Decoupling of unphysical states in the minimal pure spinor formalism I

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.


Introduction
A new superstring formalism, the pure spinor formalism, has been developed over the past ten years [1,2,3,4,5,6], see [7,8] for reviews. In this new formalism, the theory exhibits manifest super Poincaré invariance, as in the Green-Schwarz (GS) formalism, but in contrast with the GS string the worldsheet theory in flat target space is free, as in the Ramond-Neveu-Schwarz (RNS) formalism, so the theory can be quantized straightforwardly (modulo the issues with the pure spinor constraint that we discuss below). This has opened a new avenue for better understanding string perturbation theory. Indeed, the new formalism has already produced a number of interesting results in this direction, such as new non-renormalization theorems and progress towards proving finiteness of perturbative string theory [3,9], and one may anticipate more new results to appear as the formalism is developed further. On a different front, gauge/gravity dualities and flux compactifications render an urgent need for a formalism that can handle curved backgrounds with Ramond-Ramond fluxes and the pure spinor formalism is currently the best such candidate.
The new amplitude prescription has marked advantages over both the RNS and GS formalisms. Compared to the RNS formalism, the formalism does not involve worldsheet fermions, so there is no need to sum over spin structures and deal with supermoduli. Moreover, computations involving external fermions and RR fields are markedly simpler than the corresponding RNS ones and the manifest target space supersymmetry automatically leads to expressions that incorporate the entire supermultiplet. The GS formalism is also target space supersymmetry but one must use the lightcone gauge and contact term interactions [10,11] lead to complications in multi-loop computations. The pure spinor superstring is free of these problems and has already been used successfully in explicit computations [2,7,3,12,4,13,14,9,15,16,17]. However, this formalism has not been derived by gauge fixing a worldsheet diffeomorphism theory and as a result not all aspects of the formalism are fully understood. From the practical point of view, one would like to develop further the computational tools relevant for the pure spinor sector. This paper grew out of our efforts to further develop and streamline the pure spinor formalism. In this process we encountered issues with decoupling of BRST exact states which is the subject of this and of the companion paper [18].
The pure spinor superstring has two versions, the minimal [3] and the non-minimal formalism [4]. The two formalisms are formally equivalent [19] with the former being technically more intricate than the latter. The non-minimal formalism however is known to have a difficulty from genus three and higher: one of the zero mode integrals in the path integral is divergent due to poles in the composite b field [4]. Although there has been a proposal for dealing with these divergences [5], no explicit computation in g > 2 has been completed with it to date, see however [20,21] for recent work in this direction. The minimal formalism on the other hand does not appear to have such a problem: the corresponding composite b field does not have the poles that its non-minimal counterpart has. This was one of the reasons that led us to revisit the minimal formalism.
The minimal formalism contains constant spinors (C α ) and constant tensors (B mn ) in its amplitude prescription. These constant tensors enter the theory via certain operators, the picture changing operators (PCO's), which are needed to set up the amplitude prescription. It was argued in [3] that amplitudes are independent of C and B, because the Lorentz variation of PCO's is BRST exact. In this paper we show by explicit computations that the amplitudes do depend on the choice of the constant tensors and BRST exact states do not decouple. This happens already at tree level, but in this case one can show that there is a unique Lorentz invariant operator that can replace the PCO's in the tree-level amplitude prescription. With this replacement BRST exact terms do decouple and one can further show that this prescription is equivalent to the tree-level prescription obtained by integrating over C [3], which correctly reproduces known tree-level amplitudes.
Next we examine amplitudes at one loop. These should be independent of the constant tensors B and C but we find problems with Lorentz invariance and decoupling of Q exact states just like at tree level. These problems are not present when we integrate over B and C, as will be discussed in the companion paper [18]. Furthermore we prove a no-go theorem about finding new Lorentz covariant PCO's that are BRST closed inside correlators that could be used to replace the original PCO's.
Using such PCO's however one finds that all one-loop amplitudes are equal to zero.
The technical origin of the problem is that the PCO's are BRST closed only in a distributional sense and it turns out that the amplitudes are singular enough so that distributional identities do not hold. One should contrast this with the non-minimal formalism where the corresponding object, the so-called regularization factor, is BRST closed without subtleties. Indeed, we show that the problems we found at tree and one-loop level in the minimal formalism are not present in the non-minimal case.
To understand why the amplitudes are singular, let us recall that the PCO's originate from gauge fixing zero mode invariances [19]. The PCO's contain eleven delta functions of the form δ(C I α λ α ), where C I α are the constant spinors mentioned above. It turns out that for any choice of C I that give an irreducible set of eleven constraints, the solution of C I α λ α = 0 is given by λ α = 0, which is the tip of the cone that represents pure spinor space. As discussed in [22], the λ α = 0 locus should be removed from the pure spinor space. Thus this prescription corresponds to a singular gauge fixing condition and the problems we find reflect that fact. This paper is organized as follows. In the next section we review the minimal pure spinor formalism with emphasis on the tree-level and the one-loop amplitude prescription. Then in section 3 we demonstrate the dependence on the constant spinors, C α , by performing a tree-levelcomputation with two different choices for C α .
In section 4 it is shown that after integrating over C BRST exact states do decouple and we show how to formulate the prescription such that it does not contain constant spinors anymore. Section 5 examines one-loop amplitudes with unphysical states. We analyze these amplitudes both with and without integrating over B. In the final part of this section the computations are compared to their non-minimal counterparts.
Section 6 contains the no-go theorem, which states that a Lorentz invariant picture changing operator leads to vanishing of all one-loop amplitudes. In section 7 we discuss the origin of the problem as a singular fixing condition and we comment on possible modifications such that the prescription would correspond to a non-singular gauge. We conclude in section 8. The paper contains two appendices. In the first appendix we provide a comprehensive and (in some cases) pedagogical review of many technical aspects relevant for the pure spinor formalism and in appendix B we compute several integrals needed for the one-loop discussion.

Review of minimal pure spinor formalism
The worldsheet action in the minimal pure spinor formalism for the left movers in conformal gauge and flat target space is given by with m = 0, . . . , 9 and α = 1, . . . , 16. The fields p α and w α have conformal weight one and are Weyl spinors, θ α and λ α have conformal weight zero and are Weyl spinor of opposite chirality. In addition λ α is a pure spinor, i.e. it satisfies λ α γ m αβ λ β = 0, (2.2) where γ m αβ are the ten dimensional Pauli matrices, which are defined in appendix A.2. The decomposition of a Weyl spinor under the SU(5) subgroup, 16 → 1 ⊕10 ⊕ 5, which is used extensively throughout this work, is also discussed there. Since the worldsheet action consists of two βγ systems quantization seems straightforward, but λ α is a pure spinor and therefore the λw part is actually a curved βγ system [22]. To deal with this, we work on a patch in pure spinor space that is defined by λ + = 0.
On this patch the pure spinor condition expresses λ a in terms of λ ab and λ + , with a, b = 1, . . . , 5. The solution is (in SU(5) covariant components)

