The Overall Coefficient of the Two-loop Superstring Amplitude Using Pure Spinors

Using the results recently obtained for computing integrals over (non-minimal) pure spinor superspace, we compute the coefficient of the massless two-loop four-point amplitude from first principles. Contrasting with the mathematical difficulties in the RNS formalism where unknown normalizations of chiral determinant formulae force the two-loop coefficient to be determined only indirectly through factorization, the computation in the pure spinor formalism can be smoothly carried out.


Introduction
Scattering amplitudes led to the discovery of string theory more than 40 years ago.
But after all these years, explicit results for higher-loop and/or higher-point amplitudes are relatively sparse. In fact, since the publication of the famous review by D'Hoker and Phong [1] in 1988, there has been a small number of new ten-dimensional scattering computations. Using either the RNS or GS formalisms, the extensions to our knowledge in higher loops [2] or higher points [3,4,5,6] were limited to bosonic external states while the overall coefficients were not always under consideration 3 .
And the amplitudes in the pure spinor formalism were also computed up to the overall coefficients. That has changed since [26], where the precise normalizations for the pure spinor measures were determined and where it was also shown how to evaluate integrals in pure spinor space.
So in this paper we use and extend the results of [26] to obtain the coefficient of the type IIB (and IIA [27] [28]) two-loop massless four-point amplitude from a first principles computation and for the whole supermultiplet. To achieve that we use pure spinor measures which present the feature of having simple forms for all genera, in deep contrast with the complicated superstring measure for the RNS formalism [29,30]. As mentioned in [31], it is still an unsolved problem to find the precise normalizations for the chiral bosonization formulae of [32]. Therefore the two-loop coefficient can not be obtained from a direct calculation in the RNS formalism. In fact, computing the amplitude up to the overall coefficient already required several years of effort which resulted in an impressive series of papers [33,2], so the strategy adopted in [31] was to fix the two-loop coefficient indirectly by using factorization. So in this respect the calculations of this paper make it very clear how the pure spinor formalism can surpass the RNS limitations. But to present our results 3 There are however powerful approaches to discuss the coefficients which do not require direct ten-dimensional scattering computations [7] [8]. 4 The use of the pure spinor formalism however is not limited to scattering amplitudes only.
we have chosen to adopt the clear conventions of [31], which also eases the detection of any mismatches.
In section 2 the conventions and several pure spinor specific results are written down.
Emphasis is made regarding the generality and simplicity of the pure spinor setup. The computations of the three-and four-point amplitudes at tree-level are performed in section 3 to show that the conventions of section 2 match the RNS ones of [31] such that A PS 0 = A RNS 0 , where

KKC(s, t, u)
Then we use the very same machinery of the tree-level computation to obtain also the full supersymmetric one-and two-loop amplitudes -including their precise coefficients -in sections 4 and 5,

1)
A PS 2 = (2π) 10 δ (10) which explicitly shows that with the pure spinor formalism those coefficients follow directly from a first principles computation. But we find disagreement with the RNS results reported by [31], namely (1. 3) The mismatches seen in (1.3) will deserve some consideration. On one hand, the previous PS computation of the one-loop coefficient in [26] by one of the authors claimed agreement with the RNS result of [31]. But as will be pointed out in section 4, [26] made a mistake in the evaluation of the b-ghost integral which explains the difference with the computation of this paper. On the other (RNS) hand, we argue in section 4 that [31] forgot the two factors of 1/2 from the GSO projection in the left-and right-moving sectors in their measure. This observation will also explain the 1/2 4 mismatch at two-loops of section 5, as [31] fixed the two-loop coefficient using a factorization constraint which depends quadratically on the one-loop coefficient 5 .
In the appendix A we present the detailed covariant computation of the two-loop kinematic factor needed in section 5. This appendix can be regarded as a fully SO(10)covariant proof of the 2-loop equivalence 6 between the non-minimal and minimal pure spinor formalisms, and is analogous to the covariant proof of [34] for the 1-loop case. The appendix B is devoted to proving a formula mentioned en passant in [15] which is used to rewrite the two-loop amplitude in terms of integrals in the period matrix instead of in the Teichmüller parameters.

