Two-point closed string amplitudes in the BRST formalism

Two-point tree-level amplitudes in bosonic closed string theory are described by a correlation function within the BRST formalism, which respects manifest Lorentz and conformal invariance. In the derivation of the two-point amplitudes, we use the mostly BRST exact operator, which has been introduced for two-point open string amplitudes, and a closed string vertex with ghost number three, which has been explored in our recent work, in addition to the conventional one.


Introduction
In string theory, the scattering matrix element is computed by employing the Polyakov path integral expression.In [1], it is demonstrated that even the two-point open string amplitude can be computed using the same approach, despite the presence of an infinite volume for the residual symmetry after fixing two positions.The resulting amplitude agrees with standard free particle expression in quantum field theory.
In [2], it is shown that the two-point open string amplitude is calculated also in the BRST operator formalism.When the positions of two open strings are fixed in the upper half-plane, the vertex operators have the ghost number two, which is insufficient to yield non-zero amplitudes.Then, in [2], a mostly BRST exact operator is introduced to fix the residual symmetry and add ghost number one.The two-point open string amplitude is provided by calculating the three-point function in which two on-shell open string vertex operators and the supplementary operator are inserted in the worldsheet.In [3], two-point superstring amplitude is calculated in the pure spinor formalism by using a supersymmetric mostly BRST exact operator.
The three-point open string amplitudes can be also derived from the four-point function with the mostly BRST exact operator [4].Unlike the two-point amplitude, the Feynman iε prescription has an important role in this derivation.Along the similar prescription, the Veneziano amplitude, namely the four-point open string amplitude, is also calculated from the five-point function by examining the moduli space [5].
Two-point closed string amplitude is also examined in the initial paper [1], with specific details provided in [6].It appears that the amplitude has been derived, but the calculation depends on a specific coordinate frame, with two vertices placed at the origin and infinity on the complex plane.In other words, the derivation lacks manifest conformal invariance.Recently, there is an interesting study of considering a timelike Liouville direction as a regularization parameter to derive the two-point amplitude [7].This method preserves conformal invariance, and Poincaré symmetry is restored by taking a limit, although the Liouville direction breaks the symmetry.
In the framework of the BRST formalism, two-point closed string amplitude is also explored using the mostly BRST exact operator with ghost number one [4].However, it becomes necessary to adjust the unintegrated closed string vertex operator in order to achieve the required ghost number, ensuring the total ghost number of six.Consequently, a subtle issue arises where the resulting vertex operator is not annihilated by b 0 − b0 .
Recently, closed string vertex operators have been revisited by means of the Faddeev-Popov procedure and descent equations in the BRST formalism [8].As a result, it has been revealed that the b 0 − b0 condition, which implies the level-matching condition, is not necessarily required to perform the correct gauge fixing and calculate scattering amplitudes.Therefore, based on the vertex operator in [8], it should be possible to justify the derivation of two-point closed string amplitudes as discussed in [4].This is a main purpose of this paper.
In this paper, we will begin by verifying the effectiveness of the mostly BRST exact operator inserted at the bulk by computing two-point open string amplitudes with its use.Subsequently, we will demonstrate that two-point closed string amplitudes can be obtained within the BRST formalism in a similar manner.

Two-point open string amplitudes
In [2,4,5], the mostly BRST exact operator is inserted at the boundary of the worldsheet to obtain open string amplitudes.However, if the operator at the bulk, as discussed in [2] as the closed string case, proves to be effective, it can also be used to derive open string amplitudes.The operator in [2] is given by1 where X 0 (z, z) is the 0-th string coordinate and c(z), c(z) are ghost fields.This operator inserted at z 0 corresponds to choosing one of the gauge fixing conditions as which is the same one as discussed in [1] even for the open string case.
The operator can be written as the mostly BRST exact expression, where δ B denotes the BRST transformation.As discussed in [2], this expression reflects the Lorentz and conformal invariance of the amplitude with the insertion of E. It is noted that explicit calculations are simplified by employing this expression.
Let us consider open strings in 26 flat spacetime directions.The worldsheet is given by the upper half plane and the real axis corresponds to the open string boundary.First, we will illustrate the validity of the bulk operator by calculating the amplitude for two open string tachyon as follows: where g o is the open string coupling constant and the normalization constant is given by [9].The point z 0 is located within the upper half plane, while x 1 and x 2 lie on the real axis.Calculating the correlation function, (2.4) can be expressed as Here, p i 0 = ± p 2 i + m 2 by the on-shell condition p 2 = m 2 = −1/α ′ , and so the momentum conservation leads to p 1 0 + p 2 0 = 0 or p 1 0 − p 2 0 = 0. Hence, we observe that, due to the factor p 1 0 − p 2 0 , A is nonzero only when p 1 0 + p 2 0 = 0, indicating that one tachyon corresponds to incoming while the other corresponds to outgoing.In the nonzero case, the resulting amplitude is given by This provides the correct amplitude of two open string tachyons, with the exception of the sign factor, which is inevitable when working within the BRST formalism [2].
It is worth noting that in the correlation function (2.4), we fix two positions of the real axis and one position of the bulk, which seems to fix four degrees of freedom.However, this does not imply excessive fixing condition of P SL(2, R), which has three degrees of freedom.This result suggests that E merely constraints one degree of freedom within P SL(2, R), although it is located in the bulk.
For general open string states, we can establish the same result by applying analogous reasoning as discussed in [2].We consider the following correlation function, where V i (x i ) is a matter vertex operator with conformal weight one.According to the energy conservation, (2.7) is expressed as where p i 0 denotes the energy of the vertex V i , and p i 0 = ± p 2 i + m 2 i by the on-shell condition.If p 1 0 + p 2 0 = 0, q is replaced by a nonzero value after the q integration.In this case, the correlation function becomes zero, because, for q = 0, E is regarded as a BRST exact operator through the expression (2.3).If p 1 0 + p 2 0 = 0, q can be set to zero after the q integration and so A can be rewritten as (2.9) Here, the prime implies that the delta function of the energy conservation and the factor 2π are excluded.This is an analogous expression of the two-point open string amplitude in [1].By referring to the calculation in [1], we observe that this expression coincides with (2.6) for general open string states.

