone gauge superstring eld theory in the linear dilaton background are indeed CFT constructed in [9] with the identi cation cone gauge superstring symmetric X rst-quantized supersymmetric X dices C and D. Superstring eld theory in linear dilaton background background constructed in [20]. The string eld theory is given for Type II superstring lar way. from the rst-quantized approach. dilaton background its fermionic partners 1 1 Z ; 1; g^zz = dz ^ dzpg^ g^ab@aX1@bX1 2iQR^X1 transformations, are given as dz ^ dzi T X1 (z) = (@X1)2 @ ln g^zz)@X1 (@ ln gzz)2 + @2 ln gzz ; TFX1 (z) = @X1 1 + Q(@ @ ln g^zz) 1 a free theory, we concentrate on the bosonic part. De ning iQ ln(2gzz) ; ; N ) can be calculated on a Riemann given as = 2 X pr + 2Q(1 1 12Q2 [ ;gzAz] ZX gzz 24 A e S[X1;gzz] Y eiprX~ 1 (Zr; Zr) Y e prpsGA(Zr;Zs) Y Arakelov metric (C.22), and S X1; gab = = ln gzz [ ; gab] = 1 Z dz ^ dzpg gab@aX1@bX1 2iQRX1 ; dz ^ dzpg gab@a @b + 2R of the linear dilaton conformal eld theory. Notice that X~ 1 satis es c = 1 12Q2 1 p2 + Qp = (p + Q)2 @@X~ 1 = 0 ; i@X~ 1(z) = i@X~ 1(z) = where n1 and given as linear combinations of the Fock space states hp1jp2i = 2 (p1 + p2 + 2Q) ; n + 2Q n;0) ; n + 2Q n;0) ; where hpj, plane as are the BPZ conjugates of jpi, n1 ; n1 respectively. On the sphere, = 2 e S[X1;gzz] Y eiprX~ 1 (Zr; Zr) action in the background. theory with the variables ; i (i = 1; where the action for X1, 1 eld theory with central charge The string eld c = 12 12Q2 : t = x+ ; = 2p+ : and a function of L0) j (t; )i = 0 ; The action of the string eld theory is given by [7, 11] S = 1 X Z 1 2 B 1 X Z 1 d 2 F B1;B2;B3 B1;F2;F3 L0 + L0 L0 + L0 1 + Q2 1 + Q2 h V3j ( 1)i j ( 2)i j ( 3)i denotes the BPZ conjugate of j ( string vertices and gs is the string coupling constant. P ) . The third and the fourth terms are the three F denote the sums over bosonic and fermionic string interaction depicted in can introduce a complex coordinate whose real part sponding to the r-th external line so that ds2 = d d ; dilaton background eld theory in linear the vertex are given by the worldsheets depicted in gure 2. Each term in the expansion depicted in gure 3. moduli space of as [28, 29] (g) = (igs)2g 2+N C [dT ][ d ][d ] F N(g) ; [dT ][ d ][d ] = 2g 3+N 3g 3+N I=1 A's denote the can introduce a complex coordinate in the same way as the complex coordinate (2.18) is on the light-cone diagram worldsheet, as F (g) = (2 ) 2g 2+N 2 (zI ) 2 TFLC (zI ) TFLC (zI ) path integral measure dXid i = ln @ @ [ ; g^zz] = ln g^zz ; 1 Z dz ^ dzpg^ g^ab@a @b + 2R^ corresponds to the state in the (NS,NS) sector, the light-cone vertex VrLC is given as VrLC = I dwr i@X~ i1 (wr)wr n1 I dwr j1 (wr)wr s1 12 I dwr i@X~ i1 (wr)wr n1 I dwr j1 (wr)wr s1 12 iQ i1 ln(2gzz) ; local coordinate z on X nk + X sl = It is possible to calculate the right hand side of (2.23). can be given as a function of (z) = r ln E(z; Zr) r = 0 ; is the period matrix.2 The base point P0 is arbitrary. There are 2g 2 + N zeros of @ and we denote them by zI (I = 1; obtain e [ ; g^zz]. We can Taking g^zz to be the Arakelov metric [27], e in [13] to be [ ;gzAz] for higher genus surfaces is calculated [ ;gzAz] / e W Y e 2 Re N0r0r Y 2 X GA (zI ; zJ ) 2 X GA (Zr; Zs) + 2 X GA (zI ; Zr) + 3 X ln 2gzAI zI ; s ln E(Zr; Zs) + ! : (2.32) interaction points and the degenerations of the surface regularization of eld theory. BRST invariant form of the amplitudes function of the conformal gauge variables X ; becomes socalled X = Y reRe N0r0r e Q22 [ ;g^zz] S (z; Zr) ZX [g^zz]Z [g^zz] S z; Zr VrDDF(Zr; Zr) : Here X S (z; w) is de ned to be S(z; w) (z) e i Qr2 (XL+(z) XL+(w)) ; vertex operator given by VrDDF(Zr; Zr) = Ai1n(r1) i pr Npr+r X++ipirX~ i (Zr; Zr) ; Ai(rn) = Bi(sr) = product SSVrDDF is normal ordered as where X~i is the super eld for X~ i; i; i, is de ned in (B.5) and XL+ denotes the left S (z; Zr) S z; Zr VrDDF Zr; Zr w!Zr wl!imZr S (z; w) S (z; w) VrDDF Zr; Zr j lim 3 2 (zI ) 2 e 21 [gzAz; ln j@ j2] ZX [gzAz]Z [gzAz] TFLC (zI ) TFLC (zI ) S (z; Zr) S z; Zr VrDDF(Zr; Zr) : (3.7) { 10 { the amplitude (2.21) can be rewritten as [dT ][d ][ d ] dX d dbdbdcdcd d d d gzAz e Stot S (z; Zr) bzz + "K e TFLC (zI ) e TFLC (zI )i VrDDF(Zr; Zr) : (3.8) and the ghosts. It is [dT ][d ][ d ] dX d dbdbdcdcd d d d gzAz e Stot S (z; Zr) bzz + "K X (zI ) X (zI ) VrDDF(Zr; Zr) ; (3.9) X (z) = c@ [dT ][d ][ d ] dX d dbdbdcdcd d d d gzAz e Stot bzz + "K S (z; Zr) { 11 { X (zI + 2 ) X (zI + 2 ) VrDDF(Zr; Zr) : S (z; Zr) S (z; Zr) @DC (z) e i Qr2 (XL+(z) XL+(Zr)) I VrDDF(Zr; Zr) VrDDF(Zr; Zr) @DC (z) e i Qr2 (XR+(z) XR+(Zr))cce VrDDF(Zr; Zr) ; lim X (zI + 2 ) X (zI + 2 ) lim X (zI + 2 ) X (zI + 2 ) S (z; Zr) VrDDF(Zr; Zr) S (z; Zr) X (zI ) S z; Zr X (zI ) VrDDF(Zr; Zr) : lim X (zI + 2 ) X (zI + 2 ) VrDDF(Zr; Zr) VrD0DF(Zr0; Zr0) S (z; Zr) S (z; Zr0) X (zI ) S (z; Zr) S (z; Zr) S (z; Zr0) S (z; Zr0) S z; Zr0 X (zI ) { 12 { is proved in appendix D. [dT ][d ][ d ] dX d dbdbdcdcd d d d bzz + "K VrDDF(Zr; Zr) : X (zI ) X (zI ) VrDDF(Zr; Zr)cce VrD0DF(Zr0 ; Zr0 ) : su ers from the contact term divergences. The amplitudes from the rst-quantized formalism established. by the method of these papers. The prescription K times X is denoted by (m; a) with m 2 M; a 2 ' : X ! M which maps (m; a) to m. (m) and de ne a map !n(m; s(m)) ; where the integrand is schematically expressed as [24] !n(m; z1; Y (X(zi) 2 @(pggij ) bij dms BRST invariant vertex operators, m1; ; mn are the coordinates of M and the subscript detail in [26], it has been shown that 1. One can pick a dual triangulation of M such that the map ' : X ! M has a local section s over each of the codimension 0 polyhedron M depicted in gure 4. ; VN are 2. The amplitude can be given as !n(m; s (m)) + Avertical ; the vertex operators, over @M and their submanifolds which are called the vertical segments, as long as the bad points are avoided. 4. The amplitude thus de ned is gauge invariant. { 14 { S (z; Zr) bzz + "K X (zI ) X (zI ) VrDDF(Zr; Zr) : Q ! 0 limit of the light-cone gauge amplitudes A(Ng)(Q) = !n(m; s(m)) ; and we obtain A(Ng)(Q) = A(Ng)SW(Q) ; lim A(Ng)(Q) = A(Ng)SW(0) ; { 15 { !n(m; s(m)) = dmZ1 ^ dm2 ^ dX d dbdbdcdcd d d d but may su er from the spurious singularities otherwise. can de ne A(Ng)SW(Q) = X Z !n(m; s0 (m)) + Avertical ; Conclusions and discussions the theory in a linear dilaton background iQX1. The divergences of the amplitudes c = 12 supersymmetric X worldsheet theory by setting Q2 in the action of the X CFT given in (B.1). extension elsewhere. gauge closed superstring have a gauge invariant string eld theory to which our method here is applicable. The Acknowledgments { 16 { Let zz be We note that which follows from The Arakelov metric on , is de ned so that its scalar curvature RA 2gAzz@@ ln gzAz satis es and (15K05063) from MEXT. !