The conventions
The non-minimal pure spinor formalism action for the left-moving sector reads [11] with the constraints (λγ m λ) = (λγ m λ) = (λγ m r) = 0. The space-time dimensions are the following [26] [ The OPE's for the matter variables following from (2.1) can be computed to be 3) The Green-Schwarz constraint d α (z) and the supersymmetric momentum Π m (z) are which satisfy the following OPE's 6 As will be mentioned in appendix A, there is a loophole in the 2-loop equivalence proof of [18]. Some terms in the non-minimal pure spinor kinematic factor were argued to vanish using a U (5) decomposition but, as will be shown explicitly using the identities of [21], are in fact proportional to the kinematic factor of the minimal pure spinor formalism. As this loophole only affects the proportionality constant, it does not alter the conclusions of [18] but had to be taken into account here.
where D α = ∂ ∂θ α + 1 2 (γ m θ) α ∂ m is supersymmetric derivative. The composite b-ghost is given by [11] (see also [35] and satisfies [11] {Q, b(z)} = T (z) (2.6) where the BRST-charge Q and the energy-momentum tensor T (z) are Scattering amplitudes in the non-minimal pure spinor formalism use vertex operators in unintegrated and integrated forms, which for the massless states are given respectively by where A α (X, θ), A m (X, θ), W α (X, θ), F mn are the standard 10-dimensional N = 1 SYM superfields [36]. They have the following θ-expansion [37][17] Vertex operators for the closed string are V (z, z) =κV (z) ⊗Ṽ (z) and U (z, z) =κU (z) ⊗ U (z) with the understanding that only the left-moving modes carry the e ik·x factor.κ is the overall vertex operator normalization which will be fixed below toκ = κ, where κ is the normalization convention used in [31]. Therefore as in [31], its precise value in terms of α ′ and the string coupling constant [38] will not be needed here.
Finally, the string coupling constant appearing in scattering amplitude computations in the pure spinor formalism is e (2g−2)µ . As discussed below, by choosing a convenient normalization for the pure spinor tree-level measures its equality with the RNS convention of [31] e (2g−2)µ = e (2g−2)λ will follow.
They are where Ω IJ is the period matrix of the Riemann surface. It is well-known that for g = 1 the period matrix is given by the Teichmüller parameter τ .
It is convenient to consider the genus-g expectation value of the exponentials at the same time as the integration over the non-zero modes of the pure spinor variables, as the latter is equal to (det∂∂) 5 [26]. When both expressions are computed the determinant factors cancels out and one can use the following expression for their combined result. Therefore by using (2.29) the integration over non-zero modes of the pure spinor variables is already taken care of. For the sphere one has F 0 (z i , z j ) = |z ij | whereas for genus g ≥ 1 it can be written in terms of the prime form as [1] where w I (z) (I = 1, ..., g) are the holomorphic 1-forms over Σ g .
From (2.27) and (2.29) it follows that in amplitudes of closed string states the factors of A g cancel in the always-present product of, (2.31) The independence of the closed string amplitude with respect to the area of the surface follows from the fact that the number of bosonic and fermionic conformal weight-zero variables is the same.
The topological prescription [11] for computing the 4-point amplitudes at tree-level, one-and two-loops 7 is The 1 2 factor appearing in the two-loop amplitude was argued for in [39]. Every Riemann surface of genus 2 can be written like a hyperelliptic curve y 2 = h(z) where h(z) is a polynomial of degree 6 and y is the coordinate over CP 1 . This curve has the Z 2 symmetry y → −y, so the 1/2 factor is needed. We would like to thank Cumrun Vafa for this explanation.
is the fundamental domain of the Riemann surface of genus 1 (genus 2) and N is the regulator where the normalization 1/2π comes from bosonic string theory [40] because the topological prescription is based on it. With the above conventions, the space-time dimension of the genus-g four-point amplitudes is given by [A g ] = 8. In the following sections we don't keep track of the overall sign of the amplitudes.
Following [31] we use Furthermore Y s has space-time dimension −2 and is given by Mandelstam variables satisfying s + t + u = 0. Finally, the omnipresent supersymmetric kinematic factor K can be conveniently represented by the pure spinor superspace expres- where the brackets here are defined such that (λ 3 θ 5 ) = 1 [21]. While the computations of [31] did not involve the whole supermultiplet, this representation of K is convenient because its bosonic component expansion has the same normalization of the kinematic factor K of [31], K = (e 1 · e 2 ) 2tu(e 3 · e 4 ) − 4t(k 1 · e 3 )(k 2 · e 4 ) + perm + fermions (2.39) where the fermionic terms can be looked up in [21].