3)
A constraint on fields in the action induces a gauge invariance on the conjugate fields.
In this case the gauge transformations are given by In [3] this gauge invariance is dealt with by using gauge invariant quantities only. This means w α can only appear in the Lorentz current N mn , the ghost number current J and the stress energy tensor T (λw) : N mn = 1 2 w α (γ mn ) α β λ β , J = w α λ α , T (λw) = w α ∂λ α . (2.5) Since the λw part of the action is not free due to the pure spinor constraint it is not obvious what the OPE between w and λ will be. One way to proceed is by properly fixing the gauge invariance of (2.4). In [19], following [1], it was shown, by making the gauge choice w a = 0 and employing BRST methods, one can replace d 2 zw α∂ λ α by the free action, d 2 z(ω +∂ λ + + 1 2 ω ab∂ λ ab ). (2.6) One might have expected BRST ghosts associated to the gauge fixing of w α . It turns out these can be integrated out. As a check of the validity of this procedure the OPE of the Lorentz currents (N mn | wa=0 ) should give rise to the Lorentz algebra. Using (2.6) one finds J(z)J(w) ∼ −4 (z − w) 2 , J(z)N mn (w) ∼ regular, The explicit computations can be found in appendix A.4 and it should be noted that there are subtleties regarding the double poles in the OPE. Hence even though the gauge fixing condition is not Lorentz covariant the OPE's of the gauge fixed currents are. The factor of −8 of the triple pole in the JT OPE implies at tree level only correlators with total J charge -8 will be non-zero [23]. The OPE's for the matter variables can be straightforwardly derived from (2.1): x m (z)x n (w) ∼ −η mn log|z − w| 2 , p α (z)θ β (w) ∼ δ α β 1 z − w . (2.8) The action (2.1) is invariant under a nilpotent fermionic symmetry generated by where d α = p α − 1 2 γ m αβ θ β ∂x m − 1 8 γ m αβ γ m γδ θ β θ γ ∂θ δ . (2.10) The transformations it generates are given by δx m = λγ m θ, δθ α = λ α , δλ α = 0, δd α = −Π m (γ m λ) α , δw α = d α , (2.11) where Π m = ∂x m + 1 2 θγ m ∂θ is the supersymmetric momentum and again we restrict to the left movers (so in particular, the full transformation for x m contains a similar additive term with right moving fields). The cohomology of this operator (at ghost number one) indeed correctly reproduces the superstring spectrum [24].
The gauge fixed action (2.6) is no longer invariant under Q = dzλ α d α , but it is invariant underQ defined byQ On all other fieldsQ acts the same as Q. Note the second term in (2.12) is a gauge transformation with Λ a = da λ + , Λ a = 0. This implies that when acting on gauge invariant quantities Q =Q. MoreoverQw a = 0. So that for instancê Q also satisfiesQ 2 = 0, (2.14) on all fields including w, unlike Q.
It seems very natural to consider Q as a BRST operator that appeared after gauge fixing a local worldsheet symmetry that includes diffeomorphism invariance. Despite considerable work, finding such a formulation remains an open issue, see [25] for work in this direction. There has also been work in relating the pure spinor formalism to GS and RNS formalisms, see [6] and references therein.
In [19] we presented a different perspective. We considered the pure spinor action (2.1) as a σ-model action with a fermionic symmetry Q and we coupled it to topological gravity in a way that preserves Q. Gauge fixing worldsheet diffeomorphisms leads in a standard way to a second nilpotent operator, the standard BRST operator.
Then one can proceed to derive the scattering amplitude prescription following usual BRST methods. From this perspective the reason we start from an action with Q invariance is that the cohomology of Q yields the correct superstring spectrum.

Tree-level prescription
In this subsection we review the tree-level amplitude prescription of [3]. The N point open string tree-levelamplitude is given by where [Dφ] denotes functional integration over the field φ. The functional integration over x m has been studied in detail and the same correlation functions appear in the RNS formalism. We will not include this factor in the computations in this paper because they are not relevant for us. V and U are the integrated and unintegrated vertex operators, i.e. they satisfy After using the gauge invariance to set a number of components to zero the solution to these equations is given by [24] where D α = ∂ ∂θ α + 1 2 θ β γ m αβ ∂ m , a m and ξ α are the polarizations and k m is the momentum. They satisfy k 2 = k m a m = k m (γ m ξ) α = 0, there is a residual gauge invariance a m → a m + k m ω and ... contains products of k m with a m or ξ α . Y C are the picture changing operators (PCO): Y C (y) = C α θ α (y)δ(C β λ β (y)), (2.24) where C α is a constant spinor. We want to be absolutely explicit about what we mean by a delta function, since we will see the problems with decoupling of Q exact states are intimately connected with these delta functions. The definition we use in section 3 to section 5 is the usual one: The presence of the PCO's in the amplitude prescription is explained from first principles in [19] and is reviewed in section 7. In short, they come from fixing a gauge invariance due to the zero modes of the weight zero fields, λ α , θ α . Note the weight one fields do not have zero modes at tree level. At higher loops there will also be PCO's for these fields. Since the PCO's are introduced as a gauge fixing term, amplitudes should be independent of the constant tensors C α . Moreover in all computations we will choose y i = ∞ so that the PCO's have no non-zero OPE with any other field.
The functional integral (2.15) is evaluated by first using the OPE's of (2.7) and (2.8). Note that this operation reduces the total conformal dimension of the worldsheet fields involved in the OPE. For example in the p, θ OPE, the conformal weight of p β (z)θ α (w) is one and the conformal weight of δ α β is zero. Thus in the end the correlator only contains worldsheet fields of weight zero. This can be evaluated by replacing the fields by their zero modes and performing the zero mode integrations.
After integrating out the non-zero modes the amplitude reduces to where f αβγ depends on all the polarizations and momenta. Note the functional integration of x m is omitted here as will be done in all computations in this paper. A priori f αβγ also depends on z 1 , z 2 , z 3 . Of course we expect the final result to be independent of these coordinates. Also note all the fields are zero modes including those in the measure.
[dλ] is the unique Lorentz invariant measure of +8 ghost number on the space of pure spinors (cf. appendix A.5). It is given by [3] [dλ]λ α λ β λ γ = dλ α 1 ∧ · · · ∧ dλ α 11 (ǫT ) αβγ All string theory amplitude prescriptions at one loop contain a b ghost which satisfies In the RNS formalism this field appears as reparametrization antighost. In the pure spinor formalism the b ghost is composite [3], constructed out of the worldsheet fields from (2.1), as explained from first principles in [19]. However, it is not possible to solve equation (2.30) in the minimal pure spinor formalism [3], because of ghost number (J charge) conservation combined with gauge invariance of objects containing w α . The former implies b must have ghost number minus one and since there are no gauge invariant quantities with negative ghost number the latter rules out any solution. A resolution to this problem is combining the (composite) b field with a PCO, This equation ensures the Q variation of the b ghost vanishes after integrating over moduli space. The solution is given by [3] b The local b ghost, b B (u), is a composite operator, constructed out of the worldsheet (2.33) where the primes denote derivatives, BN ≡ B mn N mn and where (Bd) α ≡ B mn (γ mn d) α and G, H, K, L are given in appendix A.7.
where the polynomial h B assumes the form (λ) The integration over the zero modes of the pure spinor variables and the constant tensors is defined in [3] as for some proportionality constant c.

Tree-level amplitudes
In this section we will describe three problems with (2.15), evaluated using the def-

Lorentz invariance
The prescription of (2.26) appears to be Lorentz invariant and therefore independent of C I α because the Lorentz variation of the PCO's is BRST exact: This argument requires vanishing of QX for all X and closedness of the PCO's.
The first condition is satisfied because after integrating out the non-zero modes QX reduces to because d 16 θD α g(θ) = 0 for any function g. In order to see whether the PCO's are closed consider This seems to be zero, but if we choose C α = δ + α , we find QY C = λ + δ(λ + ). This is not zero because the measure contains 1 (λ + ) 3 . All we can use is λ + 4 δ(λ + ) = 0. This problem is made even more explicit in the computation below. It will be shown that choosing particular C's does not result into a Lorentz invariant answer.
Let us choose Note C I α has rank eleven for this choice, as it should. As is discussed in section 7, within the present formalism, the results below would be valid for any other choice, see footnote 6. The three-point tree-level function is given by The only term that contributes is the one with αβγ = + + +, in all other cases there is an integral of the form dλ ab λ ab δ(λ ab ) (no sum). There is a subtlety with these integrals, for instance [dλ](λ + ) 2 λ cd δ(λ + )δ(λ 12 ) · · · δ(λ 45 ) = dλ + d 10 λ ab λ cd λ + δ(λ + )δ(λ 12 ) · · · δ(λ 45 ) = Note however that (3.6) has N charge one (cf. (A.21)). Since the outcome of the integral (maybe after some regularization) must be a number, which does not transform under N, the integral has to vanish. In other words only integrals with zero N charge, like [dλ](λ + ) 3 δ(λ + )δ(λ 12 ) · · · δ(λ 45 ) can be non vanishing. After the integration over the λ zero modes we are left with where f +++ = A 1 + A 2 + A 3 + and this can be evaluated with the help of the explicit expressions for the gamma matrices from appendix A.3.3. We choose as external states two gauginos and one gauge boson: This answer is not Lorentz invariant and different from the expected answer, where m is an SO(10) index and all Latin letters that come before m in the alphabet are SU (5) indices. In conclusion this shows that using (2.27) and (2.25) does not lead to Lorentz invariant answers.
Under this change in C I α the tree-level three-point function changes by There is a total of four λ α 's in the numerator (one hidden in Q) one of them has to be λ 45 and the other three have to be λ + to give a non vanishing answer. The term that contributes comes from Q hitting θ + δ(λ + ), this λ + then cancels against a λ + in the denominator and the variation becomes By choosing suitable polarizations it is not difficult to see this does not always vanish.

