Constructing massive superstring vertex operators from massless vertex operators using the pure spinor formalism

The vertex operator for the first massive states of the open superstring is constructed in terms of d=10 super Yang-Mills superfields using the OPE's of massless vertex operators in the pure spinor formalism.


Introduction
The pure spinor formalism for the superstring has all spacetime symmetries manifest [1].This feature allows the construction of super-Poincaré covariant expressions for vertex operators through its quantization [2,3].These operators correspond to physical states in the cohomology [4] of the BRST charge Q = dzλ α d α , which is expressed in terms of a ten-dimensional worldsheet spinor λ α satisfying the pure spinor condition and the worldsheet variable d α for the space-time supersymmetric derivative.The knowledge of vertex operators makes it possible to establish the equivalence of superstring amplitudes in the pure spinor and RNS formalisms [5].Nevertheless, a superfield description of superstring massive vertex operators remains an open problem.
In order to construct open superstring unintegrated vertex operators of mass m 2 = 2n, one can write every possible combination of worldsheet fields with ghost number 1 and conformal weight n, and contract them with d=10 superfields.The onshell condition provides relations between these d=10 superfields [2].In the case of integrated operators, one needs to use the descent relation to constrain the d=10 superfields [7].Although straightforward, this method becomes quite involved at higher mass levels and it is convenient to resort to other ways of building the corresponding vertex operators.
In this paper, the open string unintegrated vertex operator at the first massive level will be constructed from the operator product expansion between a massless integrated and a massless unintegrated vertex operator using pure spinor formalism CFT.This massive vertex will be BRST invariant by construction and expressed in terms of super Yang-Mills d=10 superfields which have well-known theta expansion [9].This result can be generalized for any higher mass level and used to compute scattering amplitudes with massive vertex using all the machinery known for massless scattering amplitude computations [10].
In section 2, after a brief review of pure spinor formalism, the unintegrated vertex operator at the first mass level will be computed, and its BRST invariance will be verified.In section 3 the gauge symmetries will be used to find a gauge where the vertex operator superfields are related with the usual supergravity superfields [2].Note: While this work was being completed, the paper [10] appeared which contains the main results discussed here as well as other results on massive amplitudes.However, the work here presents computations which were performed independently and were not included in [10].After completing this work, the authors of [10] have informed me that they have also performed similar computations which will soon be posted on the arXiv together with further results on massive amplitudes.

Massive Vertex Operator
The pure spinor formalism for the open string has the following action where m = 0, ..., 9, and α = 1, ..., 16 are the vector and spinorial indices of SO (10), together with a nilpotent BRST operator with the GS constraint defined as and the field λ α satisfying the pure spinor property λ α γ m αβ λ β = 0.The worldsheet variables θ α , λ α have conformal weight h = 0 and their conjugate pairs p α , w α have conformal weight h = 1.There is a ghost current J = w α λ α that can be used to define the ghost number of pure spinor operators.
The integrated and unintegrated vertex operators are [1] with supersymmetric momentum Π m = ∂x m + 1 2 (θγ m ∂θ), the Lorentz current N mn = 1 2 wγ mn λ and superfields [A m ,A α ,W α ,F mn ] built out of A α , and their super Yang-Mills equations implies the onshell condition Q • V = 0 and the descent relation Q • U = ∂V .The normal ordering : • : prescription is defined as [11] : The relevant OPE's for subsequent computations are where V (w) = V (θ)e ik•x is a superfield, D α = ∂ ∂θ α + 1 2 (γ m θ)∂ m is the supersymmetric derivative, and ∂ m ≡ ∂ ∂x m .The operator algebra of string theory primary fields can be used to recover all theory higher mass resonances [8].Since unintegrated vertex operators of mass m 2 = 2n should be constructed from combinations of [Π m , d α , θ α , N mn , J, λ α ] with ghost number 1 and conformal weight n, one can define the unintegrated vertex operator corresponding to the first massive state as where U (1) and V (2) are integrated and unintegrated massless vertex operators, respectively.The onshell and descent relations of V (2) and U (1) implies m 2 =2 = 0. To write 2.10 in terms of super Yang-Mills superfields, first consider the OPE between the first term of 2.3 with 2.4, 12, one has Considering the other terms of 2.3, one obtains The vertex operator can therefore be written as It is BRST invariant by construction, as one can see by applying the onshell condition and the descent relation for U (1) and V (2) .But one can check how BRST charge acts on each term of 2.17, (2.28) (2.30) Collecting each ghost number 2 component proportional to ∂θ ξ λ α λ β , Π m λ α λ β , ∂λ α λ β , d ξ λ α λ β and N mn λ α λ β , one can see that the BRST variation of V (12) m 2 =2 vanishes.For example, the terms proportional to d ξ λ α λ β in 2.26 and 2.31 cancel each other.Using the equation of motion ik1 m (γ m W 1 ) α = 0 and the pure spinor identity (λγ n ) α (λγ n ) β = 0, one can show that the following constraint [2] : implies that the last term of 2.31 can be written as and therefore cancels all other terms proportional to ∂λ α λ β . 1 Physical information of 2.17 is obtained through a gauge fixing procedure wherein the massive vertex operator superfields are related to the spin-2 massive supermultiplet in 10 dimensions.This multiplet comprises a traceless symmetric tensor denoted as g mn , a three form b mnp and a spin-3 2 field ψ mα , all satisfying (2.34)
Using the OPE's 2.9, one finds so the vertex operator superfields have the following variations ) δ Fmnα = D α Ω 5mn . (3.10) There are additional terms proportional to ∂λ α and Jλ α coming from the gauge transformation 3.4, and the following constraint [2] : implies that 2.17 is invariant under the field redefinition Finally, after the gauge-fixing procedure 3.1, 3.2, 3.3, all vertex operator superfields will be expressed in terms of d=10 Yang-Mills superfields and will satisfy the equations: where m 2 =2 = 0 imply that 2.17 describes a massive spin-two multiplet with (mass) 2 = 2 [2].

