louise dolan and peter goddard- tree and loop amplitudes in open twistor string theory

8/3/2019 Louise Dolan and Peter Goddard- Tree and Loop Amplitudes in Open Twistor String Theory

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arXiv:hep-th/0703054v22

4Jul2007

hep-th/0703054

Tree and Loop Amplitudes

in Open Twistor String Theory

Louise Dolan

Department of Physics

University of North Carolina, Chapel Hill, NC 27599-3255

Peter Goddard

Institute for Advanced Study

Olden Lane, Princeton, NJ 08540, USA

We compute the one-loop gluon amplitude for the open twistor string model of Berkovits,

using a symmetric form of the vertex operators. We discuss the classical solutions in

various topologies and instanton sectors and the canonical quantization of the world sheet

Lagrangian. We derive the N-point functions for the gluon tree and one-loop amplitudes,

and calculate a general one-loop expression for the current algebra.
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1. Introduction

Twistor string theory [1,2,3] offers an approach to formulating a QCD string. Unlike

conventional string theory, the twistor string has a finite number of states. These include

massless ones described by N = 4 super Yang Mills theory coupled to N = 4 conformal

supergravity in four spacetime dimensions. The original theory [1] is a topological string

theory, where a finite number of states arises in the usual way. The alternative formulation

[2,3] has both Yang-Mills and supergravity particles occurring in an open string sector.

Here the absence of a Regge tower of states is due to the absence of operators involving

momentum for massive particles. Since the twistor string theory is some N=4 Yang-Millsfield theory coupled to N=4 conformal supergravity and possibly a finite number of closed

string states, we infer that this field theory system is ultraviolet finite.

In [1], gluon amplitudes with loops, and with d+1 negative helicity gluons and the rest

positive helicity gluons, were associated with a topological string theory with D-instanton

contributions of degree d. Beyond tree level, the occurrence of conformal supergravity

states is believed to modify the gluon amplitudes from Yang Mills theory. The other

version of twistor string theory [2] uses a set of first order b,c world sheet variables

with open string boundary conditions, and world sheet instantons. The target space of

both models is the supersymmetric version of twistor space, CP3|4. In [4], a path integral

construction of the tree amplitudes outlined in [2], is used to compute the three gluon tree

amplitude. Some n-point Yang-Mills and conformal supergraviton string tree amplitudes

are derived in [3], which discusses both models and general features of loop amplitudes.

Some earlier computations of conventional field theory amplitudes in a helicity basis can

be found in [5-7] for Yang Mills trees, [8,9] for gravity trees, and [10-14] for the Yang Mills

loop. General features of the twistor structure of the Yang Mills loop are described in[15,16].

In this paper, we calculate an expression for the one-loop n-gluon amplitude in Berkovits

open twistor string theory. In section 2, we review the world sheet action, establishing a

convenient notation and selecting a gauge where the world sheet abelian gauge fields have

been gauged to zero. Then the topology, or instanton number, resides in the boundary

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conditions, for an open string, or the transition functions relating different gauge patches

on the world sheet, for a closed string. We discuss the classical solutions on the sphere,

disk, torus and cylinder.

In section 3, we discuss the canonical quantization of the string, gauge invariance in the

corresponding operator formalism and the construction of vertex operators corresponding

to gluons.

In section 4, the n-point gluon open string tree amplitudes are calculated, and they match

the Parke-Taylor form [17,18], up to double trace terms from the current algebra. This

extends the three-point amplitudes found via a path integral in [4], and provides an explicit

oscillator quantization of some of the tree amplitudes found in [3].

In section 5, the gluon open string one-loop amplitudes are described as a product of

contributions from twistor fields, ghost fields, and the current algebra. We compute the

n-gluon one-loop twistor field amplitude. Both the n-point tree and loop amplitudes are

computed for the maximally helicity violating (MHV) amplitudes, where the instanton

number takes values 1, 2. This can be extended straightforwardly to any instanton number,

leading to amplitudes with arbitrary numbers of negative and positive helicity gluons.

In section 6, the general expressions for two-, three- and four-point one-loop current algebra

correlators are given for an arbitrary Lie group. They are given in terms of Weierstrass

P and functions. We expect to discuss the derivation of these expressions and theirgeneralizations, using recursion relations, in a later paper [19].

In section 7, we combine the parts and construct the four-point MHV one-loop gluon

amplitude of the open twistor string. We show how the delta function vertex operators

of the twistor string lead to a form of the final integral that is a simple product of the

current algebra loop and the twistor fields loop. We do not discuss here the infrared

regularization of the loop amplitude, nor how the gauge group of the current algebra is

ultimately determined [20].

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2. The Action and Classical Solutions

Equations of Motion and Boundary Conditions. The world sheet action for the twistor

string introduced by Berkovits can be written in the form

S = SY Z + Sghost + SG (2.1)

where SG represents a conformal field theory with c = 28 and SY Z is given by

SY Z =

i

YIDZIS + YI DZIP

g12 d2x , (2.2)

with D = iA and 1 I 8, and the fields ZS , ZP and Y are homogeneous

coordinates in the complex projective twistor superspace CP3|4 and are world-sheet scalars,

pseudo-scalars and vectors, respectively. For the action to be real, the fields must satisfy

the conditions YI = YI for the bosonic components (1 I 4) and YI = YI for

the fermionic components (5 I 8), ZIS = ZIS , ZIP = ZIP, but we must also have

A = A, i.e. A has to be pure imaginary.

The action (2.1) gives rise to the equations of motion

DZS + DZP = 0, D

Y

= 0, DY = 0 (2.3)

where D = ( + iA)g12 , to the constraint,

YZS + YZP = 0 , (2.4)

and to the end condition on the open string,

YnZS + YnZP = 0 , (2.5)

where n is a vector normal to the boundary. The end condition (2.5) will be satisfied if

Yn cos + Y

n sin = 0 , ZS sin ZP cos = 0 , (2.6)

for some function , varying over the boundary, and continuous up to multiples of . The

function changes under gauge transformations, which we shall now discuss.

In the case of a Euclidean signature for the world sheet, we can write

YDZS + YDZP = Y

zDzZ+ YzDzZ , (2.7)

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where z = x1 + ix2, z = x1 ix2, Z = ZS iZP, Z = ZS + iZP, Dz = z iAz,

Az = 12 (A1 iA2), etc. With this notation, the equations of motion become

DzZ = DzZ = 0, DzYz = DzY

z = 0, (2.8)

together with the constraints

YzZ = YzZ = 0,

the boundary conditions

Z = UZ, Y znz = U1Yznz, (2.9)

where U = e2i in terms of (2.6), and reality conditions Z = Z, Yz = Yz for bosonic

components, and Yz = Yz for fermionic components, Az = Az. The reality conditions

imply that on the boundary |U| = 1.

Gauge Invariance. The action has two abelian gauge invariances,

Yz g1Yz, Z gZ, Az Az ig1zg , (2.10)

Yz g1Yz, Z gZ, Az Az ig1zg , (2.11)

where g = e+i, g = e+i, each in GL(1,C), so that

A A + +

, (2.12)

and , need to be pure imaginary, i.e. g = g (reducing the gauge group to one copy of

GL(1,C)), if the reality condition on A is to be maintained. Az, Az, can be thought of

as components, Az, Az, taken from different gauge potentials, A, A, associated with the

transformations g, g, respectively.

The gauge invariance of the theory can be used in general to set the potential A = 0,

with the vestige of the gauge structure residing in the boundary conditions or the gauge

transformations which relate the fields on different patches, in the case of world sheets withnon-trivial global topology (and in the components Az, Az, that do not appear explicitly

in the action). In a gauge with Az = Az = 0, the equations of motion for Z, Z become

zZ = zZ = 0, i.e. Z Z(z), Z Z(z).

Solutions on the Sphere. If the world sheet is the sphere S2, corresponding to closed string

boundary conditions, mapped stereographically onto the plane, the potentials are defined

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by functions on two patches, A> on S> = {z : |z| > 1} and A

< on S

< = {z : |z| < 1+ },

for some > 0, with

A>z Az A

|z| > 1 . (2.13)

We can apply gauge transformations >, >, z , A

1 . By writing

log(znh(z)) as the sum of two functions, one regular at the origin and the other at infinity,

we can obtain gauge transformations on the two patches that will maintain A>z = A(z) = znZ(z) = znZ


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where Zm = Znm.

Solutions on the Torus. If the world sheet is a torus, T2, corresponding to a closed string

loop amplitude, we can describe it by identifying points of the complex plane related by

translations of the form z z + m1 + n1, m1, n1 Z

, for a given modulus C

. Thevarious copies of the fundamental region {z = x + y : 0 x, y < 1} have to be related by

gauge transformations, ga, ga,

Az(z + a) = Az(z) ig1a zga, Az(z + a) = Az(z) ig

1a z ga, (2.16)

where a = m1 + n1, m1, n1 Z. The gauge transformations have to satisfy

ga+b(z) = ga(z + b)gb(z) = gb(z + a)gb(z), (2.17)

and similarly for ga, and the reality condition ga(z) = ga(z).

