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Published for SISSA by Springer

Received: March 10, 2010

Accepted: April 16, 2010

Published: May 5, 2010

The overall coefficient of the two-loop superstring

amplitude using pure spinors

Humberto Gomeza and Carlos R. Mafrab

aInstituto de F́ısica Teórica, UNESP - Universidade Estadual Paulista,

Caixa Postal 70532-2, 01156-970 São Paulo, SP, Brazil
bMax-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,

Am Mühlenberg 1, 14476 Potsdam, Germany

E-mail: humgomzu@ift.unesp.br, crmafra@aei.mpg.de

Abstract: 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 formulæ force the two-loop

coefficient to be determined only indirectly through factorization, the computation in the

pure spinor formalism can be smoothly carried out.

Keywords: Superstrings and Heterotic Strings, Superspaces

ArXiv ePrint: 1003.0678

Open Access doi:10.1007/JHEP05(2010)017

mailto:humgomzu@ift.unesp.br
mailto:crmafra@aei.mpg.de
http://arxiv.org/abs/1003.0678
http://dx.doi.org/10.1007/JHEP05(2010)017


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Contents

1 Introduction 1

2 The conventions 3

2.1 The normalization of zero-modes 8

2.2 On the normalization of the holomorphic 1-forms 9

3 Tree-level 9

3.1 The tree-level normalization 10

4 One-loop 11

5 Two-loop 13

6 Conclusions 15

A Non-minimal two-loop kinematic factor 15

B Period matrix parametrization of genus-two moduli space 20

1 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 rel-

atively 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 [3–8] were limited to bosonic external states while the overall coefficients

were not always under consideration.1

Since the discovery of the manifestly space-time supersymmetric pure spinor formal-

ism [11–14] there has been progress in extending results of scattering amplitudes2 to the

whole supermultiplet [12, 17–23] by using the pure spinor superspace [24] but explicit com-

putations for genus higher than two are still missing though [25–27]. And the amplitudes

in the pure spinor formalism were also computed up to the overall coefficients. That has

changed since [28], 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.

1There are however powerful approaches to discuss the coefficients which do not require direct ten-

dimensional scattering computations [9, 10].
2The use of the pure spinor formalism however is not limited to scattering amplitudes only. For reviews,

see [15, 16].

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So in this paper we use and extend the results of [28] to obtain the coefficient of the

type IIB (and IIA [29, 30]) 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 [31, 32]. As mentioned in [33], it

is still an unsolved problem to find the precise normalizations for the chiral bosonization

formulæ of [34]. 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 [2, 35–39], so the strategy adopted in [33] 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 we have chosen to adopt the clear conventions of [33], 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 sec-

tion 3 to show that the conventions of section 2 match the RNS ones of [33] such that

APS
0 = ARNS

0 , where

APS
0 = (2π)10δ(10)(k)κ4e−2λ

( √
2

212π6α′5

)

(

α′

2

)8

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,

APS
1 = (2π)10δ(10)(k)

κ4KK

29π2α′5

(

α′

2

)8 ∫

M1

d2τ

τ5
2

4
∏

i=2

∫

d2zi

4
∏

i<j

F1(zi, zj)
αki·kj

, (1.1)

APS
2 = (2π)10δ(10)(k)κ4e2λ

√
2KK

210α′5

(

α′

2

)10∫

M2

d2ΩIJ

(detImΩIJ)5

∫

Σ4

|Ys|2
∏

i<j

F2(zi, zj)
αki·kj

(1.2)

which explicitly shows that with the pure spinor formalism those coefficients follow di-

rectly from a first principles computation. But we find disagreement with the RNS results

reported by [33], namely

APS
1 =

1

22
ARNS

1 , APS
2 =

1

24
ARNS

2 . (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 [28] by one of the authors claimed agreement

with the RNS result of [33]. But as will be pointed out in section 4, [28] 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 [33] forgot the two factors

of 1/2 from the GSO projection in the left- and right-moving sectors in their measure. This

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observation will also explain the 1/24 mismatch at two-loops of section 5, as [33] fixed the

two-loop coefficient using a factorization constraint which depends quadratically on the

one-loop coefficient.3

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 equivalence4 between the non-minimal and minimal pure

spinor formalisms, and is analogous to the covariant proof of [40] for the 1-loop case. The

appendix B is devoted to proving a formula mentioned en passant in [17] which is used to

rewrite the two-loop amplitude in terms of integrals in the period matrix instead of in the

Teichmüller parameters.

2 The conventions

The non-minimal pure spinor formalism action for the left-moving sector reads [13]

S =
1

2πα′

∫

Σg

d2z
(

∂Xm∂Xm + α′pα∂θα − α′ωα∂λα − α′wα∂λα + α′sα∂rα

)

(2.1)

with the constraints (λγmλ) = (λγmλ) = (λγmr) = 0. The space-time dimensions are the

following [28]

[α′] = 2, [Xm] = 1, [θα] = [λα] = [ωα] = [sα] = 1/2, [pα] = [ωα] = [λα] = [rα] = −1/2.

(2.2)

The OPE’s for the matter variables following from (2.1) can be computed to be

Xm(z)Xn(w) ∼ −α′

2
δm
n ln|z − w|2, pα(z)θβ(w) ∼ δβ

a

z − w
. (2.3)

The Green-Schwarz constraint dα(z) and the supersymmetric momentum Πm(z) are

dα = pα − 1

α′ (γ
mθ)α∂Xm − 1

4α′ (γ
mθ)α(θγm∂θ), Πm = ∂Xm +

1

2
(θγm∂θ) (2.4)

which satisfy the following OPE’s

dα(z)dβ(w) ∼ − 2

α′
γm

αβΠm

z − w
, dα(z)Πm(w) ∼

γm
αβ∂θβ

z − w
,

dα(z)f(θ(w), x(w)) ∼ Dαf(θ(w), x(w))

z − w
, Πm(z)f(θ(w), x(w)) ∼ −α′

2

kmf(θ(w), x(w))

z − w
(2.5)

3For a compact Riemann surface S of genus g the correct factor is 1/22g , which is the number of spin

structures over S and is in agreement with factorization.
4As will be mentioned in appendix A, there is a loophole in the 2-loop equivalence proof of [20]. 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 [23], 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 [20] but had to be taken into account here.

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where Dα = ∂
∂θα + 1

2 (γmθ)α∂m is supersymmetric derivative. The composite b-ghost is

given by [13] (see also [41])

b = sα∂λα +
1

4(λλ)

(

2Πm(λγmd) − Nmn(λγmn∂θ) − Jλ(λ∂θ) − (λ∂2θ)
)

+
(λγmnpr)

192(λλ)2

[

α′

2
(dγmnpd) + 24NmnΠp

]

−α′

2

(rγmnpr)

16(λλ)3

[

(λγmd)Nnp − (λγpqrr)NmnNqr

8(λλ)

]

,

and satisfies [13]

{Q, b(z)} = T (z) (2.6)

where the BRST-charge Q and the energy-momentum tensor T (z) are

Q =

∮

(λαdα + wαrα), T (z) = − 1

α′∂Xm∂Xm − pα∂θα + ωα∂λα + wα∂λα − sα∂rα.