Position of PCO's on the worldsheet
In the prescription of [3], PCO's are inserted at arbitrary points on the worldsheet.
The derivative of the PCO's however is Q exact: ∂Y C (y) = Q[(C∂θ(y))(Cθ(y))δ ′ (Cλ(y))], (3.13) . (3.14) and this suggests that the amplitudes do not depend on the insertion points. As we have seen, however, BRST exact terms do not decouple, so the amplitudes may depend on the insertion points. In our computations of tree-levelamplitudes we will follow [3] and insert the PCO's at y = ∞. This is equivalent to replacing the fields in the PCO's by their zero modes.

Resolution at tree level
Obtaining amplitudes which are not Lorentz invariant is a serious problem and one might ask why the tree-levelamplitude computations [3,15] in the minimal pure spinor formalism gave Lorentz invariant answers and why Q exact states decoupled. Both these points are explained in the first part of this section. In the second part we reformulate the tree-levelamplitude prescription in a way that does not contain any constant spinors.

Resolution in the literature
Lorentz invariance is restored by integrating over all possible choices of C I α , and this also results in decoupling of Q exact states as will become apparent in this section.
This prescription also ensures the decoupling of unphysical states. We will use B to denote amplitudes with unphysical states throughout this paper, while A is used for any amplitude, so at tree level with V 1 = QΩ, This can be written in the following form: where in going from the first to the second line we omitted an overall numerical factor of eleven. Such overall inconsequential factors will be neglected throughout this work.
After using the OPE's to integrate out the non-zero modes one gets: In this case one can also write out (ǫT ) explicitly and check that its trace contains a contraction of an antisymmetric tensor (ǫ) and a symmetric one (γ αβ m ).

Lorentz invariant tree-levelprescription without constant spinors
We now present a new prescription for a tree-level amplitude, which does not contain any constant spinors and is manifestly Lorentz invariant. This new prescription is equivalent to the one given in [3], when the integral over C in included. The prescription is given by In other words, we have replaced the eleven PCO's Y C by Λ αβγ (∞). After integrating out the non-zero modes and replacing the fields by their zero modes A reduces to The tensor Λ αβγ is defined by and is a function of the λ's only. More accurately, all components contain eleven delta functions or derivatives thereof. The precise form of (4.12) follows from the fact that the integral must be an invariant tensor combined with the pure spinor constraint.
Detailed arguments are provided in appendix A. 6. To see what conditions (4.12) imposes on Λ +++ note that choosing αβγ = + + + gives [dλ]λ + 3 Λ +++ = 6.  To determine whether this object is indeed part of a representation of the Lorentz group one needs to check the Lorentz algebra holds when acting on Λ +++ . First note Because of the symmetric form of Λ +++ it suffices to check Let us start with the LHS of (4.18) Note that N 12 does not contain factors of (λ 12 ) 2 (possible such factors cancel out).
This is useful when acting with N 12 in this second line. In going from the second to the last line we used xδ ′ (x) = −δ(x) twice. (4.19) and (4.20) follow along the same lines.
It is instructive to compute the next two levels (distinguished by N charge) of the components of Λ αβγ . For the components on the second (N = 11 4 ) level consider The factor of − 1 2 is consistent with N ab w + = − 1 2 w ab . Going to the next level (N = 7 4 ) This seems to leave freedom to define one of the two components, which would indeed be true if Λ αβγ was just a symmetric rank three tensor and nothing more. However Λ αβγ is gamma matrix traceless, This imposes one additional condition that relates components of equal N charge to each other. Consequently all components of Λ αβγ are uniquely fixed in terms of Λ +++ . Note that this is consistent with the discussion under (A.98), where Lorentz invariance arguments were used to come to the same conclusion.

Decoupling of Q exact states
The new insertion Λ αβγ was motivated by manifest Lorentz invariance, but it also results in a prescription in which Q exact states decouple. Indeed, the tree-level amplitude with one BRST exact state, can be written in the following form: After using the OPE's to integrate out the non-zero modes one gets: The last line vanishes because all traces of (ǫT ) vanish (cf. (4.9)).

One-loop amplitudes
In this section we investigate one-loop amplitudes with one unphysical state both in the prescription with an integral over B and without. We first show that all such amplitudes are proportional to certain zero mode integrals. Decoupling of BRST exact states would follow if these zero mode integrals vanished. However, these integrals do not vanish after the λ and N integrations have been performed as one would expect based on Q invariance of the PCO's, (3.3). We then focus on four-point functions, these being the first two non-vanishing one-loop amplitudes. We will find decoupling of unphysical states in this case. In the prescription without an integral over B, however, these amplitudes vanish because none of the remaining terms after the λ, N integrals contain precisely sixteen distinct components of the zero modes of d α .
Preliminary analysis suggests that this mechanism is not operational in higher-point functions. Furthermore, even the four-point functions are not Lorentz invariant. The four-point function containing one unphysical state with an integral over B is also analyzed and we prove it vanishes. In the companion to this paper [18] we show using a different argument that unphysical states decouple to all orders, when one integrates over B and C.
Note that the picture raising operators, Z B , are Q-closed without subtleties: This vanishes because it contains the square of a fermionic quantity, so one may anticipate that the problems are due to picture lowering operators Y not being Qclosed. Let us also record the Lorentz variation of Z B ,

Amplitudes with unphysical states without integrating over B.
A one-loop amplitude with one unphysical state is given by where λBd = B mn λγ mn d. Note that we have replaced the Y C insertions by the Lorentz invariant insertion, Λ αβγ , as in the tree-level computation. This is equivalent with inserting Y C and integrating over C. On the torus we cannot insert the PCO's such that all their OPE's would vanish. We inserted them at some arbitrary point y. For later convenience we insertedb at a different point, w.
We now integrate Q by parts. When Q acts onb we get a total derivative in moduli space, as usual. If this total derivative is non-vanishing the theory has a BRST anomaly. These total derivative terms will be suppressed below because they are not important for our discussion. More precisely, after integrating Q by parts the amplitude (5.3) becomes, where we emphasize again that we suppress the total derivative term in moduli space originating from Q acting onb. In this subsection we will evaluate B (N ) without integrating over B. The choice we make is: We demonstrate below that all such one-loop amplitudes can be written as a sum of terms proportional to a certain zero mode integral I β 2 ···β 11 . This is done by using the OPE's to remove all fields of non-zero weight, in particular N mn . This is a non trivial step because of the complicated form of the b ghost.
Had the zero mode integral I β 2 ···β 11 vanished, this would have proven that BRST exact states decouple at one loop (again modulo the total derivative term from Q acting onb). Non-vanishing of I β 2 ···β 11 does not prove that there exists a non-vanishing amplitude with a Q-exact state, because there may be additional cancellations when one performs the remaining integrals. It does show however that the PCO's are not Q closed.

Zero mode integral
We will show below that all one-loop amplitudes (5.4) can be written as a sum of terms that are proportional to the following zero mode integral, Moreover we will show all one-loop amplitudes with an unphysical state can be written as a sum of terms proportional to the trace of I α 1 β 1 ···β 11 which we call I β 2 ···β 11 : Thus (5.7) is the one-loop analog of (4.28) (or (4.8)). Note that, in spite of the notation, I α 1 β 1 β 2 ···β 11 is not manifestly Lorentz invariant. Whether it is Lorentz invariant remains to be seen. Our first task is to evaluate I β 2 ···β 11 . After using expression (A.95) for [dN] to evaluate the N integral in I β 2 ···β 11 we find In this form it becomes apparent that the problems with factors of λ + in the denominator only become bigger at one loop. At this point we can only surmise this. To find a definitive answer we have to evaluate the λ integral. This can be done by expanding the integrand by powers of λ + : The Y 's can be expressed in terms of the d's just like in the first four terms. Note that the minimal number of d a 's in Y a 1 ···a 2k is k − 1. This is the reason the series stops at k = 6. The maximum number of d a 's in Y a 1 ···a 2k is k. The λ integration of (5.9) can be evaluated term by term. I β 2 ···β 11 then becomes The integrals I k are investigated order by order in the sequel of this subsection.
For k = 0, 1, 2 one can use (4.12) and (4.9) to show the λ integrals vanish: If k > 2, however, there are also factors of λ + in the denominator. As shown in appendix B.1 the λ integrals do not vanish anymore. M charge conservation implies that I 3 can only be non vanishing if This is explained in detail in the first part of appendix B. We explicitly compute I 3 for the first case. Since Sym 31 0 ⊗ Asym 5 10 ⊗ Asym 45 contains one scalar, one finds and (ǫ 10 ) 12131415232425343545 = 1. The two permutations add terms to make the RHS symmetric under a 2i−1 a 2i ↔ a 2j−1 a 2j . The constant c 1 is computed in appendix B.2 and is given by We will not compute any components of I 4 here. Going to the next level, the only choice of β 2 , . . . , β 11 that leads to a non-zero answer for I 5 is The details are given in appendix B.2. Finally I 6 can be evaluated as: This vanished because (ǫT ) b 1 b 2 b 3 +β 2 ···β 11 = 0 and that follows from the M charge conservation rule for invariant tensors. In other words it is not possible to choose β 2 , . . . , β 11 such that the total M charge of the components is zero (cf. equation (A.25)). This concludes the computation of the pure spinor zero mode integrals that appear at one loop.