We can apply gauge transformations , to A, A to set Az = Az = 0 over the plane,

changing the gauge transition functions ga(z) ha = (z + a)ga(z)(z)1, ga(z) ha =

(z + a)ga(z)(z)1, where ha, ha are holomorphic functions of z, z, respectively. In this

gauge,

Z(z + a) = ha(z)Z(z), Z(z + a) = ha(z)Z(z). (2.18)

Writing ha(z) = eia(z), (2.17) implies

1(z + ) 1(z) (z + 1) + (z) = 2n, (2.19)

for some integer n, which describes the topology of the solution. A particular gauge

transformation that possesses this property and, more generally, satisfies (2.17), is

h0a(z) = exp

n(a a)

Im(z +

a

2) + inm1n1 + ia

(2.20)

where a+b = a + b. In fact, ifha(z) satisfies (2.19), we would need to make a further

gauge transformation to bring it into the standard form (2.20).

If we let a = m1 n1, the translation property of Z is

Z(z + 1) = eiZ(z), Z(z + ) = ei(+n(2z+))Z(z), (2.21)

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which are the defining relations for an n-th order theta function with characteristics , ,

and we must have n > 0 for non-trivial solutions. If the usual theta function is denoted by

(, ) =

mZexp

i(m + 12)

2 + 2i(m + 12)+ im + 12i

, (2.22)

the space of n-th order theta functions is spanned by the n functions

1n ( + 2p)

(nz,n), p = 0, 1, . . . n 1. (2.23)

Thus each of the components of Z has an expansion of the form

ZI(z) =n1

p=0cIp

1n( + 2p)

(nz,n), (2.24)

where 1 I 8.

Solutions on the Cylinder. For an open string loop amplitude, we need to consider the

world sheet being a cylinder, C2, which we take to be the strip region {z : 0 Re z 12}

with z identified with z + n, where is pure imaginary. The equations (2.16) and (2.17)

hold but with a, b restricted to be integral multiples of . The fields Z(z), Z(z) have to

satisfy boundary conditions on Re z = 0, 12 ,

Z(iy) = U0(y)Z(iy), Z( 12 iy) = U12 (y)Z(12 + iy), |U0(y)| = |U12 (y)| = 1,

for real y.

We can again work in a gauge in which Az = Az = 0, where Z, Z are analytic functions of

z, z, respectively. We can find the gauge transformation 0 = ei0 to take us to such a gauge

by solving the equation Az = z0 and we can impose the boundary condition 0 = 120

on Re z = 0, where U0(y) = ei0(y) with 0(y) R. Under the gauge transformation,

Z 0Z, Z 0Z, so U0(y) 0(iy)U0(y)0(iy)1 = 1 for real y. So we can choosea gauge in which Az = Az = 0 and Z, Z are analytic functions of z, z, respectively, and

Z, Z are real on Re z = 0. In this gauge we can extend the definition of the fields into

the region {z : 12 Re z 0} by reflection about Re z = 0: Z(z) = Z(z) = Z(z),

Az(z, z) = Az(z, z) = Az(z, z), thus obtaining fields defined smoothly in the

whole region {z : 12 Re z 12

}.

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Similarly we can define another gauge in which Az = Az = 0 and Z(z), Z(z) are analytic

functions real on Re z = 12 by making a gauge transformation which maps U12 1. In

this gauge we may extend the fields over the whole strip {z : 0 Re z 1} by reflection

about Re z = 12 . If Z, Z denote the fields in this second gauge, Z(z) = Z(1 z). If,

are the gauge transformation that relate the two gauges we have constructed, Z(z + 1) =

Z(z) = (z)Z(z) = (z)Z(z), for 12 < Re z < 0. So, if we assume we can extend

the definition of the gauge transformation (z) from 0 < Re z < 12 to 0 < Re z < 1, we

have, for 12 < Re z < 0,

Z(z + 1) = g1(z)Z(z), where g1(z) = (z + 1)1(z).

Thus we have constructed from the solution defined on the cylinder, C2, one defined on

the torus, T2, for which is pure imaginary, defined in a gauge in which Z(z) = Z(z).

For pure imaginary, the complex conjugate of (2.23), evaluated with z replaced by z,

is

1n ( + 2p)

(nz,n), p = 0, 1, . . . n 1. (2.25)

This is in the space of functions (2.24) if and only if is real and = 0 or 1. The general

solution for Z for the cylinder is thus given by (2.24) with these restrictions on , .

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3. Quantization and Vertices

Canonical Quantization. The quantum theory involves the twistor fields YI, ZI, 1 I 8,

the current algebra, JA, ghost fields, b, c, associated with the conformal invariance, and

ghosts u,v, associated with the gauge invariance, and the world sheet gauge fields A, A

which have no kinetic term. The conformal spins and contributions of the various fields to

the central charge of the Virasoro algebra are shown the following table:

Y Z JA u v b c

U(1) charge 1 1 0 0 0 0 0

conformal spin, J 1 0 1 1 0 2 1

central charge, c 0 28 -2 -26

The fields ZI, 1 I 8, comprise four boson fields, a, a, 1 a 2, and four fermion

fields M, 1 M 4; the gauge invariance insures that the ZI are effectively projective

coordinates in the target space CP3|4. As we saw in section 2, in a gauge in which Az =

Az = 0 and Az = Az = 0, a(z), a(z), M(z) are holomorphic, and the only further effect

of the gauge fields is through their topology. Since the contribution to the Virasoro central

charge c from the fermionic and bosonic twistor fields cancels to zero, and the ghost fields

have c = 26 2, the current algebra JA is required to have c = 28.

The mode expansion of the basic fields takes the form

(z) =

nznJ (3.1)

where stands for Y,Z,u,v,b,c, or JA, and J denotes the conformal spin of the relevant

fields. The vacuum state |0 satisfies n|0 = 0 for n > J.The canonical commutation

relations for the basic fields are

[[Zim, Yjn ]] =

ijm,n, {cm, bn} = m,n, {vm, un} = m,n, (3.2)

where the brackets [[, ]] denote commutators when either i or j is not greater than 4 and

anticommutators when both i and j are greater than 4; and

[JAm, JBn ] = if

ABCJ

Cm+n + kmm,n

AB . (3.3)

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These lead to the normal ordering relations

Zi(z)Yj() = : Zi(z)Yj() : +ij

z , (3.4)

and similar relations for c, b and v, u.

The Virasoro algebra is given by

L(z) = j

: Yj(z)Zj(z) : : u(z)v(z) : +2 : c(z)b(z) : : b(z)c(z) : +LJ(z),

(3.5)

where the moments of LJ(z) constitute the Virasoro algebra associated with the current

algebra.

The BRST current is

Q(z) = c(z)L(z) + v(z)J(z) : c(z)b(z)c(z) : + 322c(z), (3.6)

where

L(z) = j

: Yj(z)Zj(z) : : u(z)v(z) : +LJ(z).

Q(z) is a primary conformal field of dimension one with respect to L(z), and the BRST

charge Q0 satifies Q20 = 0, where Q(z) =

n Qnz

n1. L(z) generates a Virasoro algebra

with total central charge zero.

Gauge Invariance. The current associated with the gauge transformation is

J(z) = P(z) = 8j=1

Pj(z), (3.7)

where

Pj(z) =: Yj(z)Zj(z) : =m

ajmzm1 for each j; (3.8)

then

[aim, ajn] =

iijmm,n, [aim, Y

jn ] = Y

jm+n

ij , [aim, Zjn] = Z

jm+n

ij , (3.9)

where i = 1, if i 4, and i = 1, otherwise. Hence

[aim, Yj(z)] = zmYj(z)ij , [aim, Z

j(z)] = zmZj(z)ij .

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If we introduce eqi0 so that

eqi0aj0e

qi0 = aj0 iij ,

where i is as in (3.9), the normal ordered products

eX

j(z)

eqj0 exp

n0

1n

ajnzn zaj0 , (3.10)

define fermion fields, which can be identified with Yj(z), Zj(z) for j 5. In general,

formally,

Xj(z) = qj0 + aj0 log z

n=0

1

najnz

n.

Although, in the fermionic cases j 5, we have an equivalence between the spaces gener-

ated by Yjn , Zjn, on the one hand, and e

qj0 , ajn, on the other,

Yj(z) =

eXj(z)

, Zj(z) =

eX

j(z)

, j 5, (3.11)

in the bosonic cases j 4, we need to supplement eqj

0 , ajn by two fermionic fields

j(z), j(z),

{im, jn} =

ijm,n; n|0 = 0, n 0, n|0 = 0, n 1.

Note that in this case , , eq0, a generate a larger space than the fields Y, Z. Then we

can write

Yj(z) =

eXj(z)

j(z), Zj(z) =

eX

j(z)

j(z), j 4. (3.12)

Ifg(z) is an analytic function, representing a gauge transformation, defined in some annular

neighborhood of the unit circle, |z| = 1, with winding number d as z goes round the unit

circle,

g(z) = zde(z), (z) =

n=

nzn = (z), (3.13)

where0

nzn, >(z) =

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let

P[] =1

2i

P(z)(z)dz =

ann = P[],

P[0 nan.

So, if

Ug = edq0eP[] = edq0eP[

], (3.15)

noting that the an commute among themselves,

UgYj(z)U1g = g(z)

1Yj(z), UgZj(z)U1g = g(z)Z

j(z).

