Seichi Naito

Vertex operators and generating functional for one loop scattering amplitudes

With the help of a q‐number conformal mapping whose transformation function depends on a covariantly quantized string coordinate component X+(z), a generating functional for all physical vertex operators is constructed. Using these obtained vertex operators, a generating functional for one‐loop scattering amplitudes among arbitrarily excited physical states is derived.

Generating functionals of physical vertex operators in superstring

We define N operators as operators that are already normally ordered with respect to the Friedan–Martinec–Shenker spinor operator. With the help of thus defined N operators, we can give the generating functional of physical vertex operators (GFPVO) of fermionic particles (i.e., in the Ramond sector of the superstring). We also propose GFPVO of bosonic particles (i.e., in the Neveu–Schwarz sector of the superstring), which is simply obtained by supergeneralizing GFPVO in the bosonic string.

Inlaying vertex function and scattering amplitude

Scattering processes among strings are analyzed by using fundamental equations of three types, which divide the whole complex z-plane into various types of N punctured ring domains plus various unpunctured ring domains, where internal strings freely propagate. In order to calculate scattering amplitudes (among physical particles) in Witten’s quantum string field theory, we derive and apply the “Gluing theorem,’’ mathematical proof of which is given (in operator forms) by constructing various (inlint) conformal mapping operators.

Physical vertex operators in the Ramond sector

Introducing physical string coordinate’s modes as well as physical string coordino’s modes, we derive the generating functional of physical vertex operators (GFPVO) of fermionic particles (in superstring), which is just equal to the one already proposed by using superconformal mapping in the Ramond sector. Inversely solving the superconformal mapping problem, we can derive general formulas, which give physical vertex operators of various fermionic particles up to arbitrarily excited mass levels. As an example, we explicitly derive some (Gliozzi–Scherk–Olive-allowed) low-lying physical vertex operators.

Conformal mapping and vertex operators in the light‐cone gauge

Using Fourier coefficients of Neumann functions for the tree scattering in the light‐cone gauge, an open‐string field operator Ψp(X(x),{α−m}) generating all ‘‘off‐shell’’ vertex operators, which are reduced in the mass‐shell case to already proposed general vertex operators, is introduced.

Operator product expansions and stress operators

We explain in detail how to derive the operator product expansions (OPEs) among generating functionals of physical vertex operators (GFPVO) of fermionic particles (i.e., GFPVO in the Ramond sector of superstring) and the stress operators. As for OPEs among GFPVO of bosonic particles (i.e., in the Neveu–Schwarz sector of superstring) and the stress operators, they are those simply obtained by super-generalizing OPEs among GFPVO (in bosonic string) and the stress operator.

Unitarity and renormalized ’t Hooft identities

With the help of the generalized Slavnov identities in a spontaneously broken gauge model, we compactly and explicitly prove renormalized ’t Hooft identities which lead to the unitarity of the renormalized S matrix.

Quantum superstring field theory in the B0-gauge and the physical scattering amplitudes

We propose the (BRST-invariant) quantum open superstring field theory in the “B0-gauge,” based on Neveu–Schwarz (NS) strings in 1 picture and Ramond (R) strings in 12 picture. We give the propagators of these open NS and R superstrings. In order to obtain the BRST-invariant interaction terms among these superstrings, we modify the interaction terms among three superstrings (i.e., among NS–NS–NS and R–R–NS) by subtracting the infinite number of counter terms, each of which involves interaction terms among “more than four superstrings.” The modified action can be obtained successively, so that resulting amplitudes in g-loops should become BRST invariant. Thus obtained amplitudes are referred to as the “amputated scatts,” with the help of which the physical scattering amplitudes can be expressed. These physical scattering amplitudes among NB bosonic (NF fermionic) particles are calculated by using the analytic inlint gluing operator (which has already been proposed and used in the quantum bosonic string fiel...

Fierz identities in ten‐dimensional superspace

All SO(1,9)‐irreducible bases composed of tensors and tensor–spinors, which span the whole space up to 16th‐order polynomials in θ, θj ( j=1–16) being Majorana–Weyl spinors of SO(1,9), are explicitly constructed. Any superfield φ(xm,θ) in ten‐dimensional superspace can be expanded by these bases, expansion coefficients being ordinary tensor and tensor–spinor fields. Then explicit formulas are given for φ(x,θ)Φ(x,θ) (Φ being another superfield) up to sixth order in θ in terms of ordinary fields, while those for (∂/∂θ)φ(x,θ) and Γmθ ∂mφ(x,θ) are given up to eighth order in θ.

Triangle loop for the Wilson fermion

Abstract The mass contribution and the axial anomaly for the triangle loop are calculated strictly for the Wilson fermion in the continuum limit. The external momenta in the denominators of triangle loop propagators play an important role in treating the infrared behavior for the massless case and deriving the correct mass contribution for the massive case. Neglect of this situation has sometimes brought about unusual results in the past. We get, as asserted by Karsten and Smit, the same result as in the continuum theory.