The normalization of zero-modes
Since the dimension of the zeroČech cohomology group H 0 (Σ g , Ω 1 ), where Ω 1 (Σ g ) is the sheaf of holomorphic 1-forms over Σ g , is equal to the genus g of the Riemann surface we expand a generic conformal weight (1,0) field as [11] where φ i are the zero modes and {w i (z)dz} is a basis of the H 0 (Σ g , Ω 1 ) group such that where a i and b j are the generators of the H 1 (Σ g , Z) = Z 2g homology group and Ω ij is the period matrix [42]. If we expand φ over another basis {α j } related by Expanding the fields over the new basis as φ = g j=1 φ ′j α j one can show that the measure satisfies dφ ′1 · · · dφ ′g = det(B) ǫ dφ 1 · · · dφ g , (2.43) where ǫ = +1(−1) for bosonic (fermionic) fields. In the non-minimal formalism the integration measures for conformal weight-one fields is defined in terms of the φ ′ components, but it is more convenient to use the {w I } basis in explicit computations. To account for this we absorb the Jacobian (2.42) equally into each of the [dφ I ] measures as Similarly, the appearance of A g in the measures of the conformal weight-zero variables [λ α , λ α , r α , θ α ] follows from the expansion in a complete set of eigenfunctions for the Laplacian of the worldsheet [44] λ

On the normalization of the holomorphic 1-forms
The result of scattering amplitudes in the pure spinor formalism does not depend on the normalization of the holomorphic 1-forms w I (z). To see this one notes that in closed string amplitudes 8 at genus g the difference between the number of independent fermionic and bosonic conformal weight-one left-moving variables is always 16g + 11g − 11g − 11g = 5g, corresponding to d I α , s α I , w I α and w α I . As Z g appear in the conformal weight-one measures as Z 1/g g , their total contribution to closed string amplitudes is always |Z 5 g | 2 = Z 10 g . Furthermore, when saturating the 11g s α I zero modes the regulator factor N provides 11g d I α zero-modes as well -because they appear in the combination (s I d I ) in N and there is nowhere else to get s I α zero-modes from. So to complete the saturation of d I α the b-ghosts and external vertices will always provide 5g factors of |d I α w I (z)| 2 , which scales as x 10g under w I (z) → xw I (z). To finish the proof it suffices to note from (2.41) and (2.42) that Z g scales as Z g → x −g Z g and therefore |Z 5 g | 2 offsets the scaling of the |w 5g I | 2 factors from the b-ghosts and external vertices.

Tree-level
The massless four-point amplitude at tree-level is given by (2.32), (3.1) The amplitude (3.1) was computed in components by [17] and later expressed in pure spinor superspace up to an overall normalization in [21], where it was used that The normalization of the tree-level amplitude of [21] can be determined a posteriori by using the precise value for the expectation value of the exponentials, where A 0 = 4π is the area of the sphere. Doing that in the computations of [21] we obtain, where and the kinematic factor K 0 is given by the pure spinor superspace expression [21] where the last equality follows from (2.28). Using (2.27) we get and therefore where we used that R 2 = √ 2 2 16 π .