The vertex operators Vj(zj) are gauge invariant if [an, Vj(zj)] = 0, so that

eP[]Vj(zj)eP[] = Vj(zj), (3.16)

we have

0|UgV1(z1)V2(z2) . . . V n(zn)|0 = 0|edq0V1(z1)V2(z2) . . . V n(zn)|0, (3.17)

showing that the winding number, the topology of the gauge transformation, is the only

part affecting the amplitude. The winding number d can be regarded as labeling different

instanton sectors, which one needs to sum over, described by the operator edq0 .

Because the trace

tr

amV1(z1)V2(z2) . . . V n(zn)wL0

= wmtr

V1(z1)V2(z2) . . . V n(zn)wL0am

= wmtr

amV1(z1)V2(z2) . . . V n(zn)w

L0

,

we have

tr

amV1(z1)V2(z2) . . . V n(zn)wL0

= 0, if m = 0,

so that

tr

UgV1(z1)V2(z2) . . . V n(zn)wL0

= tr

edq0ua0V1(z1)V2(z2) . . . V n(zn)wL0

, (3.18)

showing that the trace depends on the instanton number d and ua0 = e0a0 . It is necessary

to sum over d and average over u = e0 with respect to the invariant measure du/u = d0.

The evaluation of such traces is discussed in Appendix C.

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Scalar Products. The reality conditions on YI, ZJ imply that

ZJn

= ZJn, (3.19)

(Y

J

n )

= Y

J

n for 1 J 4, (Y

J

n )

= Y

J

n for 5 J 8. (3.20)

It follows from (3.11) and (3.12) that

YIndedq0 = edq0YIn , Z

In+de

dq0 = edq0ZIn for 1 I 8. (3.21)

The modes YIn , ZIn satisfy the vacuum conditions

YIn |0 = 0, n 0, ZIn|0 = 0, n 1. (3.22)

If we take a single fermionic component of YI, ZI in isolation (i.e. 5 I 8), denoted

Y, Z, scalar products can be evaluated from the basic relation 0|Z0|0 = 1. With edq0

defined for this component similarly to the above for integral d, we can show that Z0|0 =

eq0 |0, and, more generally, Z1d . . . Z 0|0 = edq0 |0, for positive integers d, so that

0|edq0Zd . . . Z 0|0 = 1, (3.23)

and 0|edq0

Zn1 . . . Z nm |0 = 0 for other products Zn1 . . . Z nm (unless m = d + 1 andthe n1, . . . , nd+1 are a permutation of 0, . . . , d).

For a single bosonic component ofYI, ZI in isolation (i.e. 1 I 4), again denoted Y, Z,

scalar products can be evaluated from the basic relation

0|f(Z0)|0 =

f(Z0)dZ0, or, equivalently, 0|e

ikZ0 |0 = (k). (3.24)

In the bosonic case, the matrix elements of eq0 are specified by

0|eikZ0eq0eikZ0 |0 = 0|eq0eik

Z1eikZ0 |0 = (k)(k) (3.25)

and, analogously to (3.23),

0|edq0 exp

idj=0

kjZj

|0 =dj=0

(kj), (3.26)

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equivalently,

0|edq0f(Z0, . . . , Z d)|0 =

f(Z0, . . . , Z d)dZ0, . . . , d Z d. (3.27)

[Alternatively, instead of including the factor edq0 , we could calculate tree diagrams working

in a twisted vacuum |0 = e12dq0 |0, where Y, Z have modified conformal dimensions 1+ 12d

and 12d, respectively. Here we will take the former approach as a basis for constructing

loop contributions.]

In the cases at hand, with all eight components of YI, ZI present, the expressions have an

overall scaling invariance under ZI(z) kZI(z), YI(z) k1YI(z), which would cause

the integral over ZI0 , 1 I 4, to diverge, unless the invariant measure on this group is

divided out to leave the invariant measure on the coset space. This invariant measure can

be taken to be dS = dZj

/Zj

, with the various choices of j giving equivalent answers andvacuum expectation value has the form Z

0|f(Z0)|0 =

f(Z0)d

4Z0/dS =

f(Z0)Z

j00 d

4Z0/dZj00 (3.28)

where Z0 = (Z10 , Z

20 , Z

30 , Z

40), for any choice of j0. [Note that, before the vacuum ex-

pectation values are taken on the fermionic modes, ZI0 , 5 I 8, the integrand is a

homogeneous function of all the ZI, but after this has been done, the residual function

f(Z0) has the homogeneity property f(kZ0) = k4f(Z0).]

Vertices. The target space of the string theory is the twistor superspace CP3|4. We use

Z =

, = 12

, =

3

4

, (3.29)

where the I, 1 I 4 are anticommuting quantities and , C2, to denote homoge-

neous coordinates in this space, identifying Z and kZ for nonzero k C.

The vertex operator corresponding to the physical state | = f(Z0)JA1|0 of the string,

which corresponds to a gluon state, is

V(, z) = f(Z(z))JA(z). (3.30)

The state | is gauge invariant provided that f(Z0) is an homogeneous function of Z0,

f(kZ0) = f(Z0) for nonzero k C, and then V(, z) satisfies (3.16). f(Z0) is the wave

14


16/54

function describing the dependence of the state on the mean position of the string in

twistor superspace. Leaving aside the need to take account of the homogeneity of the

coordinates, the wave function for the string being at Z would beI(Z

I(z) ZI).

Thus, allowing for this, the wave function for Z(z) to be at Z = ( , , ) is

W(z) = 2a=1

(ka(z) a)(ka(z) a)4b=1

(kb(z) b)dk

k, (3.31)

noting that (k ) = k is the form of the delta function for anticommuting variables.

[(3.31) is invariant under Z(z) k(z)Z(z) and separately under Z kZ.]

To obtain the form of the vertex used by Berkovits and others [2-3], use the first delta

function, (k1(z) 1) to do the integration in (3.31), to obtain

W(z) = 2(z)1(z) 212

a=1

a(z)1(z) a14

b=1b(z)1(z) b1 . (3.32)

Then multiply by exp(ibb), to Fourier transform on b, and by

A() = A+ + bAb +

1

2bcAbc +

1

3!bcdAbcd +

1234A, (3.33)

(so that Abc corresponds to the six scalar bosons, Ab and Abcd to the two helicity states of

the four spin 12 fermions, and A to the two helicity states of the spin one boson, in the

(1, 4( 12 ), 6(0)) supermultiplet of N=4 Yang Mills theory) and integrate with respect to

, , yielding

1

(1)2

2

1

2

1

exp

i

aa1

1

A+ +

1

1bAb +

1

1

21

2bcAbc +

1

1

31

3!bcdAbcd +

1

1

41234A

,

where b,c,d are summed over. This is essentially the vertex used by Berkovits, except that

he omits the Ab, Abc, Abcd terms.

The form of the vertex that it will be convenient to use in what follows is that obtained

from (3.31) by Fourier transforming on a and multiplying by A() and integrating withrespect to a,

W(z) = dkk

2a=1

(ka(z) a)eikb(z)b

A+ + k

bAb +k2

2bcAbc +

k3

3!bcdAbcd + k

41234A

, (3.34)

15


18/54

where the vertex operators VA () are given by (3.36) and (3.35), Q commutes with the

integral of the product vertices as consequence of their having conformal spin 1 and U(1)

charge 0, but it does not commute with edq0 ,

[Q, edq

0 ] = dn

cnane2q0 .

To correct for this, replace edq0 by edq0edncnun in (3.37),

A = edq0edncnun nr=1

drVArr (r)w

L0 , (3.38)

ensuring that only states in the cohomology of Q contribute to the trace tr(A(1)F). A

similar instanton number changing operator is used in a different context in [21]. The

inclusion of the factor edncnun does not change the value of this trace as we can see by

expanding it in a power series.

17


19/54

4. Tree amplitudes

In this section we compute the N-gluon MHV twistor string tree amplitudes, first with

oscillators, and then compare with the path integral method, in preparation for computing

the loop amplitude, which will be given in terms of a factor times the tree amplitude. The

three-point trees were derived in [4], and various N-point trees were computed in [3] using

aspects of Wittens twistor string theory [1].

The n-point tree amplitude, corresponding to gluons, in instanton sector d is given by

Atreen =

0|edq0VA11 (z1)V

A22 (z2) . . . V

Ann (zn)|0

nr=1

dzr

dMdS (4.1)

where dM is the invariant measure on the Mobius group dS is the invariant measure onthe group of scale transformations on Z (as in (3.28)), and VA is given in (3.36) or (3.35)

as = . It follows directly from these expressions and from (3.23) that the n-point tree

amplitude for gluons vanishes unless the number of negative helicity gluons is d + 1.

The d = 0 Sector. If d = 0, we may take 1 = and all other r = +. In the vacuum

expectation value 0|VA1 (z1)VA2+ (z2) . . . V

An+ (zn)|0 the Z

I(zr) can be replaced by the

constant ZI0 , because all the ZIn, for n = 0, can be removed by annihilation on the vacuum

on either left or right. The resulting expression involves integrals over the zero modes,10,

20,

10,

20 ofZ

I, 1 I 4, and the n scaling variables kr associated with the n vertices

as in (3.36) and (3.35).