Non-zero mode integration
We now demonstrate that all one-loop amplitudes with an unphysical state can be written as a sum of terms proportional to I β 2 ···β 11 . After this proof we indicate how the argument can be modified to prove that A (N ) can be written as a sum of terms proportional to I α β 1 ···β 11 . In general the amplitude, B (N ) , becomes a sum of terms of the form where the indices in the PCO's are SU(5) indices, i j ∈ {2, . . . , N} and f m 1 ···n k does not contain any λ's or w's. The number k indicates how many vertex operators provide an N mn . The functional integrals over λ and N can be evaluated by performing the OPE's to remove all fields of non-zero weight. Then one replaces the fields by their zero modes and performs the integration over these modes. In order to perform the OPE between N mn and δ(BN) we have to Taylor expand δ(BN), as discussed in [3], whereN denotes N after omission of the zero mode. The holomorphic one form ω(y) is constant on the torus: where τ 2 is the imaginary part of the modulus τ . The b ghost also contains N mn 's which have to be taken into account if one is removing all fields of non-zero weight.
We first focus on the first term, the local b ghost, b B (u). The second term ofb(u, y), with the integration in it, will be dealt with later. After replacingb(u, y) by b(u) in the amplitude, B (N ) i 1 ···i k , becomes a sum over n, which counts the number of N mn 's the local b ghost provides, of the following objects: where δ (j) denotes the jth derivative of the delta function and the sum runs from zero to three because b does not contain δ (4) (B 1 N) or higher derivatives.
The product of the eleven delta functions, including the one from b, becomes a sum of products of eleven δ (j) (B I N 0 ) after the Taylor expansion. We start with the first term in this sum, i.e. the one withoutN's and no derivatives on the delta functions.
In this case the N m j n j (z j )'s from (5.20) have OPE's with themselves and with the λ's from the PCO's. We first concentrate on the term in which all N mn 's get contracted with an explicit λ. That term is given by 3 where and E(z, y) is the holomorphic prime form, which goes like z − y when z → y [27,28].
N mn are abstract Lorentz generators for the λ, w sector and they act to the right.
They should not be thought of as containing (zero) modes of the λ or w worldsheet fields. The N mn merely multiply every index on a λ or w they hit by a two form gamma matrix. Up to now we only considered contractions between N mn and the explicit λ's, but if two or more N mn 's contract with each other in B (N ) We still need to consider the second term inb(u, w). This was not included in the above discussion because it contains ∂N mn (v). This does not change the argument much, after the OPE's this part of the amplitude will also have the form of C  To see A (N ) can be written as a sum of terms proportional to I α 1 β 1 ···β 11 one can use the above reasoning with a slight adjustment. This consists of replacing λ β 1 (y) by λ α 1 (z 1 ) in (5.20) and adding an α 1 index to f . The only effect this has is the replacement of some F (z i , y) by F (z 1 , z i ) in (5.24), apart from the fact α 1 and β 1 are not contracted anymore.
Thus we have shown that amplitudes with unphysical states do not vanish by the λ, N integration, opposite to expectations, but nevertheless let us press on and explicitly compute a one-loop four-point amplitude with an unphysical state. Perhaps we will find some other mechanism that makes these amplitudes vanish.

Four point function without integrating over B
In this subsection we investigate two properties of the four-point one-loop function (2.38) in the minimal pure spinor formalism, namely decoupling of unphysical states and its Lorentz invariance in the formulation without integrating over B. We will find decoupling of Q exact states, in spite of the results of the previous section. The vanishing is achieved after the integral over the d zero modes. Lorentz invariance, however, does not follow in the same way.

Decoupling of unphysical states
The one-loop four-point amplitude is an example of an amplitude in which only the zero modes contribute (cf. [3]). It turns out only three terms have enough factors of d α and N mn to give a non vanishing answer. This will become clear in equation (5.27) below. Thus we can immediately replace all the fields in (5.3) by their zero modes: The only terms of b B 1 that contributes are the ones with four d's and there are only three such terms:  ) and the fourth is a d ab (M charge 1 4 ). The second term can be reduced as follows: where we used the M charge conservation property of invariant tensors together with (Bd) a = 0. After observing that (c 1 ) aba 1 ···a 8 is an SU(5) invariant tensor that is antisymmetric in the middle three pairs of indices (a 1 a 2 , a 3 a 4 , a 5 a 6 ) and there is only one invariant tensor with these symmetries [26], namely (ǫ aba 1 a 2 [a 3 ǫ a 4 ]a 5 a 6 a 7 a 8 + 5 perms), we find that the second term in the b ghost is proportional to The same logic can be applied to the third term although this case is slightly simpler.
The third integrated vertex operator must provide an N 12 zero mode. It then follows that B (4) is proportional to I β 2 ···β 11 . This integral can be written as a sum over k just as in (5.9). In this sum the k = 0, 1, 2, 6 terms vanish because of the λ integration and the k = 4, 5 terms vanish due to the d integration (note that b B 1 contains three d a 's and Y 4 , Y 5 contain at least three d a 's). The k = 3 term is given by where we used and the integral vanishes because ǫT is traceless.
Thus, for the four-point one-loop amplitudes with a BRST exact state the terms that do not vanish after the λ, N integral now vanish because they contain a square of fermionic quantity, namely d α d α (no sum). One may wonder whether the same mechanism would work in higher point functions. While we do not have a definite answer to this, preliminary results suggest that this is not the case. For example, the zero mode contribution to the 5-point function with a Q-exact state does not vanish in this way, but we should emphasize that our analysis does not exclude possible cancellations between the contributions of zero and non-zero modes.