From (2.2) it follows that [Q] = [b] = [T ] = 0.

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

V (z) = λαAα, U(z) = ∂θαAα + AmΠm +
α′

2
dαW α +

α′

4
NmnFmn (2.7)

where Aα(X, θ), Am(X, θ), W α(X, θ), Fmn are the standard 10-dimensional N = 1 SYM

superfields [42, 43]. They have the following θ-expansion [19, 44–46]

Aα(x, θ) =
1

2
am(γmθ)α − 1

3
(ξγmθ)(γmθ)α − 1

32
Fmn(γpθ)α(θγmnpθ) + · · ·

Am(x, θ) = am − (ξγmθ) − 1

8
(θγmγpqθ)Fpq +

1

12
(θγmγpqθ)(∂pξγqθ) + · · ·

W α(x, θ) = ξα− 1

4
(γmnθ)αFmn+

1

4
(γmnθ)α(∂mξγnθ) +

1

48
(γmnθ)α(θγnγpqθ)∂mFpq + · · ·

Fmn(x, θ) = Fmn − 2(∂[mξγn]θ) +
1

4
(θγ[mγpqθ)∂n]Fpq + . . .,

where am(x) = emeik·x, ξα(x) = (2/α′)1/2χαeik·x and Fmn = 2∂[man] with [em] = 0 and

[χα] = 1/2. The space-time dimensions of the superfields and the vertex operators are

[Aα] = 1/2, [Am] = 0, [W α] = −1/2, [Fmn] = −1, [V (z)] = [U(z)] = 1. (2.8)

Vertex operators for the closed string are V (z, z) = κ̃V (z) ⊗ Ṽ (z) and U(z, z) = κ̃U(z) ⊗
Ũ(z) with the understanding that only the left-moving modes carry the eik·x factor. κ̃ is

the overall vertex operator normalization which will be fixed below to κ̃ = κ, where κ is

the normalization convention used in [33]. Therefore as in [33], its precise value in terms

of α′ and the string coupling constant [47, 48] 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

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normalization for the pure spinor tree-level measures its equality with the RNS convention

of [33] e(2g−2)µ = e(2g−2)λ will follow.

The construction of the zero-mode measures for the non-minimal pure spinor variables

was given in [13] and their precise normalizations were found in [28]. It is however con-

venient to use slightly different conventions for the measures of [28] to make their genus-g

dependence (and generality) explicit, facilitating computations in different genera. The

space-time dimensionless genus-g zero-mode measures are given by

[dλ]Tα1α2α3α4α5
= cλǫα1...α5ρ1...ρ11

dλρ1 . . . dλρ11 (2.9)

[dλ]T
α1α2α3α4α5 = cλǫα1...α5ρ1...ρ11dλρ1

. . . dλρ11
(2.10)

[dω] = cωTα1α2α3α4α5
ǫα1...α5ρ1...ρ11dωρ1

. . . dωρ11
(2.11)

[dw]Tα1α2α3α4α5
= cwǫα1...α5ρ1...ρ11

dwρ1 . . . dwρ11 (2.12)

[dr] = crT
α1α2α3α4α5ǫα1...α5δ1...δ11∂

δ1
r . . . ∂δ11

r (2.13)

[dsI ] = csTα1α2α3α4α5
ǫα1...α5ρ1...ρ11∂sI

ρ1
. . .∂sI

ρ11
(2.14)

[dθ] = cθd
16θ, [ddI ] = cdd

16dI (2.15)

with the following normalizations

cλ =

(

α′

2

)−2 1

11!

(

Ag

4π2

)11/2

cω =

(

α′

2

)2 (4π2)−11/2

11!5!Z
11/g
g

(2.16)

cλ =

(

α′

2

)2 26

11!

(

Ag

4π2

)11/2

cw =

(

α′

2

)−2 (4π2)−11/2(λλ)3

11!Z
11/g
g

(2.17)

cr =

(

α′

2

)−2 R

11!5!

(

2π

Ag

)11/2

cs =

(

α′

2

)2 (2π)11/2R−1

2611!5!(λλ)3
Z11/g

g (2.18)

cθ =

(

α′

2

)4( 2π

Ag

)16/2

cd =

(

α′

2

)−4

(2π)16/2Z16/g
g (2.19)

where R is arbitrary and parametrizes the freedom in choosing the normalization of the

tree-level amplitude and Ag is the area of the Riemann surface. As will be shown in section

3, using the value

R2 =

√
2

216π
(2.20)

fixes the tree-level normalization to be the same as in the RNS computations of [33]. The

tensors Tα1...α5
, T

α1...α5 are defined as

Tα1α2α3α4α5
= (λγm)α1

(λγn)α2
(λγp)α3

(γmnp)α4α5
(2.21)

T
α1α2α3α4α5 = (λγm)α1(λγn)α2(λγp)α3(γmnp)

α4α5 (2.22)

and satisfy

Tα1α2α3α4α5
T

α1α2α3α4α5 = 5! 26(λλ)3. (2.23)

The appearance of the area Ag and of the factor Zg will be explained in the next subsection.

They are

Ag =

∫

d2z
√

g, Zg =
1

√

det(2Im(ΩIJ))
, g ≥ 1 (2.24)

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where ΩIJ 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 .

To avoid cluttering in the formulæ we define the genus g bracket 〈 〉(n,g) as

〈M(λ, λ, θ)〉(n,g) ≡
∫

[dθ][dr][dλ][dλ]
e−(λλ)−(rθ)

(λλ)3−n
M(λ, λ, θ, r) (2.25)

for an arbitrary pure spinor superfield M(λ, λ, θ, r). With the above conventions the inte-

gral over the zero modes of pure spinor space becomes [28]

∫

[dλ][dλ](λλ)ne−(λλ) =
(7 + n)!

7! 60

(

2π

Ag

)11

, n ≥ 0 (2.26)

which together with (2.23) imply that

N(n,g) ≡ 〈λ3θ5〉(n,g) = 27R

(

2π

Ag

)5/2(α′

2

)2 (7 + n)!

7!
, n ≥ 0, (2.27)

where we used the abbreviated notation (λ3θ5) = (λγrθ)(λγsθ)(λγtθ)(θγrstθ). Due to the

identities of [23] the following trick from [28] is required for the tree-level, one- and two-loop

amplitudes

〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(n,g) = − K

29 32 5
〈(λ3θ5)〉(n,g) (2.28)

where K denotes the kinematic factor of [33], which will be written down below.