Leaving aside the current algebra from the vacuum expectation value of the JAr(zr), and

dM, the amplitude is proportional to

k41

n

r=1dkrkr

n

r=12

a=1(r

a kra)eikr

ara

2

a=1dada/dS

=

k41

nr=1

dkrkr

nr=1

2a=1

(ra kr

a)2a=1

nr=1

krra 2a=1

da/dS (4.2)

involves n +1 integrals from dnkd2/dS and 2n +2 delta functions, leaving in effect n + 1

delta functions after the integrals have been done. For n > 3, this exceeds the number

required for momentum conservation and the amplitude vanishes.

18


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So for d = 0, we need only consider n = 3; then the amplitude is

Atree++ =

k41

3r=1

dkrkr

nr=1

2a=1

(ra kr

a)fA1A2A32a=1

nr=1

krra

da/dS , (4.3)

where the invariant measure of the Mobius group, dM, which replaces the three ghost

fields 0|3r=1 c(zr)|0, has cancelled against the contribution from the tree level current

algebra correlator that we have normalized as 0|JA1 (z1)JA2 (z2)J

A3 (z3)|0 = f

A1A2A3(z1

z2)1(z2 z3)1(z3 z1)1. The current algebra arises from the c = 28, SG sector of the

world sheet theory. The structure constants fABC supply the non-abelian gauge group for

the D = 4, N = 4 Yang Mills theory, as explained in [2].

Now, using the three (r1 kr1) to perform the kr integrals:

Atree++ =2a=1

3r=1

r1ra

(11)32131

3r=1

r2 r

1 2

1

d21

fA1A2A3 ,

where we have also made the choice dS = d1/1 in line with (3.28). Note that, alter-

natively, we could have used the (r2 kr2) delta functions to perform the kr integrals

obtaining

3r=1

r2ra

rather than

3r=1

r1ra

;

together these four delta functions correspond to momentum conservation. To draw this

out as an explicit overall factor, note that

3r=1

r2ra =

3r=1

r

2 r1

2

1

ra when

3r=1

r1ra = 0

and the Jacobian to change between

r=2,3 r2 r1 2

1 and a=1,2 3

r=1r2 r1 2

1 rais 2132 2231 = [2, 3].

So, writing a,b

3r=1

rbra

4(rr),

19


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Atree++ = 4(rr)[2, 3]

(1

1)3

2131

1

2 11

2

1

d2

1fA1A2A3

= 4(rr)[2, 3](1

1)2

2131fA1A2A3

= 4(rr)[2, 3]3

[1, 2][3, 1]fA1A2A3 . (4.4)

Writing the general polarization vector, r, of the r-th gluon, as the sum of positive and

negative helicity parts, r = +r +

r . These parts can be normalized by choosing vectors

sra and sra are defined such that rasra = 1 and

ar sra = 1, for each r, and setting

+r = A+r srara,

r = A

r rasra. (4.5)

Multiplying the appropriate amplitudes Ar onto Atree++, we obtain

Atree++ = 4(rr)

[2, 3]3

[1, 2][3, 1]fA1A2A3A1 A

+2 A

+3

= 4(rr)fA1A2A3

1

+2

+3 p1 +

+2

+3

1 p2 +

+3

1

+2 p3

, (4.6)

as shown in Appendix B.

The d = 1 Sector. The one-instanton sector contributes to all gluon tree amplitudes with

just two negative helicities. For d = 1, we have to evaluate

0|eq0VA1 (z1)VA2 (z2)V

A3+ (z3) . . . V

An+ (zn)|0, (4.7)

where, by (3.21) and (3.22), and because YI does not occur in VA(z), we can replace ZI(z)

by ZI0 + zZI1 (as the other terms in Z

I(z) will annihilate on the vacuum on either the left

or the right). Thus, for d = 1, by (3.27), the expression (4.7) involves integrals over the 8

bosonic variables a0, a1,

a0,

a1, a = 1, 2, as well as evaluating the dependence on the 8

fermionic variables M0 , M1, 1 M 4, using (3.23).

The vacuum expectation value of currents

0|JA1(z1)JA2(z1) . . . J

An(zn)|0 (4.8)

can be written as a sum of a number of contributions, one of which has the form

fA1A2...An

(z1 z2)(z2 z3) . . . (zn z1)(4.9)

20


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and we use this contribution in what follows [22].

Thus the d = 1 tree amplitude has the form

Atree+...+ = n

r=1dkr

kr ra (ra kra(z))eikra(zr)ra

k41k420|e

q01(z1)2(z1)

3(z1)4(z1)

1(z2)2(z2)

3(z2)4(z2)|0

a

d2ad2afA1A2...Andz1dz2 . . . d zn

(z1 z2)(z2 z3) . . . (zn z1)

dMdS

Using that, for a component of the fermion field taken alone we would have

0|eq01(z1)1(z2)|0 = (z2 z1)0|e

q01110|0 = z1 z2,

and integrating over a0, a1,

Atree+...+

=

nr=1

dkrkr

nr=1

2a=1

(ra kr

a(zr))2a=1

nr=1

krra

nr=1

krzrra

2a=1

d2a

k41k42(z1 z2)

4 fA1A2...Andz1dz2 . . . d zn

(z1 z2)(z2 z3) . . . (zn z1)

dMdS

= n

r=11

r1

n

r=1 r2 2(zr)

1

(zr)

r1

2

a=1 n

r=1r

1ra

1

(zr) n

r=1zrr

1ra

1

(zr) 2

a=1 d2a

1

121(z1 z2)

1(z1)1(z2)

4fA1A2...Andz1dz2 . . . d zn

(z1 z2)(z2 z3) . . . (zn z1)

dMdS

using the delta functions (r1 kr1(zr)) to perform the kr integrations again. Noting

that when r2 (2(zr)/1(zr))r1 = 0, for b = 1, 2

nr=1

rbra =

nr=1

b(zr)r1ra1(zr)

= b0

nr=1

r1ra1(zr)

+ b1

nr=1

zrr1ra1(zr)

,

the delta functions2a=1

nr=1

r1ra

1(zr)

nr=1

zrr1ra

1(zr)

can be replaced by the momentum conserving delta function, together with a Jacobian

factor,

(1021

11

20)

24(rarb).

21


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Then, with

r =2(zr)

1(zr)=

20 + 21z

10 + 11z

, (4.10)

using Mobius invariance, we have that

Atree+...+ = nr=1

1

r1

nr=1

(r2 rr

1)4(rarb)

d21d22

(1021

11

20)

2

112

1(1 2)4 fA1A2...And1d2 . . . d n

(1 2)(2 3) . . . (n 1)

dMdS (4.11)

Noting that we can write

d21d22

(1021

11

20)

2= dMdS ,

because the left hand side is the invariant measure on the product of the Mobius and

scaling groups, and using the (r2 rr1) to do the r integrations,

Atree+...+ = 4(r

arb)

122

1 221

14

fA1A2...Annr=1

r

2r+11 r+1

2r11

= 4(rarb)

1, 24fA1A2...An

1, 22, 3 . . . n, 1, (4.12)

with an+1 a1 .

Of course, this result could also be obtained within the path integral framework. From

this point of view, after integrating out the YI fields, the d = 1 sector path integral is

Atree =d=1

DZI ((z iAz)Z

I)

ni=1

dzi

DGe

SGJA1(z1)JA2(z2) . . . J

An(zn)

n

r=1dkr

kr r,a (ar kr

a(zr)) eikr

a(zr)ra

A+r + k4r

1(zr)2(zr)

3(zr)4(zr)Ar

dMdS . (4.13)

As discussed in section 2, we work in a gauge where the gauge potentials are zero, so

the path satisfies zZI = 0, and ZI(z) = ZI0 + Z

I1z from (2.15) . Computing the path

integral by replacing DZI ((z iA(d=1)z )Z

I) with8I=1 dZ

I0dZ

I1 , and performing the

22


24/54

integrals over ZI0 , ZI1 , where the fields

a(z), a(z), M(z) are now given by the solutions

ZI(z) = ZI0 + zZI1 , we see that evaluating the path integral gives the result obtained from

the canonical quantization. Identifying the integration variables ZI0 , ZI1 with the variables

a0,1, a0,1,

M0,1, and performing the Grassmann integration d00 = 1, d11 =1, for each M, we find that for the two negative helicities in the positions 1 , 2, we obtain

(4.12), together with a factor of A1A2A+3 . . . A+n corresponding to the polarizations in

(4.13).

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5. Loop Amplitudes

The n-point loop amplitude is a sum over contributions corresponding to instanton num-

bers d. We write the integrand of such a contribution as a product of factors:

Aloopn,d =

An,dAn,dA

JA

n Aghostd0 d

2Im

nr=1

rdr, r = e2ir , w = e2i, (5.1)

where An,d is the part of the integrand associated with the bosonic twistor fields , ;

An,d is the part associated with the fermionic twistor fields ; AJA

n is the part associated

with the current algebra JA; Aghost is the part associated with the ghost fields; and

is pure imaginary. The integration with respect to 0 is the averaging over the gauge

transformation ua0 referred to in (3.18); the normalization factor 2Im will be explained

below. We shall restrict our attention to MHV amplitudes. For these, the fermionic partAn,d will vanish unless d = 2 and so we shall use this value below.