Lorentz invariance
In this subsection we study the Lorentz invariance of the amplitudes. Recall that the Lorentz variation of the PCO's is Q exact (cf. (5.2)). Thus one expects that the amplitude is Lorentz invariance. We have seen earlier however that Q exact states may not decouple, so we will proceed to check explicitly whether the Q exact terms obtained from the Lorentz variation of the PCO's evaluate to zero. We will focus on the term obtained by the Lorentz variation of a single PCO. This term should be zero by itself because it is Q exact. Integrating Q by parts one obtains a total derivative in moduli space when Q acts onb, which will be suppressed as in our earlier discussion, and a number of terms when Q acts on the other PCO's. These terms should evaluate to zero after the λ and N integrals are performed, but we will see that they do not. As explained before the non-zero modes can be integrated out trivially in the four point one-loop function: The Lorentz variation of the four point function can be written as a sum with one term for each Z B : with similar formulas for the other terms.
Each of (A B I ) a b should be zero separately, so we focus on (5.36). After integrating Q by parts and writing out the b ghost one finds This integral can be evaluated in exactly the same fashion as the one appearing in the four-point function with a Q exact state. The first step is to perform the N integrations and then expand the integrand in powers of λ + : all other terms vanish because they contain 6 or more d a 's. The first two terms in the λ + expansion vanish by using (4.12) and (4.9). The next term reduces to To show that this contribution is non-zero, it suffices to prove that one of its components (J 3 ) α β 2 ···β 11 is non-zero. We will consider the case, Finally, we have the term containing λ α λ β 1 λ a 1 a 2 λ a 3 a 4 λ a 5 a 6 . This term however does not contain a factor of d + , so it cannot interfere with (5.42) (prior to the integration over d).
Thus we get a non-vanishing result after integrating over λ, N, opposite to expectations. Note that (5.42) contains 13 d zero modes. The remaining three d zero modes can be provided by the vertex operators, so the Lorentz variation does not vanish in a similar fashion as in the discussion in the previous subsection, although in principle there may still be a cancellation between this term and terms originating from the term with λ α λ β 1 λ a 1 a 2 λ a 3 a 4 λ a 5 a 6 in (5.39) after integrating over d.
Proportional here means in the sense of tensor multiplication: in the terms that appear after contractions, the tensor X is multiplied by gamma matrices. Evaluating the integrals in (5.43) is much easier than one might have anticipated, because we know that X must be an invariant tensor, that is symmetric and gamma matrix traceless in the α's, antisymmetric in the β's and antisymmetric in both m i ↔ n i and m i n i ↔ m j n j . To find out how many independent invariant tensors with these properties exist, we compute the number of scalars in the relevant tensor product, which is one (see also section A.3.2). As a matter of fact we already know such a nonvanishing tensor: (ǫT R) α 1 ···α 11 β 1 ···β 11 m 1 n 1 ···m 10 n 10 ≡ (ǫT ) ((α 1 α 2 α 3 β 1 ···β 11 R α 4 ···α 11 )) m 1 n 1 ···m 10 n 10 , where the double brackets denote gamma matrix traceless, see appendix A.6. We stress that Lorentz invariance has completely fixed X, there is no freedom remaining.
Starting from a correlator with an unphysical state and integrating Q by parts, it will hit a θ from a PCO (where again we suppress the total derivative in moduli space obtained when Q acts onb, which does not play a role here). This means all amplitudes with an unphysical state can be written as a sum of terms proportional to the trace of (ǫT R): There are two independent invariant tensors with indices and symmetries of the trace of (ǫT R), so one expects a non-vanishing trace. Indeed, it is proven in section 6.1 that this trace does not vanish, which implies the PCO is not Q closed. One might want to replace (ǫT R) by its traceless part to restore Q invariance, but this is not possible since all invariant tensors with the symmetries and indices of X are proportional to (ǫT R). In other words removing the trace of (ǫT R) would set the entire tensor to zero.
We conclude that the proof of decoupling of unphysical states at tree level does not generalize to one loop and one needs a new argument. Such a new argument is presented in [18], where it is shown that unphysical states decouple to all loop order.

Comparison to non-minimal formalism
In this subsection we briefly compare with the non-minimal formalism [4]. None of the problems that were found, when we examined the prescription without integration over B, are present in this case.
In the non-minimal formalism one introduces a set of non-minimal variables, the complex conjugateλ α of λ α , a fermionic constrained spinor r β satisfyinḡ λ α γ αβ mλ β = 0,λ α γ αβ m r β = 0 (5.46) and their conjugate momenta,w α and s a . Analogous to the minimal formalism these conditions induce a gauge invariance: This impliesw α and s α can only appear in the gauge invariant quantities N mn = 1 2 (λγ mnw − sγ mn r),J =λw − sr, Tλw =w α ∂λ α − s α ∂r α , (5.48) The action (2.1) is modified by the addition of the term S nm : and the generator Q by This acts on the non-minimal variables as follows These transformation rules imply that the cohomology is independent of the nonminimal variables. In other words the vertex operators can always be chosen such that they do not include these variables.
The non-minimal variables can also be understood as originating from the BRST treatment of the gauge freedom due to shifts of the zero modes of the worldsheet fields [19]. This also explains why vertex operators do not depend on the non-minimal fields and why only the zero modes of these fields appear in the path integral. Furthermore the OPE's given in section 2 still comprise a complete list, since the new fields do not have non-zero modes. Note however that in more recent work [20] that aims at dealing with divergences asλλ → 0, non-zero modes ofλ do play a role. It would be interesting to understand how this fits with the discussion in [19].
The original proposal in [3] is the special case where the function f is given by 4 This satisfies the first three conditions, but although QY ∼ λδ(λ) the fourth bullet does not hold for (6.2), as we have seen.
• For each P the total number of B P 's outside the delta functions is equal to the number of derivatives on δ(B P N) plus one. This can be inferred from the explicit form of the b ghost, (2.33), and the Taylor expansion of the delta functions. This is reflected in (6.7) because L P appears in two places.
• For a non-zero answer the total number of N zero modes must equal the total number of derivatives on the delta functions. This gives the restriction L = L P .
• One might have expected derivatives on δ(J) as well, but for a non vanishing answer there must also be enough J zero modes, so one can always reduce the amplitude to contain only δ(J).
• Compared to (2.43) the λ dependence is less general. It is possible to restrict to this class of integrands because f β 1 ···β 11 (λ) is a Lorentz tensor. To see this note the OPE's of N and J with f do not introduce derivatives: where the α index is in the i th position.
Note that the free indices on E can be either contracted among each other or with d or θ zero modes. The integral in (6.7) can be evaluated by using the definition of B integration in (2.45). Let us call the integrand of (6.7) g and write it as g(λ, N, J, B P ) = λ α 1 · · · λ α 11 h β 1 ···β 11 α 1 ···α 11 (N, J, B P ) (6.13) Using the above definition the integral in (6.7) can be evaluated as E α 1 ···α 11 p 1 q 1 ···p L q L β 1 ···β 11 m 1 n 1 ···m 10 n 10 r 1 s 1 ···r L s L = c L 1 ···L 10 δ ([p 1 s L (ǫT R) α 1 ···α 11 β 1 ···β 11 m 1 n 1 ···m 10 n 10 + symmetrization in([r L P −1 +1 , s L P −1 +1 ], . . . , [r L P , s L P ], [m P n P ]), (6.14) for some constant c L 1 ···L 10 . Note the round brackets denote symmetrization in Also note the second line above includes ten symmetrizations, one for each P . E is symmetric in these indices because they all appear on B I . (Note that by definition L 0 = 0). To get some insight how to obtain (6.14) consider the case L 1 = L = 1. In that case the rhs of (6.7) is given by where the spinor indices on (ǫT R) are suppressed. The last nine B differentiations are trivial resulting in: Now we first perform the N differentiation followed by the last two B differentiations: s 1 (ǫT R) m 1 n 1 ···m 10 n 10 + (r 1 s 1 ↔ m 1 n 1 ), (6.18) which agrees with (6.14). The above computation clarifies the appearance of the Kronecker delta's. It is a consequence of the fact ∂ ∂B 1 pq and ∂ ∂N pq appear contracted. The symmetrizations in (6.14) follow from the product rule of differentiation.
With these preliminaries we are ready to prove that if QY = 0 then all one-loop amplitudes vanish:
What remains is to show the non vanishing of this tensor implies the non vanishing of the trace of (ǫT R).
The first line is zero because we are fully antisymmetrizing seventeen indices that only take sixteen values.

Origin of the problems and possible resolutions
To understand the origin of the problems encountered when one does not integrate over C (and B), we go back to the first principles derivation of the amplitude prescription in [19]. We will see that there is a singular gauge choice implicit in the prescription of [3].