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 [28]. When both expressions are computed the determinant

factors cancels out and one can use the following expression

〈
4
∏

i=1

eik·x〉g = (2π)10δ(10)(k)
A5

g

(2π2α′)5
∏

i<j

Fg(zi, zj)
αki·kj

(2.29)

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 F0(zi, zj) = |zij |
whereas for genus g ≥ 1 it can be written in terms of the prime form as [1]

Fg(zi, zj)
αki·kj

= |E(zi, zj)|αki·kj

exp(−2π(ImΩ)−1
IJ (Im

∫ zj

zi

wI)(Im

∫ zj

zi

wJ)), (2.30)

where wI(z) (I = 1, . . . , g) are the holomorphic 1-forms over Σg.

From (2.27) and (2.29) it follows that in amplitudes of closed string states the factors

of Ag cancel in the always-present product of,

|N(n,g)|2〈
N
∏

i=1

eik·x〉g = (2π)10δ(10)(k)

√
2

22π6α′5

(

α′

2

)4((7 + n)!

7!

)2
∏

i<j

Fg(zi, zj)
αki·kj

.

(2.31)

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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 [13] for computing the 4-point amplitudes at tree-level,

one- and two-loops5 is

A0 = κ̃4e−2µ

∫

d2z4〈|N V 1(0)V 2(1)V 3(∞)U4(z4)|2〉 (2.32)

A1 =
1

2
κ̃4

∫

M1

d2τ1

4
∏

i=2

∫

d2zi〈|N (b, µ1)V
1(0)U i(zi)|2〉 (2.33)

A2 =
1

2
κ̃4e2µ

∫

M2

3
∏

I=1

d2τI

4
∏

i=1

∫

d2zi〈|N (b, µI)U
i(zi)|2〉 (2.34)

where M1 (M2) is the fundamental domain of the Riemann surface of genus 1 (genus 2)

and N is the regulator [13]

N =

g
∑

I=1

e−(λλ)−(wIwI )−(rθ)+(sIdI) (2.35)

〈 〉 denotes the integrations over the zero-modes

〈 〉 →
g
∏

I=1

∫

[dθ][ddI ][dr][dsI ][dwI ][dwI ][dλ][dλ] (2.36)

and the b-ghost insertion is [50, 51]

(b, µj) =
1

2π

∫

d2yjbzzµ
z
j z, j = 1, . . ., 3g − 3. (2.37)

where the normalization 1/2π comes from bosonic string theory [50] 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 [Ag] = 8. In the following sections we don’t keep

track of the overall sign of the amplitudes.

Following [33] we use d2τ = dτ ∧ dτ , d2z = dz ∧ dz (in particular
∫

Σ1
d2z = 2τ2).

Furthermore Ys has space-time dimension −2 and is given by

Ys = −s∆(1, 4)∆(2, 3) + t∆(1, 2)∆(3, 4), (2.38)

where ∆(i, j) ≡ w1(zi)w2(zj)−w1(zj)w2(zi) and wI(z) is the basis of holomorphic 1-forms

discussed below and s = −2(k1 · k2), t = −2(k2 · k3), u = −2(k1 · k3) are the Mandel-

stam variables satisfying s + t + u = 0. Finally, the omnipresent supersymmetric kine-

matic factor K can be conveniently represented by the pure spinor superspace expression

5The 1

2
factor appearing in the two-loop amplitude was argued for in [49]. Every Riemann surface of

genus 2 can be written like a hyperelliptic curve y2 = h(z) where h(z) is a polynomial of degree 6 and y is

the coordinate over CP 1. This curve has the Z2 symmetry y → −y, so the 1/2 factor is needed. We would

like to thank Cumrun Vafa for this explanation.

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K = 23040〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉, where the brackets here are defined such that

〈(λ3θ5)〉 = 1 [23]. While the computations of [33] 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 [33],

K = (e1 · e2)
[

2tu(e3 · e4) − 4t(k1 · e3)(k2 · e4)
]

+ perm + fermions (2.39)

where the fermionic terms can be looked up in [23].

2.1 The normalization of zero-modes

Since the dimension of the zero Čech cohomology group H0(Σg,Ω
1), where Ω1(Σg) is the

sheaf of holomorphic 1-forms over Σg, is equal to the genus g of the Riemann surface we

expand a generic conformal weight (1,0) field as [13]

φ(z) = φ̂(z) +

g
∑

i=1

wi(z)φi (2.40)

where φi are the zero modes and {wi(z)dz} is a basis of the H0(Σg,Ω
1) group such that

∫

ai

wj(z)dz = δij ,

∫

bi

wj(z)dz = Ωij i, j = 1, 2, . . . , g

(wi, wj) ≡
∫

Σg

wi wj dz ∧ dz = 2ImΩij (2.41)

where ai and bj are the generators of the H1(Σg, Z) = Z2g homology group and Ωij is the

period matrix [52]. If we expand φ over another basis {αj} related by wi = Bj
i αj then [53],

det

(

∫

Σg

wi wj dz ∧ dz

)

= det|B|2det

(

∫

Σg

αi αj dz ∧ dz

)

so that for

|detB| =
√

det(2ImΩij) = Z−1
g (2.42)

the basis {αj} is orthonormal, (αi, αj) = δij . Expanding the fields over the new basis as

φ =
∑g

j=1 φ′jαj one can show that the measure satisfies

dφ′1 · · · dφ′g = det(B)ǫdφ1 · · · dφg, (2.43)

where ǫ = +1(−1) for bosonic (fermionic) fields. In the non-minimal formalism the in-

tegration measures for conformal weight-one fields is defined in terms of the φ′ compo-

nents, but it is more convenient to use the {wI} basis in explicit computations. To ac-

count for this we absorb the Jacobian (2.42) equally into each of the [dφI ] measures as

(det(B)ǫ/gdφ1) · · · (det(B)ǫ/gdφg), which explains the factors of Zg in (2.16)–(2.19).

Similarly, the appearance of Ag in the measures of the conformal weight-zero vari-

ables [λα, λα, rα, θα] follows from the expansion in a complete set of eigenfunctions for the

Laplacian of the worldsheet [54]

λα(z) = λα
0 Λ0 +

∑

j

λα
j Λj(z, z) (2.44)

and Λ0 = 1 is the generator of the cohomology group H0(Σg,O) = C, where O is the sheaf

of holomorphic functions over Σg. Because the norm of Λ0 is ||Λ0||2 = Ag the measures of

the scalars must have the Jacobian A
ǫ/2
g (where ǫ = +1(−1) for bosonic (fermionic) fields),

explaining the factors of Ag in (2.16)–(2.19).