(z) : dz ^ dz i zz = 1 ; ! ^ ! = dsA2 = 2gzAzdzdz ; gzAzRA = de ned to satisfy4 @z@zGA(z; z; w; w) = 2 Let F (z; z; w; w) be the which satis es @z@z ln F (z; z; w; w) = 2 which can be given by F (z; z; w; w) = exp 2 Im GA(z; z; w; w) = ln F (z; z; w; w) 4The delta function 2(z w) is normalized by w) = 1. 2gzAz = lim exp GA(z; z; w; w) Supersymmetric X 1 Z Here the supercoordinate z is given by the super eld X is de ned as z = (z; ) ; supersymmetric X CFT whose action is given by Ssuper X ; g^zz = + DX super X +; g^zz : (B.1) (z; z) = X (z) + i (z) + i d (Rez) d (Imz) d d : The interaction term super is given by super X +; g^zz = (z; z) = ln + (z) = 1 Z ln g^zz ; which is the super Liouville action de ned for variable with the background metric T X (z) = S(z; XL+) ; where S(z; XL ) denotes the super Schwarzian derivative S(z; XL ) = { 18 { Y e ipr+X (Zr; Zr) Y e ips X + (ws; ws) ZsXuper[g^zz] 2 the theory (B.1) with the identi cation ZsXuper[g^zz] = [dX ]g^zz exp 1 Z d2zDX DX Zr = (Zr; r) ; ws = (ws; s) : version of (z) in (2.30) which is de ned by s (z) = (z) + f (z) ; r rS (z; Zr) ; S (z; w) = E (z; w) all the external lines are in the NS-NS sector and is an even spin structure, S (z; w) is equal to the so-called Szego kernel 0 + 00 ( 0; 00 2 Rg), given by #[ ]( j ) = 2 Cg=(Zg + Zg) . The right hand side of (B.8) can be calculated to be Y e ps s +2 s (ws; ws) e d 810 super[ s+ s;g^zz] ; (B.15) Y e ipr+X (Zr; Zr) Y e ips X + (ws; ws) = (2 ) { 19 { super [ s + s; g^zz]) = exp [ ; g^zz] r = I = 1 @f f (zI(r) ) + c:c: ; 1 @3f @2f @f f @f f + (zI ) + c:c: ; functions of the X the correlation functions of fermions , which is useful in appendix D. (B.15) can be rewritten as5 = (2 ) Y e ipr+X +pr+( r )(Zr; Zr) Y e ips X++ps ( s ++ s +)(ws; ws) Y e ps 12 ( + )(ws; ws)e d 10 16 Z [g^zz] )(Zr; Zr) Y eps ( s ++ s +)(ws; ws) ; (B.19) Sint = + c:c: : = (2 )2 the following identity Y e ipr+X (Zr; Zr) Y e ips X+(ws; ws) + (u1) Z [g^zz] 2 Z + (u~1) + (u~n) Y e ps 12 ( + )(ws; ws)e d1610 e 1 R d2z( @ ++ @ +) Sint + (u~1) + (u~n) (v~m) : (B.21) 4 @3 +@2 +@ + + ) CFT coincide Let us consider the conformal eld theory with the action 1 Z dz ^ dzpg brzc + brzc ; 2 Z or We de ne = 1 to be where the elds b; c are with conformal weight ( ; 0) ; (1 ; 0) and b; c are their antiholomorphic counterparts with conformal weight (0; ); (0; 1 ). Here we consider the case if b; c are Grassmann odd 1 if b; c are Grassmann even { 21 { There exist local operators eq (z; z) q 2 Z2 , which satisfy b (z) eq (w; w) c (z) eq (w; w) b (z) eq (w; w) c (z) eq (w; w) X qi = Y e qi (zi; zi) ; on a genus g Riemann surface. qi should satisfy spin structure 000 . Namely, the elds b(z); c(z); b(z); c(z) transform as ; g) cycle once, and they transform as c(z) ! e2 i 0 c(z) ; b(z) ! e2 i 0 b(z) ; c(z) ! e2 i 0 c(z) ; b(z) ! e2 i 0 b(z) ; c(z) ! e2 i 00 c(z) ; b(z) ! e2 