Derivation of the amplitude prescriptions
In [19] the minimal and non-minimal amplitude prescriptions were derived by coupling the pure spinor sigma model to topological gravity and then proceeding to BRST quantize this system. Following [29], the BRST treatment included the gauge invariance due to zero modes. The singular gauge fixing refers to the gauge fixing of the invariance due to pure spinor zero modes.
The BRST quantization led to the following generating functional of scattering amplitudes where ρ i are sources that couple to vertex operators, dµ σ is the path integral measure for the original sigma model fields, dµ is the path integral measure for the fields introduced in the BRST quantization procedure and S is the worldsheet action with two dimensional coordinate invariance. L 1 contains the gauge fixing terms due to the diffeomorphism and Weyl symmetry and L 2 contains the gauge fixing terms for the invariances due to the zero modes of the ghost fields. In the case of the bosonic string [29], L 1 leads to the usual ghost action and L 2 to the usual ghost and antighost insertions in the path integral. In our case these contributions cancel out. To understand the cancellations recall that the pure spinor sigma model has a fermionic nilpotent symmetry generated by Q, the pure spinor BRST operator. After coupling to topological gravity and gauge fixing all symmetries, there is a second nilpotent operator Q V , the standard BRST operator related with gauge fixing local symmetries.
Q V in particular contains the standard terms related to diffeomorphisms and Weyl transformations and it also has terms related to the invariance due to zero modes of the worldsheet fields. Since we want to keep the Q symmetry manifest, all fields are introduced in Q-pairs. In particular, together with the b, c ghosts we also introduce their Q partners, β, γ. These have opposite statistics and it turns out these fields can be integrated out and the b, c part cancels against the β, γ part. Even though all terms related to gauge fixing of worldsheet diffeomorphism cancel out, this procedure explains why the pure spinor amplitude prescription is so similar to the bosonic string amplitude prescription.
The main object of interest here is L 3 , which is related to gauge fixing of the invariance due to the zero modes of λ α and its conjugate momentum w α . The part relevant to the λ zero modes is given by The field π α is the BRST auxiliary field that enforces the gauge fixing condition for the invariance due to zero modes of λ. Since there are eleven zero modes we need eleven gauge fixing conditions and the BRST auxiliary field π α must contain only eleven independent components. The gauge condition implicit in (7.2) will be discussed shortly. (b α , c α ) and their Q-partners (b α , γ a ) are the corresponding BRST ghosts. These fields can be integrated out and cancel each other. Then we are left with L ′ 3 = π α λ α +π α θ α .
Minimal formalism. To express the fact that π α andπ α have eleven independent components we parametrize them as follows π α = p I C I α ,π α =p I C I α , I = 1, . . . , 11. (7.4) where C I α is a matrix that must have maximal rank. Thus the gauge fixing condition is given by We will shortly show that this is a singular gauge condition.
The eleven constant spinors C I α are the ones that enter in the minimal pure spinor prescription. Indeed, using (7.4) we find that the path integral contains which are the eleven picture changing operators Y C we discussed earlier.
where f (λ) contains λ but not its complex conjugate. For this expression to be welldefined f (λ) should not contain any (C I λ) poles and moreover there should not be any poles that obstruct the analytic continuation of λ to real values.
At higher loops the conjugate momentum has zero modes as well and gauge fixing this invariance leads exactly to the insertion of PCO's Z B , Z J , where the tensors B mn enter through the gauge fixing condition, see [19] for the details. In addition, one needs a composite b field satisfying (2.30). In the minimal formulation, a solution of (2.30) is given by [30] where G α is given in (A.111). This is however too singular to be acceptable. One can obtain a non-singularb field by combining the b field with the PCO and solving instead (2.31). Note that thisb field now depends on the B mn constant tensors but not on C α .
Non-minimal formalism. We now show that the same expression (7.3) leads to the so-called regularization factor in (5.52). This time we choose π α to be a pure spinor of opposite chirality to λ α , usually calledλ α . This indeed has 11 independent components, as required. The fieldπ α , usually called r α , automatically follows because it is the Q variation of π α , r α = Qλ α . (7.9) This leads to the non-minimal formalism. To see this explicitly note that the factor e −L 3 in (7.1), which is given by is precisely N . The additional factors N mnN mn + 1 4 S mn λγ mn d + JJ + 1 4 Sλd originate from gauge fixing the zero modes of w α , see [19] for the details.
Note that λ is now holomorphic and π α ≡λ α is considered as its complex conjugate variable. Typical integrals one encounters at tree level in the non-minimal formalism are therefore (7.11) At higher loop order we also need the b field. In the non-minimal formalism, equation (2.30) has a solution that depends on both λ andλ. It is however singular asλλ → 0 and this causes problems starting from three loops. Note that the b field does not depend on how we treat the gauge invariances due to the zero modes of w α . This is similar to the b field in (7.8) but different thanb which depends on the gauge fixing of the invariance due to zero modes of the conjugate momentum through B mn .
To summarize, the minimal and non-minimal are related by field redefinitions and an analytic continuation in field space. In particular, starting from the non-minimal formalism one obtains the minimal formalism by takingλ α = C I α π I and analytically continuing π I to be imaginary while at the same time analytically continuing λ to be real. There are similar redefinitions and analytic continuations in the sector related with the conjugate momentum. Furthermore, the non-minimal b field combined with part of N is related tob. Clearly, the two formalisms would be equivalent if the analytic continuations had not been obstructed by singularities in the amplitudes.
Finally, note that the underlying gauge choice for the invariance due to pure spinor zero modes is the same: the gauge fixed action is the same, only the reality condition of the fields is different.

Toy example
Given the formal equivalence between the minimal and non-minimal formalisms one may wonder why we found problems at one loop in the one formalism but not the other. We discuss this issue here by analyzing a toy example that has almost all features of the actual case. Consider the following integral To compare with the expressions in the previous subsection p corresponds to the BRST auxiliary field and x to the pure spinor.
If one wants to evaluate the above integral, contours have to be chosen for x and p. If we choose p = ip 1 and x = x 1 with p 1 , x 1 real, we get Another choice is to consider x complex and take p = x * . In this case I becomes This agrees nicely with the general property of contour integrals, that one is free to deform them as long as no poles are encountered. Note that (7.13) resembles a zero mode integral in the minimal formalism and (7.14) a non-minimal one.
The difference between the two prescriptions is exposed by considering the integral I with a function f in the integrand.
Now rotate the contour, p = x * , so that the integral becomes

16)
I min is the analogue of (7.7) and I non−min the analogue of (7.11). I min and I non−min give exactly the same answer if f (x) is non singular but (7.15) is ill defined for any choice of singular f (x) whereas (7.16) may be well defined. For example, for the function x , (7.17) (7.15) yields ∞ but (7.16) gives 0. More precisely, (7.16) is well defined for all functions f (z) = n c n z n , with c n = 0 for n < −1. For the n < −1 terms the θ integral vanishes and the r integral diverges, which makes I non−min ambiguous for these kind of functions.
A third representation is obtained by noticing that the θ integral can be rewritten as a contour integral where z = re iθ and the contour C is a circle of radius r. Thus for any meromorphic function f (z) the integral over theta is independent of r and The expression (7.19) are well-defined for all meromorphic functions f (z) whereas (7.15) and (7.16) are not.
Going back to pure spinors and working on the patch with λ + = 0 we see that because of the factor (λ + ) −3 in the measure (cf. (A.91)) the minimal formalism is expected to have a singularity unless the integrand provides a factor of (λ + ) 3 , but the expressions (7.16) and (7.19) are not necessarily singular.

Singular gauge and possible resolution
We show in this subsection that the gauge (7.5) is singular for any choice of the constant spinors C I α . To see this, recall that the space of pure spinors can be covered with 16 coordinate patches and on each patch at least one of the components of λ α is non-zero. Let us call this component λ + and solve the pure spinor condition as in (2.3). Then, 0 = C I α λ α = C I + λ + + C I,ab λ ab + C I a λ a = C I + λ + + C I,ab λ ab + 1 8 C I a ǫ abcde λ bc λ de 1 λ + ⇒ C I + (λ + ) 2 + C I,ab λ + λ ab + 1 8 C I a ǫ abcde λ bc λ de = 0. (7.20) This system of equations however does not have a solution with λ + = 0 and the gauge is singular. To see this, we first solve ten of the above equations to obtain λ ab as a function of λ + . A scaling argument implies that these functions are linear in λ + . Then we plug the relation λ ab = b ab λ + in the eleventh equation to find that λ + vanishes. Thus we find that for any choice C I α of maximal rank , the path integral localizes at the λ α = 0 locus 6 , which is the point that should be excised from the pure spinor space for the theory to be non-anomalous [22].
As discussed above, the minimal and non-minimal formalisms are related by analytic continuation in field space. In the toy example in the previous subsection, we saw that the analytic continuation from the "minimal variables" x 1 , p 1 to the "non-minimal variables" x, x * sets to zero certain singular contributions (functions f (x) ∼ x −1 ) but the integral still localizes at x = 0. One would thus expect that the zero mode integrals in the non-minimal formalism localize at the λ α = 0 locus, as the minimal ones do, and the problems with theλλ poles at 3 loops and higher are a manifestation of this fact.
To avoid these problems 7 one must find a way to gauge fix the zero mode invariances such that the zero mode integrals do not localize at λ α = 0. Let us discuss how to achieve this in the minimal formulation. First, in order to avoid the unnecessary analytic continuation to real λ one should work with the analogue of the contour representation of the delta function in (7.19) which is appropriate for holomorphic λ (and is less singular than (7.15) and (7.16)) In this language the choice of C's translates into a choice of position of poles. Secondly, one must take global issues into account.
In particular, as mentioned above, the space of pure spinors can be covered with sixteen coordinates patches. In order to avoid landing in the singular gauge discussed above, one should arrange such that the expression for the path integral insertions valid in any given patch always contains at least one pole that lies in another patch.
Work in this direction is in progress [31]. We also refer to [32] for related relevant work.