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2.2 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 wI(z). To see this one notes that in closed string

amplitudes6 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 dI
α, sα I , wI

α and wα I . As Zg appear in the conformal weight-one

measures as Z
1/g
g , their total contribution to closed string amplitudes is always |Z5

g |2 = Z10
g .

Furthermore, when saturating the 11g sα I zero modes the regulator factor N provides 11g

dI
α zero-modes as well — because they appear in the combination (sIdI) in N and there

is nowhere else to get sI α zero-modes from. So to complete the saturation of dI
α the b-

ghosts and external vertices will always provide 5g factors of |dI
αwI(z)|2, which scales as

x10g under wI(z) → xwI(z). To finish the proof it suffices to note from (2.41) and (2.42)

that Zg scales as Zg → x−gZg and therefore |Z5
g |2 offsets the scaling of the |w5g

I |2 factors

from the b-ghosts and external vertices.

3 Tree-level

The massless four-point amplitude at tree-level is given by (2.32),

A0 = κ̃4e−2µ

∫

d2z4〈|N V 1(0)V 2(1)V 3(∞)U4(z4)|2〉. (3.1)

The amplitude (3.1) was computed in components by [19] and later expressed in pure spinor

superspace up to an overall normalization in [23], where it was used that 〈∏4
i=1 eikix(zi,zi)〉 =

|z4|−
1

2
α′t|1 − z4|−

1

2
α′u. The normalization of the tree-level amplitude of [23] can be deter-

mined a posteriori by using the precise value for the expectation value of the exponentials,

〈
4
∏

i=1

eikix(zi,zi)〉0 = (2π)10δ(10)(k)

(

A0

2π2α′

)5

|z4|−
1

2
α′t|1 − z4|−

1

2
α′u, (3.2)

where A0 = 4π is the area of the sphere. Doing that in the computations of [23] we obtain,

A0 = (2π)10δ(10)(k)κ̃4e−2µ

(

4π

2π2α′

)5(α′

2

)4

K0K0C(s, t, u), (3.3)

where

C(s, t, u) = 2π
Γ(−α′s

4 )Γ(−α′t
4 )Γ(−α′u

4 )

Γ(1 + α′s
4 )Γ(1 + α′t

4 )Γ(1 + α′u
4 )

(3.4)

and the kinematic factor K0 is given by the pure spinor superspace expression [23]

K0 = 〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(3,0) = − K

29 32 5
〈(λ3θ5)〉(3,0) (3.5)

where the last equality follows from (2.28). Using (2.27) we get

K0 = K
N (3,0)

(29 32 5)
=

R√
2

(

α′

2

)2

K, (3.6)

6The analysis can be trivially modified to the open string.

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and therefore

A0 = (2π)10δ(10)(k)κ̃4e−2µ R2

2

(

2

πα′

)5(α′

2

)8

KKC(s, t, u) (3.7)

= (2π)10δ(10)(k)κ̃4e−2µ

( √
2

212π6α′5

)

(

α′

2

)8

KKC(s, t, u),

where we used that R2 =
√

2
216π .

3.1 The tree-level normalization

To fix the normalizations at tree-level to match those of [33] we need two conditions [47, 48],

therefore we also evaluate the three-point amplitude, which is given by

At = κ̃3e−2µ〈|NV (0)V (1)V (∞)|2〉. (3.8)

Using (2.29), the component expansion found in [40] and the fact that (ki · kj) = 0

At = (2π)10δ(10)(k)κ̃3e−2µ A5
0

(2π2α′)5
|Kt|2

hence,

At = (2π)10δ(10)(k)κ̃3e−2µ

√
2

26π6α′5

(

α′

2

)4

W3W 3 (3.9)

where we used that

|Kt|2 = |〈(λA1)(λA2)(λA3)〉(3,0)|2 =
|N(3,0)|2
28802

W3W 3 =

√
2

26π

(

2π

A0

)5(α′

2

)4

W3W 3

and W3 = (e1 · e2)(k2 · e3) + (e1 · e3)(k1 · e2) + (e2 · e3)(k3 · e1) is the 3-pt kinematic factor

in the RNS computation of [33].

In the normalization conventions of [33] the tree-level tree- and four-point amplitudes

were shown to be given by7

ARNS
t = (2π)10δ(10)(k)κ3e−2λ

( √
2

26π6α′5

)

(

α′

2

)4

W3W 3, (3.10)

ARNS
0 = (2π)10δ(10)(k)κ4e−2λ

( √
2

212π6α′5

)

(

α′

2

)8

KKC(s, t, u). (3.11)

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

of (3.9) and (3.7) it follows that

κ̃ = κ, e−2µ = e−2λ, (3.12)

so the PS and RNS tree-level normalization conventions are the same. The numerical value

of the parameter R in (2.20) was chosen precisely for this match to happen. After this

tree-level matching is done there remains no more freedom to adjust conventions.

7Note that [At] = 6 and [A0] = 8, so in [33] the factors of (α′/2) were forgotten.

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4 One-loop

The one-loop massless four-point amplitude is given by (2.33),

A1 =
1

2
κ4

∫

M1

d2τ1

4
∏

i=2

∫

d2zi〈|N (b, µ1)V
1(0)U i(zi)|2〉. (4.1)

The regulator in (2.35) becomes N = e−(λλ)−(w1w1)−(rθ)+(s1d1), 〈 〉 denotes the integra-

tions over the zero-modes of [θα, dα, rα, sα, wα, wα, λα, λα] and the b-ghost insertion written

in (2.37) reads

(b, µ1) =
1

2π

∫

d2zbzzµ
z
z. (4.2)

As discussed in [13], there is an unique way to saturate the zero-modes of all variables.

The b-ghost must provide two d1
α zero-modes with 1

263
(α′

2 )(λγmnpr)(d1γmnpd
1)w1w1, where

w1 = 1 is the holomorphic 1-form in the torus. Therefore the integral (4.2) is easily

computed to give

(b, µ1) =
1

273π

(

α′

2

)

(λγmnpr)(d1γmnpd
1)

(λλ)2
,

because
∫

d2zw1w1µ1 = 1. The integrated vertices contribute three d1
α zero-modes via

(α′

2 )3(d1W 2)(d1W 3)(d1W 4), so (4.1) becomes

A1 =
1

21532 π2
κ4

(

α′

2

)8 ∫

M1

d2τ

4
∏

i=2

∫

d2zi|K1|2〈
4
∏

i=1

eikX(zi)〉1, (4.3)

where the computation of the zero-mode integrations in

K1 =

∫

[dd1][ds1][dw1][dw1] e−(w1w1)+(s1d1) ×

×〈(λγmnpr)(d1γmnpd
1)(λA1)(d1W 2)(d1W 3)(d1W 4)〉(1,1) (4.4)

is straightforward and goes as follows. Using the measures (2.11) and (2.12) and the results

of [28] one gets
∫

[dw][dw]e−(ww) =
(λλ)3

(2π)11Z22
1

. (4.5)

Hence,

K1 =
1

(2π)11Z22
1

∫

[dd1][ds1]e(s1d1)〈(λγmnpr)(dγmnpd)(λA1)(dW 2)(dW 3)(dW 4)〉(4,1).