Twistor bosonic contribution to the loop integrand. Using the vertices (3.35) and (3.36),

the part of the integrand for the n-particle loop amplitude associated with the bosonic

twistor fields, , , is

An,2 =

tr

e2q0ua0

nr=1

exp {ikra(r)ra + ikr

a(r)ra} wL0

nr=1

dkrkr

2a=1

eiraradra/dS, (5.2)

Here we have rewritten the delta functions (kra(r) ra) as Fourier transforms on ra

in order that we can perform the trace on the bosonic variables a, using the relation (C.1)

from Appendix C

tr

edq0ua0

n

j=1eijZ(j)wL0

= u(d+1)/2

d

i=1

n

j=1Fdi (j, w)j

, (5.3)

where j = u 1

2 j = e2ij , j = j + i0/4, and

Fdk (, w) =

n=

w12 (n1)(d(n2)+2k)

w

d(1n)k= d/2wd/8k/4

2k/d 1

0

(d, d) ,

(5.4)

24


26/54

using the notation of (2.22), so that

F21 (, w) = 3(2, 2), F22 (, w) = w

14 2(2, 2), (5.5)

From this we have

An,2 = u6

2i,a=1

nr=1

krF2i (r, w)ra

nr=1

krF2i (r, w)ra

nr=1

dkrkr

2a=1

eirara

dra/dS. (5.6)

for, putting d = 2 in (5.3), we see that we get a factor of u32 for each of the four bosonic

components of Z. Expressing the second delta functions as Fourier transforms on ai ,

An,2 = u6 2

i,a=1

r

krF2i (r, w)ra

exp

i

nr=1

2a=1

2i=1

krai F

2i (r, w)ra irar

a

2a=1

d2anr=1

dkrkr

2a=1

dra/dS

= u6 nr=1

2a=1

kra(r, w) r

a 2i,a=1

nr=1

krF2i (r, w)ra

2a=1

d2anr=1

dkrkr

/dS,

where we have performed the integrals over ra and set

a(r, w) = a1F

21 (r, w) +

a2F

22 (r, w). (5.7)

Performing the kr integrations using the

kr1(r, w) r1

delta functions,

An,2 = u6

nr=1

1

r1

2(r, w)

1(r, w)r

1 r2

2

i,a=1

nr=1

F2i (r, w)

1(r, w)r

1ra

2a=1

d2a/dS .

As before, given the constraints 2(r, w)r1 = 1(r, w)r

2, we can replace

2i,a=1

nr=1

F2i (r, w)

1(r, w)r

1ra

by (1122

12

21)

24(rr),

so that

An,2 = 4(rr)u

6

(11

22

12

21)

2nr=1

1

r1

(r, )r1 r

2 2a=1

d2a/dS , (5.8)

25


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where

(, ) =2(, w)

1(, w)=

21F21 (, w) +

22F

22 (, w)

11F21 (, w) +

12F

22 (, w)

=

21(, ) + 22

11(, ) + 12

, for (, ) =

F21 (, w)

F22 (, w) . (5.9)

Now we use the delta functions

(r, )r1 r2

to do the integrations over r in (5.1):

Aloopn,2 = 4(rr)

u6

nr=1

1

(r1)2

(r, )

(1122

12

21)

2An,dA

JA

n Aghostd0 d

2Im

2a=1

d2a/dS , (5.10)

where = nr=1 r and(, )

(, )

.

It is to be understood that in the rest of the integrand kr and r are determined by

kr =r

1

1(r, w), (r, ) =

r2

r1, (5.11)

so the second equation determines r as a function of ai , , 0 and

2r/

1r .

Twistor fermionic contribution to the loop integrand. Now consider the fermionic part of

the integrand,

An,2 = k41k

42tr

e2q0ua0(1)a01(1)2(1)

3(1)4(1)

1(2)2(2)

3(2)4(2)w

L0

.

(5.12)

If we consider one component of the fermion field M() taken in isolation,

tr

e2q0ua0(1)a01(1)1(2)w

L0

= u32 F21 (1, w)F

22 (2, w) F

22 (1, w)F

21 (2, w)

= u 3

2 ((1, ) (

2, ))

1(1, w)1(2, w)

1122

12

21

= u

32 1, 2

k1k2(1122

12

21)

so that

An,2 =u61, 24

(1122

12

21)

4. (5.13)

26


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Thus, from (5.10), we see that the factors of u cancel and

Aloopn,2 = 1, 244(rr)

nr=1

1

(r1)2

(r, )

AJ

A

n Aghost

(1122

12

21)

2

d0 d

2Im

2a=1

d2a/dS .

(5.14)

Since

(, ) =21(, ) +

22

11(, ) + 12

,

nr=1

(r, )

(r, ) (r+1, )=

nr=1

(r, )

(r, ) (r+1, )

=

n

r=1(r

1)2(r, )

r2r+11 r+12r1=

n

r=1(r

1)2(r, )

r, r + 1(5.15)

Using this in (5.14),

Aloopn,2 =1, 244(rr)

1, 22, 3 . . . n, 1

nr=1

(r, ) (r+1, )

(r, )

AJ

A

n Aghost

(1122

12

21)

2

2a=1

d2a

dS

d0d

2Im. (5.16)

which separates out from the rest of the amplitude the kinematic factor present in the tree

amplitude.

From (5.5) we have that

(, ) = w14

3(2, 2)

2(2, 2).

Using the relations,

23(2, 2)3(0, 2) = 3(, )2 + 4(, )

2,

22(2, 2)2(0, 2) = 3(, )2 4(, )

2,

we see that (, ) is related by a bilinear transformation, whose coefficients are functions

of , to 3(, )2/4(, )2. Further, using the relation

1(, )22(0, )

2 = 4(, )23(0, )

2 3(, )24(0, )

2,

we see that (, ) is also related by a bilinear transformation to the Weierstrass P function

P(, ) =2

3

2(0, )

4 4(0, )4

+

1(0, )3(, )

3(0, )1(, )

2. (5.17)

27


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Using the bilinear invariance of the integrand, this implies


1, 22, 3 . . . n, 1

n

r=1

P(r, ) P(r+1, )

P(r, ) AJA

n Aghost

(1122

12

21)

2

2

a=1

d2

a

dS

d0d

2Im.(5.18)

as an alternative symmetric expression.

Regarding 21

22

11 12

as the matrix defining the bilinear transformation that takes r = (r, ) to r

2/r1 for

1 r 3, the invariant measure

d2a

(1122

12

21)

2= d

1d2d3(1 2)(2 3)(3 1)

dS,


1, 22, 3 . . . n, 1

(3 4)

(3 1)

nr=4

(r r+1)

r

AJ

A

n Aghostd1d2d3

d0d

2Im.

(5.19)

For the first non vanishing amplitude, n = 4, noting that

1, 23, 4

1, 43, 2=

s

t

and (1 2)(3 4)

(1 4)(3 2)

(1 2)(3 4)

(1 4)(3 2)+

s

t

d4 =

(3 4)(4 1)

(3 1)4,

Aloop4,2 = 1, 244(rr)

1, 22, 33, 44, 1

s

t

(1 2)(3 4)

(1 4)(3 2)+

s

t

AJ

A

4 Aghost

4r=1

rdrd0d

2Im.

(5.20)

Similarly, we can derive a corresponding expression for the n-point loop,

Aloopn,2 =1, 2n4(rr)

2, 3n23, 1n4

nr=4

1

r, 12

(r 3)(2 1)

(r 1)(2 3)

r, 32, 1

r, 12, 3

AJ

A

n Aghost

nr=1

rdrd0d

2Im. (5.21)

Because r = r + i0/4 and the integral is invariant as r r + 12, we see that it

is periodic under 0 0 + 2Im. So to average over 0, we integrate over the range

28


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0 0 2Im and divide by 2Im, which explains the normalization factor introduced

in (5.1).

Path Integral Derivation. In the path integral approach, working in the gauge Ai = 0,

we use the paths given in (2.24) with n = 2 and a

() = Za

(), a

() = Za+2

(),a = 1, 2; M() = ZM+4(), 1 M 4.

The path integral on the cylinder includes, up to normalization,

An,2

nr=1

dr =

4I=1

dcI0dcI1

1=0

20

d

nr=1

exp {ikra(r)ra + ikr

a(r)ra}

n

r=1dkrkr

2

a=1eirar

a

dra(n1

r=1dr)

dS .

Performing the integrations over cI0, cI1, 1 I 4, we find a formula analogous to the

canonical expression (5.6), save that the functions F21 (r, w) and F22 (r, w) are replaced

with [12

](2r, 2), and [12+1

](2r, 2) , respectively, and the 0 integration is exchanged

for a sum over and the integration.

The calculation proceeds as before to a formula similar to (5.8), except that (, ) is

replaced with

(, ) = 21(, ) + 22

11(, ) + 12

where (, ) = [12 ](2, 2)

[12 +1

](2, 2)=

3(2+ 12 + 12 , 2)2(2+ 12

+ 12 , 2).

After integrating with respect to , the resulting integrand depends on the differences

i j , and is independent of . Then (, ) is essentially (, ) from (5.9), since the

factor w14 in (, ) can be absorbed in the a integrations, and /2 can be replaced by

i0/2, d = d0/Im (0, real, pure imaginary).

In analogy with (5.10), we use the delta functions (r, )r1 r2 to do the integra-tions over r:

An,2

nr=1

dr

= 4(rr) 2

20

d

nr=1

1

(r1)2 (r, )

(11

22

12

21)

2

2a=1

d2a

dS .