Conclusion
In this paper we have studied tree-level and one-loop amplitudes in the minimal pure spinor formalism, in particular those with a Q exact state. The amplitude prescription includes constant spinors C I α and constant tensors B J mn that are used to define the picture changing operators which are necessary to absorb the zero modes of the worldsheet fields. Amplitudes should be independent of these constant tensors because the Lorentz variation of the PCO's is Q exact. The first computation we performed demonstrated that this argument does not hold, because a tree-level amplitude does depend on the choice of C's and a certain amplitude for a given choice of C I α is not Lorentz invariant. In the subsequent section it was shown that integrating over the C's, which was originally done to make the formalism manifestly Lorentz invariant, results in a prescription that decouples Q exact states. We also introduced a formulation of the minimal formalism at tree level in which the insertions of the picture changing operators are replaced by a (unique) Lorentz tensor, so the formalism is manifestly Lorentz invariant. BRST exact states are shown to decouple and it also turned out that this formulation is equivalent to the formulation in which one integrates over C I α . At one loop we found similar problems in the case we did not include the integral over B. Although the Lorentz variation of the PCO's is Q exact, the Lorentz variation of a one-loop amplitude does not vanish. At least not after the λ, N integrations, as one would expect. One expects the Lorentz variation to vanish after the λ, N integrals because the formal argument for decoupling of Q-exact states uses that picture changing operators are BRST closed. In the minimal formalism however the picture changing operators are BRST closed in a distributional sense, QY ∼ xδ(x) with x that depends on λ and N, so the amplitudes should vanish if distributional identities hold and this requires performing the integrations of λ, N but none of the other integrations. The case with an integral over B is dealt with in the companion paper [18]. That paper contains a proof of decoupling of unphysical states in the minimal pure spinor formalism including an integral over B.
In the tree-level case one could reformulate the prescription so that the picture changing operators are replaced by a BRST closed Lorentz tensor, as mentioned above. So one may wonder whether something similar can also be done at one loop.
We showed in section 6 that this is not possible. More precisely, we showed that if the picture changing operators are Lorentz invariant and BRST closed then all one-loop amplitudes vanish.
Note that the problems we find in the minimal formulation at tree level and one loop are not present in the non-minimal formalism. In this formalism the PCO's are replaced by the regularization factor N . In contrast to the PCO's, N is Q closed without subtleties. Hence the non-minimal formalism does not suffer from such problems.
In [19] we showed that both the PCO's and the regularization factor N come from a proper BRST treatment of fixing the gauge invariance generated by shifting the zero modes of the worldsheet fields. The difference between the minimal and non-minimal formalism can be understood as choosing different contours for the zero modes integrations. As became apparent in this work the choice that leads to the minimal formalism gives rise to anomalies. Moreover we saw that the gauge condition implicit in the current formulation of the amplitude prescriptions is singular and localizes the pure spinor zero mode integrals at the λ α = 0 locus, which should be excised from the pure spinor space for the theory to be non-anomalous. We suspect that the three-loop problems in the non-minimal formalism are also due to this singular gauge choice. To avoid these problems one should reformulate the theory in a non-singular gauge. We hope to report on this in the future.

A Definitions, conventions and technical results
This appendix contains detailed derivations of arguments we used in the main text. subgroup. The following two subsections deal with the Lorentz generators and measures for the pure spinor sector. Their main purpose is to set the conventions, however they contain more than just that. The fifth subsection is about gamma matrix traceless invariant tensors. Finally there is a subsection on the chain of operators that is used in the construction of the composite b ghost.

A.1 Invariant tensors
Before we give the definition of an invariant tensor it is useful to recall what a rep- where A ∈ SO(N) and g(A) is a linear map from C d → C d . The fundamental representation is given by d = N and g is the identity map (g(A) = A): In fact this can be generalized to construct a second representation from any given This is called the conjugate representation. Note the position of the indices on the conjugate representation is opposite to the original representation. This is very convenient because together with the rule that indices can only be summed over if one is up and one is down, tensors transform as indicated by their free indices. In particular combinations without free indices are invariant. For example for an arbitrary representation and its conjugate An invariant tensor is a tensor that transforms into itself under all elements of the group. For example δ a b is an invariant tensor for any representation. Note the range of a and b depends on the (dimension of the) representation. Its transformation is given by For SO(N) δ ab is also an invariant tensor where a, b denote the vector representation, hence they run from 1 to N. Invariant tensors can be used to construct invariants from tensors. Objects that consist of (covariant) tensors and invariant tensors transform according to their free indices. In particular combinations without free indices are where the definition of SO(N) was used.
The complex conjugate of a representation, g(A), is given by g * (A). One can check this always defines a representation if g(A) did. If a representation is equivalent to its complex conjugate it is real. For SU(N) the conjugate of the fundamental representation is equivalent to the complex conjugate because A −1T = A * .

A.2 Clifford algebra and pure spinors
The Clifford algebra in ten dimensions with Euclidian signature is given by is an invariant tensor. A proof for the four dimensional case can be found in [33]. The Clifford algebra has a representation in which the 32 by 32 components Γ matrices are off diagonal: where α, β = 1, · · · , 16. The Lorentz generators Σ become This implies the representation of the Lorentz group is reducible: The sixteen dimensional representation is the Weyl representation and it is not equivalent to its conjugate, hence the prime. The γ m αβ are invariant tensors with respect to Weyl representation.
Note the Clifford algebra now reduces to In particular (γ m ) αβ is the inverse of (γ m ) αβ . An explicit solution to (A.12) is given in the next section.  From (A. 15) we see that the SO(2N) algebra has an N 2 dimensional subalgebra. This subalgebra contains a U(1) generated by M ≡ M a a and the other N 2 − 1 generators, The generators (M S ) a b are traceless and generate an SU(N), The In our case we will be interested in decomposition of representations the (Wick rotated) Lorentz group SO(10) under SU (5). For several cases of interest the decomposition reads, where the subscripts are the U(1) charges.

A.3.1 Charge conservation and tensor products
The In order to find the constant of proportionality, computing a single component on both sides suffices.

A.3.2 Dynkin labels and gamma matrix traceless tensors
Throughout this work we denote irreducible representations by their dimensions. This is slightly ambiguous, therefore we clarify what we mean exactly by specifying the Dynkin labels of the highest weight state of the representation. where the Dynkin labels are specified. These representations are discussed in more detail in [35]. There are three gamma matrix traceless tensors that interest us in particular: [ For the three tensors above the computer algebra program LiE can be used to conclude there is only one independent invariant tensor. Note this is consistent with the arguments in [36], where it is argued that a tensor which is symmetric and gamma matrix traceless, let us say in some indices α i , is completely specified by the components where the α's are all +. In order to see that this implies there is only one independent invariant tensor of the form of T 1 note that for an invariant tensor the components (X 1 ) +++  [36] implies that the entire invariant tensor is completely specified by a single component and therefore the space of invariant tensors of the form of T 1 is one dimensional. The above argument applies equally well to T 2 and T 3 .

