String Two-Point Amplitude Revisited by Operator Formalism
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
String Two-Point Amplitude Revisited by Operator Formalism Shigenori Seki* Faculty of Sciences and Engineering, Doshisha University, 1-3 Tatara-Miyakodani, Kyotanabe, Kyoto, 610-0394 Japan *e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—Although one has trusted that the tree-level string two-point amplitudes vanish due to the infinite volume of residual gauge symmetry, recently the non-zero string two-point amplitudes have been pointed out in the path integral formalism. We consider them in the operator formalism. We have succeeded in obtaining the non-zero two-point amplitude in an open bosonic string theory by introducing a novel mostly BRST exact operator. We then show some trials in a closed string theory. DOI: 10.1134/S1547477120050313
1. INTRODUCTION
In terms of this Fadeev–Popov determinant, they have concluded that the two-point amplitude is,
So far it has been believed that tree-level two-point string amplitudes vanish. The two-point amplitude is evaluated by a correlation function of two vertex operators with conformal weight one, i.e., V1V2 . In an open string theory, the automorphism PSL(2, Z) of the upper-half plane is partially fixed by the locations of two vertex operators, but the residual gauge volume is still infinity. Therefore the amplitude vanishes as ! = finite ∞ = 0 . Recently Erbin et al. [1] have pointed out that the two-point string amplitude has a non-zero value, which is same as a two-point amplitude of standard free particles,
! = 2 p0(2π)D −1δD −1( p1 − p2 ), p = p1 = p2 (on shell) 0
0
0
(1)
due to the cancelation of the infinite gauge volume by δ(0). This delta function originates from the onshell energy-momentum conservation, that is, δ( p10 − p20 )δD −1( p1 − p2 ) = δ(0)δD −1( p1 − p2 ) with pi0 = 2 2 pi + m (i = 1, 2 ). In [1] they have proved Eq. (1) in the path integral formalism with the Fadeev–Popov trick. Indeed, for open strings, substituting the gauge fixing functions, f0 = X 0(zˆ0) , f1 = z1 − zˆ1 and
∏
2
f2 = z2 − zˆ2 , into 1 = ΔFP Dg i =0 δ( fi (z g )), they have obtained the Fadeev–Popov determinant, 0 0 ΔFP = zˆ01zˆ02 zˆ12∂X + zˆ01zˆ02 zˆ12 ∂X , zˆi = zˆi (i = 1, 2).
(2)
2 ΔFPδ( X 0(z0, z0 ))V1(z1)V2 (z2 ) = 2α' p0 zˆ12 V1V2 ' , (3)
where
' stands for excluding the integration over the
zero mode of the time direction, and V1V2 ' is zˆ12−2(2π)D −1δD −1( p1 − p2 ) . In this calculation of the twopoint amplitude, the correlation function,
∂X (z0 )V1(z1)V2 (z2 ) = i α' p 0
0
z12 V1V2 , z01z02
(4)
shows that ∂X 0 provides the factor p0 to the amplitude. Now we are interested in reproducing the non-zero two-point string amplitudes in the operator formalism [2]. The operator formalism has an advantage in, for example, studying a string field theory in future. 2. OPEN STRING In the open bosonic string theory, the energy conservation is
0 p10 − p20 = 2πδ( p10 − p20 ),
(5)
where 0 is the SL(2, R) invariant vacuum and p is
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