The tree-level normalization
To fix the normalizations at tree-level to match those of [31] we need two conditions [38], therefore we also evaluate the three-point amplitude, which is given by Using (2.29), the component expansion found in [34] and the fact that (k i · k j ) = 0 where we used that and W 3 = (e 1 · e 2 )(k 2 · e 3 ) + (e 1 · e 3 )(k 1 · e 2 ) + (e 2 · e 3 )(k 3 · e 1 ) is the 3-pt kinematic factor in the RNS computation of [31].
In the normalization conventions of [31] the tree-level tree-and four-point amplitudes were shown to be given by 9

10)
(3.11) Comparing the RNS results of (3.10) and (3.11) with the corresponding PS amplitudes of

Two-loop
The two-loop massless four-point amplitude in the non-minimal pure spinor formalism is given by where denote the zero-mode integrations The 32 (22) zero-modes of d α (s α ) are denoted by d I α (s α I ) for I = 1, 2. As shown in [11], they are saturated by the different factors of (5.1) as where f ij (y) ≡ w i (y)w j (y), i, j = 1, 2 is the basis of holomorphic quadratic differentials for the genus-2 Riemann surface [46]. It follows from a short computation that, d 2 y j µ j (y j )∆(y 1 , y 2 )∆(y 2 , y 3 )∆(y 3 , y 1 ) 1 (λλ) 6 (λγ abc r)(λγ def r)(λγ ghi r)( In the computation of (5.4) one can check that combinations containing a different number of d 1 α and d 2 α zero modes e.g., vanish trivially due to the index symmetries, confirming the zero mode counting of (5.3).
Using the period matrix parametrization of moduli space the b-ghost insertions become where d 2 Ω IJ = d 2 Ω 11 d 2 Ω 12 d 2 Ω 22 and we used the identity of the appendix B.
The integration over [dw I ][dw I ] can be done using the results of [26] taking into account the different normalizations for the measures (2.11) and (2.12), It is straightforward to use the measure (2.14) to integrate over [ds 1 ][ds 2 ], and the amplitude (5.1) becomes where the only non-vanishing contribution from the external vertices contains two d 1 and two d 2 zero-modes coming from (α ′ /2) 4 (dW ) 4 . Integrating the d α zero-modes in (5.6) using (2.15) and (4.8) -(4.10) one gets where the non-minimal kinematic factor K is given by K 2 = (λγ m 1 n 1 p 1 r)(λγ def r)(λγ m 2 n 2 p 2 r)(λγ m 1 def m 2 λ) and we defined In the Appendix A we will show that where the second equality follows from (2.28). Hence (5.7) is given by From the formula (2.31) we get which together with Z 10 2 = 2 −10 det(ImΩ IJ ) −5 implies that which is the final result for the 2-loop amplitude 12 . And we have shown that the computation of the whole supersymmetric amplitude including its coefficient is straightforward using the non-minimal pure spinor formalism. 12 The coefficient obtained here is 1/16 times the result reported by [31]. This difference can be accounted for by the missing factor of 1/4 in their 1-loop result which is used as input in their fixing of the 2-loop coefficient through factorization.

Conclusions
We used the genus-g measures in the non-minimal pure spinor formalism to find the overall coefficient of the two-loop amplitude and have shown that there are no major differences in carrying out the computations when compared against the analogous calculations for the tree-level and one-loop amplitudes. In fact, this task is significantly simplified by the pure spinor superspace identities of [21] linking the four-point kinematic factors.
These observations must be compared against the unsolved difficulties in the RNS formalism, which besides having no explicit computations for the whole supermultiplet has to rely on a factorization procedure to find the two-loop coefficient. Furthermore, we argued that the mismatch of 1/16 found in the two-loop amplitude compared with the result of [31] is due to a missing factor of 1/4 from the GSO projection in their one-loop amplitude.
Acknowledgements: CRM and HG would like to thank Eric D'Hoker, Nathan Berkovits and Stefan Theisen for discussions. CRM acknowledges support by the Deutsch-Israelische Projektkooperation (DIP H52). HG acknowledges support by FAPESP Ph.D grant 07/54623-8.