Fixing B and H
In this subsection, the 42 degrees of freedom of Ω 1β , Ω ξ 2 , Ω 3m will be used to impose the following constraints on Bαβ and Hmβ Using Super Yang-Mills equations of motion 2.5, 2.6, 2.7, and Fierz decomposition A.19 the bi-spinor 2.18 can be written as where To obtain the algebraic condition 3.22, one can choose ) and 3.23 is therefore implied by, In this gauge, B ′ mnp = 1 96 γ αβ mnp ( Bαβ + δ Bαβ ) is and H ′ mα = Hmα + δ Hmα is which is traceless, as one can verify by using γ mβα (γ pqm ) αξ = 8(γ pq ) β ξ .To understand the relation between 3.31 and 3.32, one can define the tensor It can be expressed from 3.31 as and has a non-vanishing trace It will be useful to note that the traceless part (H B ′ )

satisfies the relation
Nevertheless, the expression 3.31 for B ′ mnp does not satisfy the transversality condition.This is a necessary condition to remove the extra degrees of freedom at the zeroth order in θ expansion of B ′ mnp and H ′ mα [6].

Additional gauge-fixing
In this subsection, it will be shown that ∂ m B mnp = 0, when Ω 1β is written as In this gauge, B αβ and H mα are related as 3.17.
The additional contribution Ω 1β .After this additional gauge-fixing, the resulting B mnp is To obtain Λ in terms of SYM superfields, H B mα := (γ np ) β α D β B mnp will be required to satisfy γ mαβ H B mα = 0. Indeed, if H B mα is assumed to be traceless, 3.35 implies that Hitting both sides of 3.39 with D β , one finds that But (Dγ mnp D)(Dγ mnp D) = 96 • 48 at the first massive level, then Λ is given by and the additional gauge fixing Ω (1) In the gauge γ αβ m B αβ = 0, γ mαβ H mα = 0, one has and and B mnp is transverse to To demonstrate 3.17, one can write where 3m is the variation of 3.32, and δH B ′ mα is the variation of 3.33 which is implied by 3.33, 3.38 and 3.47.Using 3.36, one can write a statement equivalent to 3.17, with F β defined in 3.35.One finds from the identity that the variation 3.49 is and 3.50 is therefore satisfied, So it has been proven that in the gauge 3.22, 3.23 and ∂ m B mnp = 0, the equation 3.17 is satisfied.
In this gauge, the superfield H mα is

Fixing C
In this subsection, the 46 degrees of freedom of Ω 4 and Ω 5mn will be used to impose the algebraic constraint From the Fierz decomposition A.20, one finds ) (3.57) Using 3.44, Ω 5mn is The γ (4) component of C β α is one therefore obtains from 3.44, 3.56, 3.58 that Finally, the equation implies that 3.38 and 3.60 are related as (3.62)

Fixing F
In this subsection, the gauge invariance 3.15 with will be used to impose the following algebraic constraint To obtain 3.65, the trace part of 3.63 should be so the constraints 3.64, 3.65 imply In this gauge, 1 2 F mnα can be written as (3.68) Using the equation γ p[m γ n]p = 16γ mn , one obtains where So one can define the following tensors whose combinations satisfy the relation 3.71.Expanding 3.70, it is straightforward to check that where Finally, it will be shown that Ḡα + δ Λ G α = 0. Using 3.14, G α can be written as

( 1 )
1β = D β Λ does not change the five-form part of B αβ because of the identity γ αβ mnpqr D α D β = 0.So the previous subsection gauge fixing leaves gauge invariances parameterized by Ω (1) .70) To show the relation 3.19, one can add F (0)