(4.6)

The integration over [ds] using the measure (2.14) leads to

K1 =
(2π)−11/2

26(11!5!)Z11
1 R

(

α′

2

)2 ∫

[dd1]Tα1...α5
ǫα1...α5δ1...δ11dδ1 . . .dδ11

× 〈(λγmnpr)(d1γmnpd
1)(λA1)(d1W 2)(d1W 3)(d1W 4)〉(1,1). (4.7)

Using the identities
∫

d16ddρ1
. . .dρ16

=ǫρ1...ρ16
, ǫρ1...ρ16

ǫα1...α5ρ1...ρ11=11!5!δα1 ...α5

ρ12 ...ρ16
,(4.8)

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(γabc)ρ12ρ13(γm1n1p1
)ρ12ρ13

=−253 δabc
m1n1p1

, (4.9)

(λγm1)[α1
(λγn1)α2

(λγp1)α3
(γm1n1p1

)α4α5]=(λγm1)α1
(λγn1)α2

(λγp1)α3
(γm1n1p1

)α4α5
(4.10)

the integration over [dd1] is easily performed and (4.7) becomes

K1 =
3(2π)5/2Z5

1

2R

(

α′

2

)−2

〈(λγmnpD)(λA1)(λγmW 2)(λγnW 3)(λγpW
4)〉(1,1) (4.11)

where we also used that [20]
∫

e−(rθ)rα(. . .) =
∫

Dαe−(rθ)(. . .). Using the identity [40]

〈(λγmnpD)(λA1)(λγmW 2)(λγnW 3)(λγpW
4)〉(1,1) = 40〈(λA1)(λγmW 2)(λγnW 3)F4

mn〉(2,1)

=
K

26 32
〈(λ3θ5)〉(2,1)

where in the last line we used (2.28), the kinematic factor (4.11) can be written as

K1 =
(2π)5/2Z5

1K

3R 27

(

α′

2

)−2

〈(λ3θ5)〉(2,1) (4.12)

Using the definition (2.27) one concludes from (4.12) that

|〈K1〉|2 =
(2π)5Z10

1

21432R2
KK|N(2,1)|2

(

α′

2

)−4

. (4.13)

The amplitude (4.3) therefore is given by

A1 =
(2π)5

22934R2π2
KKκ4

(

α′

2

)4 ∫

M1

d2τZ10
1

4
∏

i=2

∫

d2zi|N(2,1)|2〈
4
∏

i=1

eikX(zi)〉1

which upon using (2.31),

|N(2,1)|2〈
4
∏

i=1

eikX(zi)〉 = (2π)10δ(10)(k)
22534R2

(2π)5α′5

(

α′

2

)4
∏

i<j

F1(zi, zj)
αki·kj

and Z10
1 = (2τ2)

−5 finally becomes

A1 = (2π)10δ(10)(k)
κ4KK

29π2α′5

(

α′

2

)8 ∫

M1

d2τ

τ5
2

4
∏

i=2

∫

d2zi

4
∏

i<j

F1(zi, zj)
αki·kj

. (4.14)

It should be pointed out that the previous computation in [28] claimed that the 1-loop

computation in the pure spinor formalism agreed with the RNS result of [33], but it was

incorrectly used that
∫

d2zw1w1µ
z
z = 2 instead of = 1. And to compare with the result

of [33] one takes into account the translation invariance of the torus to integrate the “extra”
∫

d2z1

τ2
= 2 integral in their equation (2.22) to conclude that (4.14) differs8 by 1

4 from the

RNS result reported in [33]. We argue that the one-loop result of [33] is missing the

two factors of 1/2 from the GSO projection for both the left- and right-moving sectors,

explaining the 1/22 discrepancy.9

8There is a missing factor of (α′/2)8 in [33].
9We thank Eric D’Hoker for kindly confirming to us their missing 1/4 factor [55].

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5 Two-loop

The two-loop massless four-point amplitude in the non-minimal pure spinor formalism is

given by

A2 =
1

2
κ4e2λ

4
∏

i=1

3
∏

j=1

∫

M2

d2τj

∫

d2zi〈|N (b, µj)U
i(zi)|2〉 (5.1)

where 〈〉 denote the zero-mode integrations
∏2

I=1

∫

[dθ][ddI ][dr][dsI ][dwI ][dwI ][dλ][dλ] and

(b, µj) =
1

2π

∫

d2yjbzzµ
z
j z. (5.2)

The 32 (22) zero-modes of dα (sα) are denoted by dI
α (sα

I ) for I = 1, 2. As shown in [13],

they are saturated by the different factors of (5.1) as

N → (s1d1)11(s2d2)11
3
∏

j=1

(b, µj) → (d1)3(d2)3 U1U2U3U4 → (d1)2(d2)2, (5.3)

so that each b-ghost contributes only zero-modes with the term (α′

2 ) (λγmnpr)

192(λλ)2
(dγmnpd). The

expansion dα(yi) = d̂α(z) + d1
αw1(yi) + d2

αw2(yi) implies a zero-mode contribution of

(dγmnpd)(y) = (d1γmnpd
1)f11(y) + 2(d1γmnpd

2)f12(y) + (d2γmnpd
2)f22(y)

where fij(y) ≡ wi(y)wj(y), i, j = 1, 2 is the basis of holomorphic quadratic differentials for

the genus-2 Riemann surface [56]. It follows from a short computation that,

3
∏

j=1

(b, µj) = cb

3
∏

j=1

∫

d2yjµj(yj)∆(y1, y2)∆(y2, y3)∆(y3, y1)

× 1

(λλ)6
(λγabcr)(λγdefr)(λγghir)(d

1γabcd1)(d1γdefd2)(d2γghid2) (5.4)

where cb = 2
(384π)3

(α′

2 )3 and ∆(y, z) = w1(y)w2(z) − w2(y)w1(z). In the computation

of (5.4) one can check that combinations containing a different number of d1
α and d2

α zero

modes e.g.,

(λγabcr)(λγdefr)(λγghir)(d
1γabcd2)(d1γdefd2)(d2γghid2)

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

∫

M2

d2τ1d
2τ2d

2τ3|
3
∏

j=1

(b, µj)|2 =

= c2
b

∫

M2

d2ΩIJ |
1

(λλ)6
(λγabcr)(λγdefr)(λγghir)(d

1γabcd1)(d1γdefd2)(d2γghid2)|2

where
∫

d2ΩIJ =
∫

d2Ω11d
2Ω12d

2Ω22 and we used the identity of the appendix B.