Appendix A. Non-minimal two-loop kinematic factor
The non-minimal two-loop computation of section 5 leads to the kinematic factor In [18] it was shown 13 that (A.1) is proportional to (λγ mnpqr λ)(λγ s W )F mn F pq F rs (0,2) , the kinematic factor obtained in the minimal pure spinor formalism [15], whose equivalence with the RNS result of [2] was established in [16,21]. We will now evaluate all the terms in (A.1) to find the exact coefficient announced in (5.10). 13 There is a loophole in the proof of [18] though. In that proof the terms in (A.1) which are of the form kW W W F where argued to vanish after summing over the permutations. However we show here that by using the identities of [21] those terms are actually proportional to W F F F , so the conclusions of [18] still hold true. CM would like to acknowledge a question made by I. Park which sparked the motivation to revisit that proof.
To simplify the covariant computation of (A.1) we use (λγ def D)(λγ adef g λ) = 48(λλ)(λγ ag D) − 48(λγ ag λ)(λD) and drop the last term because (λγ m W I ) is BRSTclosed. And for the same reason we can use (λγ a γ g D) instead of (λγ ag D) in the first term.
Therefore (A.1) becomes The strategy to evaluate and simplify 14 (A.2) is straightforward due to the identities obeyed by the pure spinor λ α . One uses the SYM equation of motion for W α in the form of (λγ abc D)(λγ m W 1 ) = 1 4 (λγ m γ m 1 n 1 γ abc λ)F 1 and uses gamma matrix identities 15 in such a way as to get factors which vanish by the pure spinor property of (λγ m ) α (λγ m ) β = 0. For example, one gets identities like and Following the above steps (A.2) becomes The last line of (A.7) vanishes. To see this note that the factor inside brackets is BRSTclosed, so that we can replace (λγ a γ g D) by (λγ ag D). Furthermore (λγ gai D)(λγ ga D) = −(λγ ga γ i D)(λγ ga D) − 2(λγ a D)(λγ ia D) and the last term vanishes when acting on 14 These kind of computations confirm the observations made long ago that pure spinors simplify the description of super-Yang-Mills theory [47]. 15 The package GAMMA [48] is often very useful for these manipulations.
and (λγ i ) α (λγ i ) β = 0 due to the pure spinor property. Therefore by using the gamma matrix identity of and dropping the term proportional to the BRST charge and using momentum conservation (so that D α and D β effectively anti-commute) we get The first term in the RHS of (A.9) is proportional to k i and vanishes by momentum conservation, while the last term vanishes when acting on F 3 bc (λγ i W 4 )(λγ b W 1 )(λγ c W 2 ) for the same reason as explained above.
For convenience we write (A.7) as while K a 2 and K a 3 can be obtained by permuting the labels in K a 1 . Using the SYM equations of motion and a few gamma matrix identities we get After a long and tedious computation using straightforward manipulations and identities like (λγ mnpqr λ)F I mn F J pq = (λγ mnpqr λ)F J mn F I pq and [15] (λγ mnpqr λ)(λγ one gets To simplify the (−1,2) terms in (A.13) it is convenient to have λ α in the combination (λλ) by using the identities, and similarly 16) In [21] it was proved that (λγ mnpqr λ)(λγ s W 4 )F 1 mn F 2 pq F 3 rs (n,g) = −16(k 1 · k 2 ) (λA 1 )(λγ m W 2 )(λγ n W 3 )F 4 mn (n,g) (A. 17) and that (λA 1 )(λγ m W 2 )(λγ n W 3 )F 4 mn (n,g) is completely symmetric in the particle labels, hence where we also used the momentum conservation relation of (k 1 · k 3 ) + (k 1 · k 4 ) = −(k 1 · k 2 ).