Two-point closed string amplitudes
Let us now consider the derivation of two-point closed string amplitudes by inserting the operator E.
According to the conventional approach, the correlation function consisting of two closed strings and the operator E possesses ghost number five, which is insufficient for deriving non-zero amplitudes.This is because the conventional fixed vertex operator of closed strings has ghost number two.
Hence, we replace one of the closed string vertices with the ghost number three operator, where V (z, z) is a matter vertex.This vertex operator is derived in [8] through the Faddeev-Popov procedure for gauge fixing of P SL(2, R).As discussed in [8], the state corresponding to (3.1) is not annihilated by b 0 − b0 .However, this operator still yields the correct amplitude.In particular, it is noted that the factor 1/(4π) is determined by the Faddeev-Popov procedure.
First, we will examine a two-point closed string tachyon amplitude, which should be obtained from the following correlation function: where the correlation function is defined over the whole complex plane, and the bra vacuum 0| is taken as the hermitian conjugate to |0 , as defined in [8]. 2 g c denotes the closed string coupling constant and C S 2 is the normalization constant given in [9].When computing the correlation function, A can be expressed as follows: Applying the on-shell condition and employing reasoning analogous to that in the open string case, A becomes non-zero only when p 1 0 + p 2 0 = 0.In this case, the resulting amplitude is We can confirm that the numerical constant agrees with the correct one given by the standard quantum field theory.Using C S 2 = 8π/(α ′ g 2 c ) in [9], we find that Thus, we can derive the two-point closed string tachyon amplitude by means of E, excluding the sign factor.
For general closed string vertices V i (z i , zi ) (i = 1, 2), the amplitude should be given by the following correlator Analogous to the open string case, this correlator becomes zero owing to the BRST invariance if p 1 0 + p 2 0 = 0.When p 1 0 + p 2 0 = 0, similar to (2.9), it can be rewritten as a correlator without the delta function of energy conservation: 2 As in [8], the vacuum is normalized as Then, the unintegrated closed string vertex operator is defined as iccV (z, z), (3.4) which naturally corresponds to the integrated vertex, d 2 zV (z, z) = idz ∧ dzV (z, z).
Since ∂X 0 is a holomorphic current, we find that It is noted that the factor iα ′ /2 is different from the open string two-point case, where z 1 and z 2 are on the boundary.We can compute the remaining correlation of (3.10) as follows: (3.11) Here, the second factor arises from the ghost expectation value, and the third one is attributed to the matter contribution, which is determined by conformal symmetry.This computation closely follows the approach presented in [1], where two-point open string amplitudes are derived using a similar method.As a result, this correlation is independent of the positions of the vertices.Finally, substituting this result and its anti-holomorphic counterpart into (3.8), it becomes evident that the two-point closed string amplitudes are accurately obtained from (3.8), with the exception of the sign factor.In other words, the resulting amplitude agrees with (3.6), including the numerical factor, for general closed string vertices.

Concluding remarks
We have shown that, in the BRST formalism, two-point closed string amplitudes can be derived from the correlation function of two closed string vertices and the mostly BRST exact operator.In this derivation, we have used closed string vertices with ghost number three, which are introduced in the previous study [8].The advantage of our derivation is evident in its preservation of Lorentz symmetry and conformal symmetry.
In the case of introducing E at the boundary for open string amplitudes, as discussed in [4], the overall sign factor can be identified as the signed intersection number associated with the graph u = X 0 (y), where y represents the boundary position.It is worth noting that the same identification applies when inserting E within the bulk.In this case, the signed intersection number is associated with the graph of u = X 0 (z, z), where z represents the position along an arbitrary path connecting two unintegrated vertex operators on the worldsheet.By following our formulation, it becomes relatively straightforward to explore the two-point amplitude of closed superstrings.Furthermore, by the conformal invariance of our approach, we can derive general amplitudes through the correlation function with the insertion of the operator E. We will elaborate on the specifics of this study in the future.