A.3.3 Explicit expression for gamma matrices and pure spinors
A solution to (A.12) is given by Note these matrices are skew diagonal, this is a consequence of the charge conservation property of invariant tensors.
A pure spinor is a Weyl spinor that satisfies The explicit expression of the three form gamma's is: (A.42)

A.4 Pure spinor Lorentz generators
In our computations we often need the explicit form of the pure spinor Lorentz generators, N mn , which are given by In this subsection these components are given in terms of the SU(5) components of λ and w in the gauge w a = 0. The SU(5) components of (A.43) are given by The explicit gauge invariant form is obtained by plugging in the expressions for γ After using the pure spinor solution and setting w a to zero J in terms of the free variables is given by In the gauge w a = 0 this becomes

A.4.1 Lorentz currents with unconstrained spinors
As mentioned in section 2 the action for the eleven independent components of λ and their conjugate variables can be used to prove the pure spinor OPE's from (2.7).
This will be demonstarted below. First we consider two unconstrained bosonic Weyl spinors. The OPE for such fields is given by The OPE of the pure spinor Lorentz current with itself is given by Tr(γ m 1 m 2 γ n 1 n 2 ) (z − w) 2 .
Using identities, We can read off the algebra of the Lorentz charges from the single pole in the OPE In case the worldsheet fields are fermionic, the OPE remains the same: The Lorentz generator for the fermionic variables has a minus sign: This sign is necessary to reproduce the commutation relation (A.63). As a consequence the sign in the double pole in the OPE changes from -4 to +4. This coefficient is called the level. We would like the Lorentz current of the combined p, θ and λ, w sector to have level one, since this is the level of the ψ sector in the RNS formalism.
This implies the N (λw) generators must have level −3. In the next subsection we explain how such currents can be obtained from the pure spinor action after gauge fixing.
All other components of the Nλ OPE can be checked along the same lines. The N mn N pq OPE is a different story. The single pole always leads to the correct Lorentz algebra, but the coefficient of the double pole depends on which SU(5) components we choose to take. For instance The first OPE would imply a Lorentz current level of − 7 4 and the second one −1. It will be shown below that it is possible to deform the currents in equations (A.52)-(A.55) by conserved quantities such that the level of the N mn N pq OPE is minus three [7]. This fixes the total derivatives one has to add to (A.57) in order for the JN OPE to be regular. Demanding N mn is a primary field of weight one determines the total derivatives in the stress energy tensor. If one now computes the JT OPE, a ghost number anomaly value of minus eight follows. This cannot be adjusted.
The deformations are most easily given after bosonization of λ and w, which is given by where φ, χ are chiral bosons satisfying The OPE's for these new variables are Note that the field equations imply the∂ operator annihilates these deformations.
Hence the deformed charges are still conserved. Furthermore the deformations do not modify the Nλ OPE, which is manifest in the s, t variables.

A.5 Lorentz invariant measures
The Lorentz invariant measures for both the weight zero field, λ α , and the weight one field, N mn , are discussed below. Both these measure were first introduced in [3] and the λ zero mode measure is also discussed in [32].
N mn ) zero modes at tree level, the measure for the λ zero modes must have ghost number +8. In addition it must be Lorentz invariant. This results in [dλ]λ α λ β λ γ = X αβγ β 1 ···β 11 dλ β 1 ∧ · · · ∧ dλ β 11 (A.89) for some invariant tensor X. The number of invariant (3,11) tensors with spinor indices that are symmetric in the upper indices and antisymmetric in lower ones is one [26]. In other words there is only one possibility for X which is given in (2.28).
Because the LHS of (A.89) is zero when contracted with γ m αβ , the RHS should vanish too. It does because there are no scalars in 10 ⊗ 16 ⊗ Asym 11  where we used (ǫT ) +++ β 1 ···β 11 is only non-zero if β 1 , . . . , . This is a consequence of the M charge conservation property of invariant tensors.

A.6 Gamma matrix traceless projectors
The operator Λ αβγ is introduced in (4.12). This equation has a special form and in this subsection we explain it. First note that I αβγ are contracted with covariant objects the total object is Lorentz invariant. After performing the integral the object is of course still Lorentz invariant and therefore I must be an invariant tensor. Furthermore I αβγ α ′ β ′ γ ′ must be symmetric in both its upstairs and downstairs indices and since λ is a pure spinor I must satisfy γ m αβ I αβγ α ′ β ′ γ ′ = 0. The SO(10) invariant tensors of the form T (αβγ) (α ′ β ′ γ ′ ) form a vector space which is two dimensional as can be computed by counting the number of scalars in Sym 3 16 ⊗ Sym 3 16 ′ [26]. A basis of this vector space is given by Since λ is a pure spinor where we used (cf. (A.27)) We could have anticipated ending up with one equation for c 1 , c 2 because 10 ⊗ 16 ⊗ Sym 3 16 ′ contains one scalar.
In summary the number of scalars in Sym 3 16 ⊗ Sym 3 16 ′ determined the number of degrees of freedom (c i ) and the number of scalars in 10 ⊗ 16 ⊗ Sym 3 16 ′ determined the number of relations between them.

A.6.1 Arbitrary rank
The tensor in equation (A.98) can be denoted as There is a unique such tensor because the number of scalars in Gam 3 16⊗(16 ′ ) 3 is one (cf. (A.29) for the meaning of Gam). In fact there is one scalar in Gam n 16 ⊗ (16 ′ ) n for any n. In order to write an explicit expression for δ ((α 1 β 1 · · · δ αn)) βn for any n we look for a basis of rank (n, n) invariant tensors that are symmetric in both their upper and lower indices. For even n the number of scalars in Sym n 16 ⊗ Sym n 16 ′ is n 2 + 1. For odd n the number of scalars in Sym n 16 ⊗ Sym n 16 ′ is n−1 2 + 1. Since odd n is of more relevance to this work we explicitly give the basis for odd n. The n−1 2 + 1 basis elements are given by where k = n−1 2 . In order to see these tensors are independent compute the following components: We can conclude for some coefficients c i , which can be explicitly computed as we did for the n = 3 case. Note the above is for odd n. Even n works very much in the same way, the only difference is the last δ in all the T 's. If one removes this, the T 's form a basis for the even case.
A.7 Chain of operators for b ghost The following chain of operators plays an important role in the b ghost: The text below is essentially a summary of section 3 of [37]. The primary fields of weight two that solve the above equations are given by : Only the antisymmetric part of L αβγδ is given because in [37] the full L αβγδ is not given in terms of gauge invariant objects. An explicit expression is known within the Y formalism [37,38,39] and it is also proven all Y dependence from L αβγδ disappears when contracted with Z αβγδ . In [3] L αβγδ is given as NB2: the coefficients of the total derivative terms depend on the normal ordering prescription and the ones above are only consistent with the prescription of [37].

B Detailed computations of I k
This appendix contains the details of the λ integrals that appear at one loop. We are especially interested in those that appear in computations involving a Q exact state.
A typical integral one encounters in an amplitude in subsection 5.1 is given by By charge conservation we can conclude at most two choices for β 2 , . . . , β 11 lead to a non vanishing I ′ k for any k. This follows from There are only two choices we can make. For example for k = 3 equation (B.2) implies only the components with N(β 2 · · · β 11 ) = − 1 2 are non vanishing. Thus β 2 · · · β 11 must consist of either seven 10 indices and three5 or a +, five 10's and four5's.
In section B.1 we first compute all integrals of the form Since I k vanishes for k < 3 (cf. (5.12)-(5.14)), we are only interested in I ′ k for k ≥ 3. By a similar argument the I ′ k 's are also only non vanishing for at most two choices of δ 1 δ 2 δ 3 . In the last subsection half of the non vanishing components of I 3 and all components of I 5 are computed.

B.1 Coefficients in λ integrals
For a given k at most two components of Λ give non vanishing results. We can make three choices for β 1 in I ′ k , all three choices lead to an integral of the form (not necessarily for the same k): After some algebra one finds the only non vanishing components of the I ′′ k 's are: The first step to obtain these results is finding the number of invariant tensors with the appropriate symmetries, this is one in all cases but the second. Finding the coefficients requires more work, this is done in subsection B.1. All these coefficients are fixed by (4.12), including the overall factor. Two corollaries are (d 1 ǫ d 2 )a 1 a 2 a 3 a 4 ) + 2 perms, (B.9) [dλ] 1 λ + λ a 1 a 2 · · · λ a 7 a 8 Λ +d 1 d 2 = c 3 ǫ a 1 a 2 a 3 a 4 (d 1 ǫ d 2 )a 5 a 6 a 7 a 8 + 2 perms (B.11) and [dλ]  If we now use equations (B.11) and (B.12) to evaluate the LHS of the above integrals we completely determine the values of c 3 , c 4 , c 5 . In fact we find more than three equations, but they include only three independent conditions as they should. To obtain c 3 one has to write out λ a and λ b in (B.13) and then perform all the contractions of the two ǫ's with the RHS of (B.11): Finding c 4 and c 5 is more involved. The LHS of (B.14) can be evaluated as  B.2 Computing the I k 's The idea of this section is simple, use the explicit form of the gamma matrices and the λ integrals (B.5)-(B.10) to evaluate I k . In practice this involves a lot of computation.
We already know I 0 , I 1 , I 2 and I 6 all vanish. By the charge conservation property there is only one choice of β 2 · · · β 11 for which I 5 does not vanish. For I 3 and I 4 we can make two choices. We explicitly compute I 3 for β 2 , · · · , β 11 = +, c 1 , c 2 , c 3 , c 4 , b 1 b 2 , · · · , b 9 b 10 .