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The integration over [dwI ][dwI ] can be done using the results of [28] taking into account

the different normalizations for the measures (2.11) and (2.12),

∫

[dw1][dw1][dw2][dw2]e−(w1w1)−(w2w2) =
(λλ)6

(2π)22
Z−22

2 (5.5)

It is straightforward to use the measure (2.14) to integrate over [ds1][ds2], and the ampli-

tude (5.1) becomes

A2 =
κ4e2λ

256π2636(11!5!)4

(

α′

2

)8 ∫

M2

d2ΩIJ |Z−11
2

∫

[dθ][dd1][dd2][dr][dλ][dλ]

×e−(λλ)−(rθ)

(λλ)6
(λγabcr)(λγdefr)(λγghir)(d

1γabcd1)(d1γdefd2)(d2γghid2)

×(λγm1)α1
(λγn1)α2

(λγp1)α3
(γm1n1p1

)α4α5
(λγm2)β1

(λγn2)β2
(λγp2)β3

(γm2n2p2
)β4β5

×ǫα1...α5ρ1...ρ11ǫβ1...β5δ1...δ11d1
ρ1

. . .d1
ρ11

d2
δ1 . . .d

2
δ11

×
[

(d1W 1)(d1W 2)(d2W 3)(d2W 4)w1(z1)w1(z2)w2(z3)w2(z4)

+(d1W 1)(d2W 2)(d1W 3)(d2W 4)w1(z1)w2(z2)w1(z3)w2(z4)

+(d1W 1)(d2W 2)(d2W 3)(d1W 4)w1(z1)w2(z2)w2(z3)w1(z4)

+(d2W 1)(d2W 2)(d1W 3)(d1W 4)w2(z1)w2(z2)w1(z3)w1(z4)

+(d2W 1)(d1W 2)(d1W 3)(d2W 4)w2(z1)w1(z2)w1(z3)w2(z4)

+(d2W 1)(d1W 2)(d2W 3)(d1W 4)w2(z1)w1(z2)w2(z3)w1(z4)
]

|2 × 〈
4
∏

i=1

eik·x〉2 (5.6)

where the only non-vanishing contribution from the external vertices contains two d1 and

two d2 zero-modes coming from (α′/2)4(dW )4. Integrating the dα zero-modes in (5.6)

using (2.15) and (4.8)–(4.10) one gets

A2 =
π6

2432

(

α′

2

)6 ∫

M2

d2ΩIJZ10
2

∣

∣K2

∣

∣

2 × 〈
4
∏

i=1

eik·x〉2 (5.7)

where the non-minimal kinematic factor K is given by

K2 = 〈(λγm1n1p1
r)(λγdefr)(λγm2n2p2

r)(λγm1defm2λ)

×
[

+ (λγn1W 1)(λγp1W 2)(λγn2W 3)(λγp2W 4) (H1234 + H3412)

+(λγn1W 1)(λγp1W 3)(λγn2W 2)(λγp2W 4) (H1324 + H2413)

+(λγn1W 1)(λγp1W 4)(λγn2W 2)(λγp2W 3) (H1423 + H2314)
]

〉(−3,2) (5.8)

and we defined

Hijkl = w1(zi)w1(zj)w2(zk)w2(zl). (5.9)

In the appendix A we will show that

K2 = 212 33 5Ys〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2) = 23 3Ys K〈(λ3θ5)〉(0,2) (5.10)

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where the second equality follows from (2.28). Hence (5.7) is given by

A2 = κ4e2λ22π6KK

(

α′

2

)6 ∫

M2

d2ΩIJZ10
2 |Ys|2|N(0,2)|2〈

4
∏

i=1

eik·x〉2. (5.11)

From the formula (2.31) we get

|N(0,2)|2〈
4
∏

i=1

eik·x〉2 = (2π)10δ(10)(k)

√
2

22π6α′5

(

α′

2

)4
∏

i<j

F2(zi, zj)
αki·kj

(5.12)

which together with Z10
2 = 2−10det(ImΩIJ)−5 implies that

A2 = (2π)10δ(10)(k)κ4e2λ

√
2KK

210α′5

(

α′

2

)10 ∫

M2

d2ΩIJ

(detImΩIJ)5

∫

Σ4

|Ys|2
∏

i<j

F2(zi, zj)
αki·kj

(5.13)

which is the final result for the 2-loop amplitude.10 And we have shown that the compu-

tation of the whole supersymmetric amplitude including its coefficient is straightforward

using the non-minimal pure spinor formalism.

6 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 [23] 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 [33] is due

to a missing factor of 1/4 from the GSO projection in their one-loop amplitude.

Acknowledgments

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.

A Non-minimal two-loop kinematic factor

The non-minimal two-loop computation of section 5 leads to the kinematic factor

K = 〈(λγabcD)(λγghiD)(λγdefD)(λγadefgλ)
[

(λγbW 1)(λγcW 2)(λγhW 3)(λγiW 4)
]

〉(−3,2).

(A.1)

10The coefficient obtained here is 1/16 times the result reported by [33]. 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.

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In [20] it was shown11 that (A.1) is proportional to 〈(λγmnpqrλ)(λγsW )FmnFpqFrs〉(0,2),

the kinematic factor obtained in the minimal pure spinor formalism [17], whose equivalence

with the RNS result of [2] was established in [18, 23]. We will now evaluate all the terms

in (A.1) to find the exact coefficient announced in (5.10).

To simplify the covariant computation of (A.1) we use (λγdefD)(λγadefgλ) =

48(λλ)(λγagD)−48(λγagλ)(λD) and drop the last term because (λγmW I) is BRST-closed.

And for the same reason we can use (λγaγgD) instead of (λγagD) in the first term. There-

fore (A.1) becomes

K = 48〈(λγghiD)(λγaγgD)(λγabcD)
[

(λγbW 1)(λγcW 2)(λγhW 3)(λγiW 4)
]

〉(−2,2). (A.2)

The strategy to evaluate and simplify12 (A.2) is straightforward due to the identities obeyed

by the pure spinor λα. One uses the SYM equation of motion for W α in the form of

(λγabcD)(λγmW 1) =
1

4
(λγmγm1n1γabcλ)F1

m1n1
(A.3)

(λγaγgD)(λγmW 2) =
1

4
(λγagm2n2mλ)F2

m2n2
(A.4)

and uses gamma matrix identities13 in such a way as to get factors which vanish by the

pure spinor property of (λγm)α(λγm)β = 0. For example, one gets identities like

(λγbγm1n1γabcλ)(λγaγgD)
[

F1
m1n1

(λγcW 2)
]

= 48(λλ)(λγaγgD)
[

F1
ac(λγcW 2)