29


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In the rest of the integrand kr and r are determined by kr = r1/1(r, w) and (r, ) =

r2/r1. The one-loop integrand for the fermionic fields, for one permutation of helicities,

is

An,2 = k41k

42

4

M=1 dcM+40 dc

M+41

M(1)M(2) =

1, 24

(11

22

12

21)4

.

Thus we obtain (5.16) as before, apart from a factor of 4.

Ghost contribution to the loop integrand. The ghost fields in the theory are the custom-

ary ghost fields, b, c, with conformal spin (2, 1), associated with the reparametrization

invariance, and the ghosts for the U(1) gauge fields u, v with conformal spin (1, 0).

The partition function for a general fermionic b, c system with conformal dimensions

and 1 respectively is

tr

b0c0L0

c24 (1)F

=

c24

12(1)

n=1

(1 n)2 ,

where the central charge is 12(1)2, and L(z) =

b(z)c(z)

+(1)

b(z)c(z)

.

So

tr

b0c0L0

c24 (1)F

=

112

n=1

(1 n)2 = ()2 .

and the reparametrization and U(1) ghosts each contribute this factor to the integrand:

The factor of (1)F is included so that the ghosts have the same periodicity as the original

coordinate transformations. See Freeman and Olive [23]. The b0, c0 insertion projects onto

half the states in the b, c system; without this projection, the (1)F would force the trace

to vanish.

So the total ghost contribution is

Aghost = ()4. (5.22)

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6. Current Algebra Loop

For the final piece of the integrand of the loop amplitude, we compute the one-loop am-

plitude,

tr(Ja1

(1)Ja2

(2) . . . J an

(n)wL0

), (6.1)

for the current algebra of an arbitrary Lie group, G,

[Jam, Jbn] = if

abcJcm+n + kmabm,n, J

a() =n

Jann1. (6.2)

We calculate this using a recursion relation, and will present the derivation in a future

publication [19]. Relevant discussion is also given in [24,25]. The amplitudes can be

expressed in terms of the -dependent invariant tensors,

tr(Ja10 Ja20 . . . J

an0 w

L0), (6.3)

which themselves can be expressed as products of (constant) invariant tensors and functions

of. We note that (6.1) is symmetric under simultaneous identical permutations of the j

and aj . The structure of the loop amplitude for n 4 is best understood by considering

first the zero, two and three-point functions. (The one-point function necessarily vanishes.)

In what follows L0 denotes the zero-mode generator for the Virasoro algebra associated

with the current algebra.

Fermionic Representations. We begin by considering the case where Ja() is given by a

fermionic representation,

Ja() =i

2Taijb

i()bj() Jan =i

2

r

Taijbirbjnr (6.4)

[Ta, Tb] = fabcTc, tr(TaTb) = 2kab, bj() =r

bjrr 12 , (6.5)

where Ta, 1 a dim G, are real antisymmetric matrices providing a real dimension

D representation of G, and the bir, r Z + 12 , are Neveu-Schwarz fermionic oscillators,{bir, b

js} =

ijr,s, bjr|0 = 0, r > 0, (b

jr) = bjr.

For this representation, the zero-point function,

() = tr

wL0

=

s= 12

(1 + ws)D, (6.6)

31


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while the one-point function vanishes, tr(Ja()wL0), as it does for any representation.

The n-point one-loop current algebra amplitude is computed by using the usual recurrence

relation, for calculating tr(bi1r1 . . . binrnw

L0) in the free fermion representation, obtained by

moving binrn around the trace,

tr(bi1r1 . . . binrnw

L0) =wrn

1 + wrn

n1m=1

(1)m+1rm,rnimintr(bi1r1 . . . b

im1rm1

bim+1rm+1 . . . bin1rn1

wL0).

Using this, we obtain the two-point loop,

tr(Ja(1)Jb(2)w

L0) =k()

12abF(1 2, )

2, (6.7)

where

F(, ) =

r= 12

e2ir + wre2ir

1 + wr =i

2 2(0, )4(0, )3(, )

1(, ) . (6.8)

The three-point loop,

tr(Ja(1)Jb(2)J

c(3)wL0) =

ikfabc()

12321F

32F

13F , (6.9)

and the four-point loop,

tr(Ja(1)Jb(2)J

c(3)Jd(4)w

L0)

=

()

1234 [abcd

12F

23F

34F

14F

abdc

12F

24F

34F

13F

acbd

13F

23F

24F

14F

+ k2 abcd (12F )2 (34F )

2 + k2 acbd (13F )2 (24F )

2 + k2ad bc (14F )2 (23F )

2].

(6.10)

where ijF = F(j i, ) and

abcd = tr(TaTbTcTd). (6.11)

General Representations. The amplitude (6.1) can be evaluated by means of recurrence

relations between functions

tr

Ja10 Ja20 . . . J

am0 J

b1(1)Jb2(2) . . . J

bn(n)wL0

that can be established using the cyclic properties of the trace and the algebra (6.2), from

which it follows that

[Jam, Jb(z)] = izmfabcJc(z) + kmzm1ab.

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For example, in this way it follows that

tr

JamJa1(1) . . . J

an(n)wL0

(1 wm)

= in

j=1 faaja

j trJa1(1) . . . J aj1(j1)Jaj (j)Jaj+1(j+1) . . . J an(n) mj+ km

nj=1

aaj tr (Ja1(1) . . . J aj1(j1)J

aj+1(j+1) . . . J an(n))

m1j (6.12)

and, hence,

tr

Ja()Ja1(1) . . . J an(n)w

L0

= 1tr

Ja0 J

a1(1) . . . J an(n)w

L0

+ i nj=1

1(j, )

faajaj

trJa1(1) . . . J aj1(j1)Jaj (j)Jaj+1(j+1) . . . J an(n)wL0+ k

nj=1

2(j , )

jaaj tr

Ja1(1) . . . J

aj1(j1)Jaj+1(j+1) . . . J

an(n)wL0

(6.13)

where

1(, ) =

m=0e2im

1 wm=

m=1e2im wme2im

1 wm=

i

2

1()

1()

1

2, (6.14)

2(, ) =m=0

me2im

1 wm=

m=1

me2im + wme2im

1 wm=

1

2i1(, ). (6.15)

These functions relate to the Weierstrass elliptic functions by

P(, ) = 422(, ) 2(), () = 1

6

1 (0, )

1(0, ), (6.16)

(, ) = 2i1(, ) i + 2(), P(, ) = (, ). (6.17)

Using (6.13) and similar relations we can establish general formulae for the two-, three-

and four-point one-loop functions.

We write the zero-point (partition) function as

tr

wL0

(). (6.18)

33


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Because it is isotropic, we can write

tr

Ja0 Jb0wL0

= ab(2)() where (2)() =1

dim Gtr

Ja0 Ja0 w

L0

.

Then the general form of the two-point current algebra loop is

tr(Ja(1)Jb(2)w

L0) =abk()

12

F(2 1, )

2 + f()

, (6.19)

where

f() =(2)()

k()+

3 (0, )

423(0, ). (6.20)

Noting that we can write

tr Ja0 Jb0Jc0wL0 = 12tr [Ja0 , Jb0 ]Jc0wL0+ 12tr {Ja0 , Jb0}Jc0wL0we have

tr

Ja0 Jb0Jc0wL0

= 12 ifabc(2)() + 12d

abc(3)(),

where dabc is a totally symmetric isotropic tensor, which may vanish, as it does for SU(2),

but not SU(3). Among the simple Lie groups, only SU(n) with n 3 has a symmetric

invariant tensor of order 3 [26]. Then the general form of the three-point current algebra

loop is

tr(Ja(1)Jb(2)J

c(3)wL0)

=ikfabc()

123

21F

32F

13F

i

2

21 + 32 + 13

f()

+

dabc(3)()

2123, (6.21)

where ij = (j i). The recurrence relations do not manifestly maintain the permu-

tation symmetry of the loop amplitudes but the final result is necessarily symmetric and

can be put into this form.

The general form of the four-point loop can be put into the symmetric form

tr(Ja(1)Jb(2)J

c(3)Jd(4)w

L0)1234

=

abcd

k2() [(12F )2 + f()][(34F )

2 + f()] (2)()2/()

+ acbd

k2() [(13F )2 + f()][(24F )

2 + f()] (2)()2/()

+ adbc

k2() [(14F )2 + f()][(23F )

2 + f()] (2)()2/()

34


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+ tr(Ja0 Jb0Jc0Jd0 w

L0)S

1

96

abcd + adcb + acdb + abdc + adbc + acbd

()42(0, )

44(0, )

abcd + adcb

12()

12F 23F

34F

41F

1

162f()

P24 P32P24 P32

P24 P41P24 P41

+

1

42f()P24

acdb + abdc

12()

13F

34F

42F

21F

1

162f()

P24 P32P24 P32

P21 P

32

P21 P32

+

1

42f()P32

adbc + acbd

12()

14F

42F

23F

31F

1

162 f()P24 P

32

P24 P32P14 P

31

P14 P31+ 142 f()P34

i

4(3)()

abcd

21 + 32 + 43 + 14

+ adcb

41 + 34 + 23 + 12

+ acdb

31 + 43 + 24 + 12

+ abdc

21 + 42 + 34 + 13

+ adbc

41 + 24 + 32 + 13

+ acbd

31 + 23 + 42 + 14

,

(6.22)

where Pij = P(j i, ), Pij = P(j i, ), abcd is given by (6.11) and

tr(Ja0 Jb0Jc0Jd0 wL0)S is the symmetrization of the trace tr(Ja0 Jb0Jc0Jd0 wL0) over permutationsof a,b,c,d.