i 00 b(z) ; c(z) ! e2 i 00 c(z) ; b(z) ! e2 i 00 b(z) ; Q e qi (zi; zi) is evaluated in [35] to be i gAzz@z@z R d2zpgA Ye qiqjGA(zi;zj)5 ; d (q) = q (q + 1 and the characteristics ab is de ned so that ( a + b) = surface by 24 = K. { 22 { A formula for the superghosts Substituting (A.8), (A.9) and e = Therefore there exists a holomorphic g2 form (g 1)z where S is independent of z. (z) has no zeros or poles, and it should transform as (z) such that cycles. These properties x (z) and it should coincide with the (z) in [30, 33] up to a = 4 gAzz@z@z R d2zpgA 2 exp @z@z ln Using (A.5), we can see = 4 gAzz@z@z R d2zpgA (ej ) Y E (zi; zj )qiqj Y { 23 { = 4 gAzz@z@z R d2zpgA 2 exp 4 Im 2@@ ln gzAz = X qi = 1) zz = 0 ; 1) = 0 ; 1)2 Im with S; A independent of z, we get (C.17). the determinant factor and e 3(2 1)2S. The determinant factor can also be recast into a = 1; = 1. For arbitrary and we get (zi; zi) gAzz@z@z R d2zpgA jdet ! zi j gAzz@z@z R d2zpgA Qi>j E (zi; zj ) Q Qi E (zi; R) e3S ; (C.21) gAzz@z@z R d2zpgA Qi E (zi; R) (R) det ! zi # 00 (ej ) Qi>j E (zi; zj ) Q Therefore (C.17) can be rewritten as Qi E (zi; R) (R) det ! zi # 00 (ej ) Qi>j E (zi; zj ) Q (ej ) Y E (zi; zj )qiqj Y e ( 3(2 1)2+1)S : gAzz@z@z R d2zpgA With vanishing central morphic parts. gAzz@z@z R d2zpgA #[ L] (0j ) #[ R] (0j ) : The correlation function be evaluated to be X Z Zr X Z Zr I;r E (zI ; Zr) QI>J E (zI ; zJ ) Qr>s E (Zr; Zs) Q e 12S : (C.25) meromorphic one-form @ (z) dz respectively, 2 + N ) are the zeros and the poles of the { 25 { holds in the divisor sense. Therefore we obtain On the other hand, (A.8), (A.9) and (C.15) imply QI;r E (zI ; Zr) QI>J E (zI ; zJ ) Qr>s E (Zr; Zs) Q and from (2.31) we get QI;r E (zI ; Zr) QI>J E (zI ; zJ ) Qr>s E (Zr; Zs) Q = exp 4 X GA (zI ; zJ ) + X GA (Zr; Zs) X GA (zI ; Zr)5 = const. A formula for the reparametrization ghosts In [13] it was shown that the following identity holds: re2 Re N0r0r e [gzAz; + ] ZX [gzAz] dbdbdcdc gzAz e Sbc Y cc(Zr; Zr) dz ^ dz i K b + K b :(C.29) 6 + 2N ) de2 + N ) so that Re (z1) Re (z2) Re (z2g 2+N ) ; { 26 { TI Re (zI+1) Re (zI ) (I = 1; : : : ; 2g 3 + N ) : d (Im ) (b + b ) = There are 2g 3 + N insertions of this kind. The twist to the other. The antighost insertion should be d (Im ) (b b ) = insertions of this kind. = const: [gzAz; + ] ZX [gzAz] dbdbdcdc gzAz e Sbc Ycc(Zr; Zr) bzz +"K bzz : (C.35) Here "K = { 27 { A proof of (3.9) fermionic charge @ (z) c i@X+ @ (z) c i@X+ One can show that X (zI ) can be expressed as X (zI ) = (w) e (zI ) + b @ e2 + (zI ) ; (D.2) O = +2Q2i 4 (@X+)3 2@2X+ (@X+)2 2 (@X+)2 and Q^ satis es the following identities: hQ^; cce VrDDF(Zr; Zr) = 0 ; (w) e (zI ) = Q^; I (w) e (zI ) = 0 ; (zI ) = 0 ; { 28 { Q^; e @ Q^; e @ = 0 : 6g 6+2N The antighost insertions bzz + "K bzz is a product of the contour integral of the type (C.32) becomes dzi@X+ dzi@X+ the contour integrals (C.33), (C.34), we obtain which vanishes because X+and should be singlevalued. 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Nobuyuki Ishibashi, Koichi Murakami. Multiloop amplitudes of light-cone gauge NSR string field theory in noncritical dimensions, Journal of High Energy Physics, 2017, 34, DOI: 10.1007/JHEP01(2017)034