]

(A.5)

and

F3
rs(λγhγrsγabcλ)(λγa)α(λγb)β(λγc)γ =

= 16(λλ)(δh
b F3

ac − δh
c F3

ab − δh
aF3

bc)(λγa)α(λγb)β(λγc)γ . (A.6)

Following the above steps (A.2) becomes

K = 576〈(λγghiD)(λγaγgD)
[

F1
ab(λγbW 2)(λγhW 3)(λγiW 4)

−1

3
F3

ab(λγbW 1)(λγhW 2)(λγiW 4)− 1

3
F4

ab(λγbW 1)(λγhW 2)(λγiW 3)+(1 ↔ 2)
]

〉(−1,2)

−192〈(λγgaiD)(λγaγgD)
[

F3
bc(λγiW 4)(λγbW 1)(λγcW 2) + (3 ↔ 4)

]

〉(−1,2). (A.7)

The last line of (A.7) vanishes. To see this note that the factor inside brackets is BRST-

closed, so that we can replace (λγaγgD) by (λγagD). Furthermore (λγgaiD)(λγgaD) =

−(λγgaγiD)(λγgaD) − 2(λγaD)(λγiaD) and the last term vanishes when acting on

11There is a loophole in the proof of [20] though. In that proof the terms in (A.1) which are of the form

kWWWF where argued to vanish after summing over the permutations. However we show here that by

using the identities of [23] those terms are actually proportional to WFFF , so the conclusions of [20] still

hold true. CM would like to acknowledge a question made by I. Park which sparked the motivation to

revisit that proof.
12These kind of computations confirm the observations made long ago that pure spinors simplify the

description of super-Yang-Mills theory [57, 58].
13The package GAMMA [59] is often very useful for these manipulations.

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F3
bc(λγiW 4)(λγbW 1)(λγcW 2) because (λγiaD) = (λγiγaD)−δi

a(λD) and (λγi)α(λγi)β = 0

due to the pure spinor property. Therefore by using the gamma matrix identity of

(γmn) δ
α (γmn) σ

β = −8δσ
αδδ

β − 2δδ
αδσ

β + 4γm
αβγδσ

m (A.8)

and dropping the term proportional to the BRST charge and using momentum conservation

(so that Dα and Dβ effectively anti-commute) we get

(λγgaγiD)(λγgaD) = 8(λλ)(DγiD) + 4(λγmD)(λγmγiD). (A.9)

The first term in the r.h.s. of (A.9) is proportional to ki and vanishes by momentum

conservation, while the last term vanishes when acting on F3
bc(λγiW 4)(λγbW 1)(λγcW 2)

for the same reason as explained above.

For convenience we write (A.7) as

K = 576Ka1
− 192Ka2

− 192Ka3
+ (1 ↔ 2) (A.10)

where

Ka1
≡ 〈(λγghiD)(λγaγgD)

[

F1
ab(λγbW 2)(λγhW 3)(λγiW 4)

]

〉(−1,2)

while Ka2
and Ka3

can be obtained by permuting the labels in Ka1
. Using the SYM

equations of motion and a few gamma matrix identities we get

Ka1
= +〈(λγghiD)

[

6k1
c (λγgW 1)(λγcW 2)(λγhW 3)(λγiW 4)

−1

4
(λγmnpqgλ)F1

mnF2
pq(λγhW 3)(λγiW 4) − 1

4
(λγagmnhλ)F1

acF3
mn(λγcW 2)(λγiW 4)

−1

4
(λγagmnhλ)F1

acF4
mn(λγcW 2)(λγiW 3)

]

〉(−1,2). (A.11)

After a long and tedious computation using straightforward manipulations and identities

like (λγmnpqrλ)FI
mnFJ

pq = (λγmnpqrλ)FJ
mnFI

pq and [17]

(λγmnpqrλ)(λγsW 4)
[

F1
mnF2

pqF3
rs + F3

mnF1
pqF2

rs + F2
mnF3

pqF1
rs

]

= 0 (A.12)

one gets

Ka1
= −1

2
〈k1

m(λγghiγnW 1)F2
pq(λγmnpqgλ)(λγhW 3)(λγiW 4)〉(−1,2) + (1 ↔ 2)

−1

4
〈
(

2F3
rsk

1
[a(λγghiγc]W

1) + 2k3
r (λγghiγsW 3)F1

ac

)

(λγagrshλ)(λγcW 2)(λγiW 4)〉(−1,2)

−1

4
〈
(

2F4
rsk

1
[a(λγghiγc]W

1) + 2k4
r (λγghiγsW 4)F1

ac

)

(λγagrshλ)(λγcW 2)(λγiW 3)〉(−1,2)

+〈(λγmnpqrλ)
[ (

F1
mnF3

pqF2
rs−4F1

mnF2
pqF3

rs

)

(λγsW 4)−3F3
mnF4

pqF1
rs(λγsW 2)+(3 ↔ 4)

]

−72k1
m(λγmW 2)

[

F1
hi(λγhW 3)(λγiW 4) + F3

hi(λγhW 1)(λγiW 4) + F4
hi(λγhW 1)(λγiW 3)

]

+24k1
m(λγmW 4)F2

hi(λγhW 1)(λγiW 3) + 24k1
m(λγmW 3)F2

hi(λγhW 1)(λγiW 4)〉(0,2) (A.13)

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To simplify the 〈 〉(−1,2) terms in (A.13) it is convenient to have λα in the combination (λλ)

by using the identities,

(λγghiγnW 1)(λγmnpqgλ)(λγhW 3)(λγiW 4) = 2(λλ)(W 3γgiγnW 1)(λγmnpqgλ)(λγiW 4)

(A.14)

and similarly

(λγghiγaW 1)(λγagrshλ)(λγcW 2)(λγiW 4) = 2(λλ)(W 4γahiW 1)(λγahirsλ)(λγcW 2) (A.15)

(λγghiγcW 1)(λγagrshλ)(λγcW 2)(λγiW 4) = 2(λλ)(W 4γghcW 1)(λγagrshλ)(λγcW 2). (A.16)

In [23] it was proved that

〈(λγmnpqrλ)(λγsW 4)F1
mnF2

pqF3
rs〉(n,g) = −16(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4

mn〉(n,g)

(A.17)

and that 〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(n,g) is completely symmetric in the particle labels,

hence

〈(λγmnpqrλ)
[ (

F1
mnF3

pqF2
rs − 4F1

mnF2
pqF3

rs

)

(λγsW 4) − 3F3
mnF4

pqF1
rs(λγsW 2)

]

〉(0,2)

+(3 ↔ 4) = +240(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2),

where we also used the momentum conservation relation of (k1 ·k3)+ (k1 ·k4) = −(k1 ·k2).