We will now specialize to the case of a general representation of SU(2). In this case,

tr(Ja(1)Jb(2)J

c(3)Jd(4)w

L0)1234

=

abcd

k2() [(12F )2 + f()][(34F )

2 + f()] (2)()2/()

+ acbd

k2() [(13F )

2 + f()][(24F )2 + f()] (2)()2/()

+ adbc k2() [(14F )2 + f()][(23F )2 + f()] (2)()2/() [abcd + acdb + adbc] ()

1

4842(0, )

44(0, )

1

6

(4)()

k()

abcd ()

12F

23F

34F

41F +

f()

82

P13 + P24

13 + 32 + 21

13 + 34 + 41

24 + 41 + 12

24 + 43 + 32

35


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acdb ()

13F 34F

42F

21F +

f()

82

P14 + P23

14 + 42 + 21

14 + 43 + 31

23 + 31 + 12

23 + 34 + 42

adbc ()14F 42F 23F 31F +f()

82 P12 + P34

12 + 23 + 31

12 + 24 + 41

34 + 41 + 13

34 + 42 + 23

.

(6.23)

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7. Twistor String Loop

We now assemble the parts for the one-loop MHV gluon amplitude of the twistor string.

The fact that the twistor string has delta function vertices leads to the form for the final

integral that is a simple product of the loop for the twistor fields and the current algebraloop. We have provided several equal expressions for the twistor field loop Aloopn,2 in (5.16),

(5.18) - (5.21). These forms were given for particles 1, 2 having negative helicity, and we saw

that the expressions naturally divide into two factors: a piece equal to the kinematic factor

of the tree amplitude multiplied by a function of s and t. For the four-gluon amplitude, if

we consider the form given in (5.20), we have

Aloop4,2 = 1, 244(rr)

1, 22, 33, 44, 1

s

t

(1 2)(3 4)

(1 4)(3 2)+

s

tAJ

A

4 ()4

4

r=1rdr

d0d

2Im,

where we can take r = 3(2r, 2)/2(2r, 2), r = r + i0/4 and AJA

4 is given in

(6.22) for a general compact Lie group. We expect the gluon loop amplitude Aloop to be

related to the field theory loop amplitude for gluons in N = 4 Yang Mills theory coupled

to N = 4 conformal supergravity. It is believed this field theory requires a dimension four

gauge group to avoid anomalies [20]. We hope that a better understanding of that may

follow from further analysis of the loop expressions computed in this paper.

37


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Acknowledgements:

We thank Edward Witten for discussions.

LD thanks the Institute for Advanced Study at Princeton for its hospitality, and was

partially supported by the U.S. Department of Energy, Grant No. DE-FG01-06ER06-01,Task A.

38


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Appendix A. Gauge Potentials

An example of a potential on S2, for which Az = Az = 0, is

A

z = 0,

Az = 0, A>z =

in

(1 + zz)z.

Then A> A< = ig

1g, A> A< = ig

1g for g = zn, g = zn.

And an example of a potential on T2, for which Az = Az = 0, is

Az(z, z) =in

Im(z z), Az(z, z) = 0,

Az(z, z) = 0, Az(z, z) = in

Im(z z),

ThenA(z + a, z + a) A(z, z) = ig

1a (z)ga(z),

A(z + a, z + a)

A(z, z) = ig

1

a (z)ga(z)for

ga(z) = en(aa)

Im (z+a2 )+inm1n1+ia ,

ga(z) = en(aa)

Im (z+a2 )inm1n1ia .

39


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Appendix B. Twistors

A Riemann surface world sheet, which we use in this paper, corresponds to complex target

space fields ZI that transform under the conformal group SU(2, 2). The spacetime metric

is = diag(1, 1, 1, 1), and 12

= 21

= 12 = 21 = 1. The coordinates arex = x0 x , with x = x, and det x = xx. Then

x (ax + b)(cx + d)1,

a bc d

0 12

12 0

a bc d

=

0 12

12 0

,

where

a bc d

SU(2, 2).

The two 2-spinors (a, a) define complex 2-planes in complexified Minkowski space by

= x. Under a conformal transformation, the twistor

a bc d

.

Real Lorentz transformations are given by x axa, a, d where d = (a)1

and a, d SL(2,C). For any vector p, we can write

p = p =

p0 p3 p1 + ip2

p1 ip2 p0 +p3

(paa), p

= p,

where ab

ab

paapbb = 2 det(p) = 2pp

2p2

, and ab

ab

paaqbb = 2p q. Then real Lorentztransformations are given by p apa, a SL(2,C); and the complex Lorentz transfor-

mations are p apb, a, b SL(2,C).

If p2 = 0, we can write p = T, i.e. paa = aa. Similarly, if q2 = 0, qbb = bb,

then 2p q = abab abab = , [, ], where , = abab and [, ] =

abab.

Ifnr=1pr = 0, with praa = rara, then

nr=1 rara = 0; and

r=ssrra = 0 and

r=s[sr]ra = 0, where [rs] = [r, s], and rs = r, s. E.g., for n = 3, 1a/2a =

[2, 3]/[1, 3].

Under conformal transformations,

a bc d

,

a bc d

,

where =

1

2

and =

3

4

.

40


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Polarizations

To compute the polarization dependence of the amplitudes as in (4.6), we note that sa and

sa are reference vectors that scale as the momentum: sa, a u scale up and sa, a u1

scale down. The polarization vector aa r does not scale, so A r goes as u2 and A+ r as

u2. Hence the vertex operator (1)2 A+ r + (11 )4 1234 A r does not scale. Then1

+2

+3 p1 +

+2

+3

1 p2 +

+3

1

+2 p3

= A1A+2A+3 (b1sa2 s3b1a

3b2b) = A1A+2A+3[23]3

[12][31].

To derive this we use momentum conservation3r=1 rara = 0, and properties of the

Penrose spinors such as

2a=1

arra = 0, where ra = ab

br and

ar =

abrb. In particular,

consider s3b 3r=1 brbr = 0 to find s3bb1 = [23]/[12], and similarly sa21a = [23]/[13].

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Appendix C. Trace Calculations

Bosonic Trace

To calculate the M-point trace, as in (3.18) let

(, k) = tr

eq0eik1Z0eq0eik2Z0 . . . eq0eikdZ0ua0ei1Z(1)ei2Z(2) . . . eiMZ(M )wL0

,

so

(0, k) =di=1

(ki),

because only the states with no non-zero modes contribute to this trace. Let

n(, k) = treq0eik1Z0eq0eik2Z0 . . . eq0eikdZ0ua0Zne

i1Z(1)ei2Z(2) . . . eiMZ(M )wL0

,

so

k1 = iu1d, . . . kdm = ium, . . . kd = iu0,

using

eq0Zneq0 = Zn+1, u

a0Znua0 = u1Zn.

Then, using the cyclic property of trace,

n = uwnnd = u

(n+m)/dw(n+dm)(n+m)/2dm, 0 m < d, n+m a multiple of d

and

j = i

n=

nnj = i

d1i=0

n=

dnidn+ij

= i

d1i=0

i

n=

unw12n(d(n+1)2i)dn+ij

= i

d1i=0

i

n=

w12 (n1)(dn2i)uni/d jw dn+i ui/d

= id1i=0

iFddi(u

1/dj , w)ui/d

=

dn=1

Fdn(u1/dj, w)u

n/dkn

42


44/54

where

Fdi (j, w) =

n=

w12 (n1)(d(n2)+2i)

jw

d(1n)iSo depends on ki, j, 1 i d, 1 j M, through

ui/dki +Mj=1

Fdi (j , w)j, where j = u1/dj ,

so that

(, k) = u(d+1)/2di=1

ui/dki + Mj=1

Fdi (j , w)j

and the M point function

tr

edq0ua0ei1Z(1)ei2Z(2) . . . eiMZ(M )wL0

= (, 0)

= u(d+1)/2di=1

Mj=1

Fdi (j , w)j

(C.1)

= u

(d+1)/2

Fd1 (1, w) Fd1 (2, w) . . . F

d1 (d, w)

Fd2 (1, w) Fd2 (2, w) . . . F

d2 (d, w)

. . . . . .. . . . . .Fdd (1, w) F

dd (2, w) . . . F

dd (d, w)

1

d

m=1 (m), (C.2)where the last equality (C.2) holds provided that M = d.

Comparison with Bosonic Tree Amplitudes

We can compare these results for corresponding results for tree amplitudes,

0|edq0eiMZ(M ) . . . ei0Z0 |0.

We start with

0|edq0eikdZd . . . eik0Z0)|0 =di=0

(ki).

Let

= 0|edq0di=0

eikiZiMj=0

eijZ(j)|0

43


45/54

j = i

dk=0

0|edq0di=0

eikiZiMj=0

eikZ(j)Zk|0kj

=

d

i=0ijki.

These provide M + 1 linear partial differential equations in the M + d + 2 variables k, ,

implying that depends only on the d + 1 variables

ki +Mj=0

ijj

so that

=d

i=0

ki +M

j=0ijj

.

Thus

0|edq0eiMZ(M ) . . . ei0Z(0)|0 =di=0

Mj=0

ijj

=

1 1 . . . 10 1 . . . d. . . . . .. . . . . .

d0 d1 . . .

dd

1

dj=0

(j)

=

0i


46/54

=

dnj=0

nd...n1n0ndd . . .

n11

n00

= 1 1 . . . 1

0 1 . . . d

. . . . . .

. . . . . .d0

d1 . . .

dd

=

0i


47/54

where = (1)N+d1u and, again,

tr

edq0ua0(1)Na0Z(j)(Z)wL0

=

n=nnj =

d1

m=0

n=dnmdn+mj

=d1m=0

m

n=

nunw12n(d(n+1)2m)dn+mj

=d1m=0

m

n=

nw12 (n1)(dn2m)unm/d

jw

dn+mum/d

=

d1m=0

um/dtr

edq0ua0(1)Na0Zm(Z)wL0

Fddm(u1/dj , w)

where = (1)N+d1 and

Fdm (, w) =

n=

nw12 (n1)(d(n2)+2m)

w

d(n1)mWriting j = u

1/dj , it follows that

tr

edq0ua0(1)Na0Z(d) . . . Z (2)Z(1)wL0

=d

mj=1 tr edq0ua0(1)Na0Zmd . . . Z m2Zm1w

L0d

j=1 u(mj1)/dFddmj+1(j, w)

= u(d+1)/2(1)Ndd

mj=1

dj=1

md...m2m1Fddmj+1(j, w)

= u(d+1)/2d

mj=1

dj=1

md...m2m1Fddmj+1(j, w), (C.5)

provided that N = 0 when d is odd and N = 1 when d is even, because then = 1 and

Fd1m (, w) = Fdm(, w). i.e. a factor (1)

a0 is included when d is even and no factor of

(1)Na0 is included if d is odd. In particular, in this case, there is a precise cancellationbetween (C.2) and (C.5) in the case M = d. More generally, for M = d, the overall

factors of u cancel and the functions Fdm(, w) involved are the same. We shall use this

factor in computing the trace for the twistor string loop, and find agreement with the path

integral derivation. Since [Q, ai0] = 0, it still holds that only states in the cohomology of

Q contribute to the trace following the discussion at the end of section 3.

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Appendix D. Current Algebra Loop from Recurrence Relations

We outline the derivation of the three and four-point current algebra one-loop amplitude

from the recurrence relations described in section 6. For the three-point function, from

(6.12)and (6.13),

tr(Ja(1)Jb(2)J

c(3)wL0)

= ifabc(123)1

(2)()

1(12, ) + 1(23, ) + 1(31, )

+ k()

12(23, ) + (1(12, ) + 1(31, ))2(23, )

+dabc

2123(3)()

= ifabc(123)1

(2)() + k()2(23, )1(12, ) + 1(23, ) + 1(31, )

1

4ik()2(23, )

+

dabc

2123(3)()

where ij = j i, and from (6.12) we encounter propagators in addition to those in

section 6:

1(, ) = 1(, ) +1

2

2(, ) =1

2i1(, ) =

1

421()

1()

= 2F(, ) +

1

423 (0, )

3(0, )

11(, ) =m=0

e2imwm

(1 wm)2=m=0

e2im

(1 wm)2 1(, )

= 122(, ) 12 (1(, ))2 +

1

12

1

2421 (0, )

1(0, )

12(, ) =m=0

me2imwm

(1 wm)2=

1

2i1(, ) =

1

2i

11(, )

= i163

1(, )1 (, ) 1 (, )1(, )21(, )

= 14i

2(, ) 1(, )2(, ) .

47


49/54

Let

P() = 422F(, ) 13 (0, )

3 (0, ) 4

2 (2)()

(),

which depends on the partition function () and the propagator (2)() of a given rep-

resentation. It is related to the standard Weierstrass P function P(, ) by an additivefunction of . Then

tr(Ja(1)Jb(2)J

c(3)wL0)

= ifabc(123)1k()(

i

163)

2P23

23 + 31 + 12

+ P23

+

dabc

2123(3)().

Using the Weierstrass addition formulae, which hold for both P(, ) and P(, ):

23

= 13

+ 21

+1

2

P13 P21

P13 P21

P23 =P23[P

21 P

13] + P21P

13 P

21P13

P13 P21,

we can write

tr(Ja(1)Jb(2)J

c(3)wL0)

= ifabc(123)1k()(

i

163)

P23P13 P

21

P13 P21+ P23

+

dabc

2123(3)()

= ifabc(123)1k()( i163

) P21 P13 P21 P13P13 P21

+ dabc2123

(3)()

= ifabc(123)1k()

21F

32F

13F

i

2

21 + 32 + 13

f()

+ (123)

1 dabc

2123(3)()

where the last line follows from standard theta function addition formulae [27].

48


50/54

In a similar way, we compute from the recurrence relation the four-point loop as

tr(Ja(1)Jb(2)J

c(3)Jd(4)w

L0)

= (1234)1

abcd

(2)()

()

+ k2(12, ) (2)() + k()2(34, )

1()((2)())2

+ acbd(2)()

()+ k2(13, )

(2)() + k()2(24, )

1()((2)())2

+ adbc

(2)()()

+ k2(14, )

(2)() + k()2(23, )

1()((2)())2+ tr(Ja0 Jb0Jc0Jd0 wL0)

+ fabefcde

12(2)()1(34, )

+

11(23, ) 11(24, )

(2)() + k()2(34, )

1(12, ) [

(2)()(1(23, ) + 1(34, ) + 1(42, ))

+ k()(12(34, ) + (1(23, ) + 1(42, ))2(34, ))]

+ facefbde

(2)()11(34, ) + k()3(34, )

(2)()1(34, )1(24, ) + k()12(34, )1(24, )

+ k()12(34, )

+ 1(13, ) [(2)()(1(23, ) + 1(34, ) + 1(42, ))

+ k()(12(34, ) + (1(23, ) + 1(42, ))2(34, ))]

+ fadefbce

(2)()11(34, ) + k()3(34, )

+ (2)()1(34, )1(23, ) k()12(34, )1(23, )

1(14, ) [(2)()(1(23, ) + 1(34, ) + 1(42, ))

+ k()(12(34, ) + (1(23, ) + 1(42, ))2(34, ))]

+i

2(3)()

fabed

ecd1(12) + facedbed1(13) + f

adedbce1(14)

+ fbc edaed1(23) + f

bdedace1(24) + f

cdedabe1(34)

,

49


51/54

where an additional propagator occurs,

3(, ) =m=0

me2imw2m

(1 wm)3= 12

12(, ) +

1

4i

11(, )

= 1212(, ) 121(, )12(, ) + 18i 12(, ) + i963

1 (0, )1(0, ) .This form of the four-point current algebra loop can be expressed in terms of the Weier-

strass P functions:

tr(Ja(1)Jb(2)J

c(3)Jd(4)w

L0)1234

=

abcd

1

162k2() P12P34

1()((2)())2

+ ac

bd 1162 k2() P31 P24 1()((2)())2

+ adbc

1

162k2() P14P32

1()((2)())2

+ tr(Ja0 Jb0Jc0Jd0 w

L0)

+ fabefcde 1

644k()

P32P24 P32P24P24 P32

P24 P32

P24 P32+

P24 + P14

P24 P14

+ 4P34P32

+ k()

(2)()

k()+

1

42

3 (0, )

3(0, )2

+1

484

2(0, )4

4(0, ) +

1

3

(2)()

k()

1

6

(2)()

k()+

1

423 (0, )

3(0, )

42(0, )

44(0, )

+ facef

bde 1

644k() {

P32P24 P

32P24

P24 P32

P14 P

31

P14 P31+

P24 P32

P24 P32

4P34(P24 + P32)}

+ k()

2

(2)()

k()+

1

423 (0, )

3(0, )

2

1

2442(0, )

44(0, )

1

6

(2)()

k()

+1

3 (2)()k() + 142 3 (0, )3(0, )42(0, ) 44(0, )+

i

2(3)()

fabed

ecd1(12) + facedbed1(13) + f

adedbce1(14)

+ fbc edaed1(23) + f

bdedace1(24) + f

cdedabe1(34)

.

50


52/54

We introduce traces over the group matrices to express the combinations of structure

constants and d-symbols that occur in the four-point loop. For abcd tr(TaTbTcTd),

abcd adcb + acdb abdc = ifcd edabe,

abcd adcb + acdb + abdc = 2k fabefcde.

The symmetrization of the trace tr(Ja0 Jb0Jc0Jd0 w

L0) over permutations of a,b,c,d is

tr(Ja0 Jb0Jc0Jd0 w

L0)

=

tr(Ja0 Jb0Jc0Jd0 wL0)S

+i

4

fcded

abe + fbd edace + fbc ed

ade

(3)() +1

6

fbc ef

ade fcd efabe

(2)().

Then the four-point loop can be written in the symmetric form (6.22).

51


53/54

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louise dolan and peter goddard- tree and loop amplitudes in open twistor string theory

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