The last two lines of (A.13) can be simplified by using (λγmW ) = QAm − km(λA) and

by noticing that the terms of the form Q(Am)Fpq(λγpW )(λγqW ) are BRST exact and

therefore vanish. Doing that one gets

−72〈k1
m(λγmW 2)

[

F1
hi(λγhW 3)(λγiW 4) + F3

hi(λγhW 1)(λγiW 4) + F4
hi(λγhW 1)(λγiW 3)

]

+24k1
m(λγmW 4)F2

hi(λγhW 1)(λγiW 3) + 24k1
m(λγmW 3)F2

hi(λγhW 1)(λγiW 4)〉(0,2)

= +240(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2). (A.18)

Feeding the results above into the expression for Ka1
in (A.13) one can write it as

Ka1
= Ka11

+ Ka12
, where

Ka11
= −〈k1

r (λγmnpqrλ)(W 3γmnsW
1)(λγsW 4)F2

pq〉(0,2) + (1 ↔ 2) (A.19)

−
[

〈(F3
rsk

1
[a(W

4γgh|c]W
1)+k3

r (W
4γghsW

3)F1
ac)(λγagrshλ)(λγcW 2)〉(0,2)+(3 ↔ 4)

]

and

Ka12
= +480(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4

mn〉(0,2) (A.20)

Furthermore, by using the gamma matrix identities γmnp = γmnγp − ηmnγp + ηamγn and

(γmn) δ
α (γmn) σ

β = −8δσ
αδδ

β + 4γm
αβγδσ

m − 2δδ
αδσ

β ,

the pure spinor identities (λγamnpqλ)(λγa)β = (λγm)α(λγm)β = 0, the equation of motion

kI
m(λγmW I) = 0 and the results above, Ka11

(and its permutations Ka21
and Ka31

) can be

further simplified. In fact, one can show that

−〈k1
r(λγmnpqrλ)(W 3γmnsW

1)(λγsW 4)F2
pq〉(0,2)

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= 32〈k1
m(λγmW 4)(λγpW 3)(λγqW 1)F2

pq〉(0,2) + (3 ↔ 4)

= −32
(

(k1 · k3) + (k1 · k4)
)

〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2). (A.21)

From γmnp
αβ γγδ

mnp = 48(δγ
αδδ

β − δδ
αδγ

β) and the equation of motion for W α
3 it follows that,

−k3
r (λγagrshλ)(W 4γghsW

3)F1
ac(λγcW 2) = 48(k3 · k4)(λA1)(λγmW 2)(λγnW 3)F4

mn

and

1

2
F3

rsk
1
c (W

4γghaW
1)(λγagrshλ)(λγcW 2) = 48(k1 · k2)(λA1)(λγmW 2)(λγnW 3)F4

mn.

From (A.21) one also gets

− 1

2
F3

rsk
1
a(W

4γghcW
1)(λγagrshλ)(λγcW 2) = 16(k1 · k3)(λA1)(λγmW 2)(λγnW 3)F4

mn.

(A.22)

Plugging the identities (A.21)–(A.22) in (A.19) and summing over the indicated permuta-

tions leads to

Ka11
= 240(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4

mn〉(0,2) (A.23)

hence

Ka1
= Ka11

+ Ka12
= 720(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4

mn〉(0,2). (A.24)

From (A.10) and (A.24) and their permutations one arrives at the final result14 for (A.1),

K = +720〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2) ×

×
[

576(k1 · k2)−192(k3 · k2)−192(k4 · k1)+576(k2 · k1)−192(k3 · k1)−192(k4 · k2)
]

= 3 · 27 · 2880(k1 · k2)〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2). (A.25)

The complete kinematic factor (5.8) is obtained using the result (A.25) and permuting

its labels. The first line of (5.8) is given by (A.25) while the second and third are obtained

by replacing s → u and s → t respectively. The final result is therefore

K2 = −3 · 26 · 2880〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2)

×
[

s(H1234 + H3412) + u(H1324 + H2413) + t(H1423 + H2314)
]

= 212 33 5Ys〈(λA1)(λγmW 2)(λγnW 3)F4
mn〉(0,2) (A.26)

where we used the Mandelstam variables and u = −t − s together with

H1234 + H3412 − H1324 − H2413 = ∆(1, 4)∆(2, 3)

H1423 + H2314 − H1324 − H2413 = −∆(1, 2)∆(3, 4).

and the definition (2.38). With (A.26) the expression for the kinematic factor (5.8) is

finally demonstrated.

14To check results we performed explicit component expansion computations with especially-crafted pro-

grams using FORM [60, 61].

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B Period matrix parametrization of genus-two moduli space

Let µ z
i z (i = 1, 2, 3) be the Beltrami differentials, τi (i = 1, 2, 3) the Teichmüller parameters

and wI(z) (I = 1, 2) the holomorphic 1- forms over Σ2, then [17]

∫

d2τ1d
2τ2d

2τ3

∣

∣

∣

3
∏

i=1

∫

d2ziµi(zi)∆(1, 2)∆(2, 3)∆(3, 1)
∣

∣

∣

2
=

∫

d2Ω11d
2Ω12d

2Ω22 (B.1)

where ∆(i, j) = w1(zi)w2(zj) − w1(zj)w2(zi). To prove this one uses the identity15 [1, 65]

∫

d2z wI (z)wJ (z)µi(z) =
δΩIJ

δτi
(B.2)

and expands ∆(1, 2)∆(2, 3)∆(3, 1) to get

3
∏

i=1

∫

d2ziµi(zi)∆(1, 2)∆(2, 3)∆(3, 1) = −δΩ11

δτi

δΩ12

δτj

δΩ22

δτk
ǫijk. (B.3)

So

dτ1 ∧ dτ2 ∧ dτ3

3
∏

i=1

∫

d2ziµi(zi)∆(1, 2)∆(2, 3)∆(3, 1) = −δΩ11

δτi

δΩ12

δτj

δΩ22

δτk
ǫijkdτ1 ∧ dτ2 ∧ dτ3

= −δΩ11

δτi

δΩ12

δτj

δΩ22

δτk
dτi ∧ dτj ∧ dτk

= −δΩ11 ∧ δΩ12 ∧ δΩ22.

Multiplying the last expression by its complex conjugate we get (B.1).

Open Access. This article is distributed under the terms of the Creative Commons

Attribution Noncommercial License which permits any noncommercial use, distribution,

and reproduction in any medium, provided the original author(s) and source are credited.

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	Introduction
	The conventions
	The normalization of zero-modes
	On the normalization of the holomorphic 1-forms

	Tree-level
	The tree-level normalization

	One-loop
	Two-loop
	Conclusions
	Non-minimal two-loop kinematic factor
	Period matrix parametrization of genus-two moduli space

		2010-05-05T13:56:40+0200
	Preflight Ticket Signature