Fermionic superstring loop amplitudes in the pure spinor formalism
arXiv:0704.0015v2  [hep-th]  10 Mar 2008
Preprint typeset in JHEP style - HYPER VERSION
Fermionic superstring loop amplitudes in the
pure spinor formalism
Christian Stahn
Department of Physics, University of North Carolina
Chapel Hill, NC 27599–3255, USA
E-mail: stahn@physics.unc.edu
Abstract: The pure spinor formulation of the ten-dimensional superstring leads to
manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor super-
space. This paper explores different methods to evaluate these integrals and then uses
them to calculate the kinematic factors of the one-loop and two-loop massless four-point
amplitudes involving two and four Ramond states.
Keywords: Superstrings, Pure Spinors.
http://arxiv.org/abs/0704.0015v2
mailto:stahn@physics.unc.edu
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 1
2. Zero mode integration 2
2.1 Symmetry considerations and tensorial formulae 3
2.2 A spinorial formula 5
2.3 Component-based approach 7
3. One-loop amplitudes 7
3.1 Review: four bosons 8
3.2 Four fermions 10
3.3 Two bosons, two fermions 10
4. Two-loop amplitudes 12
4.1 Review: four bosons 13
4.2 Four fermions 14
4.3 Two bosons, two fermions 15
5. Discussion 16
A. Reduction to kinematic bases 17
A.1 Four bosons 17
A.2 Four fermions 18
A.3 Two bosons, two fermions 20
B. A gamma matrix representation 21
1. Introduction
The quantisation of the ten-dimensional superstring using pure spinors as world-sheet
ghosts [1] has overcome many difficulties encountered in the Green-Schwarz (GS) and
Ramond-Neveu-Schwarz (RNS) formalisms. Most notably, by maintaining manifest space-
time supersymmetry, the pure spinor formalism has yielded super-Poincaré covariant multi-
loop amplitudes, leading to new insights into perturbative finiteness of superstring theory
[2, 3].
Counting fermionic zero modes is a powerful technique in the computation of loop
amplitudes in the pure spinor formalism and has for example been used to show that
at least four external states are needed for a non-vanishing massless loop amplitude [2].
Furthermore, the structure of massless four-point amplitudes is relatively simple because all
– 1 –
fermionic worldsheet variables contribute only through their zero modes. In the expressions
derived for the one-loop [2] and two-loop [4] amplitudes, supersymmetry was kept manifest
by expressing the kinematic factors as integrals over pure spinor superspace [5] involving
three pure spinors λ and five fermionic superspace coordinates θ,
K1-loop =
(λA)(λγmW )(λγnW )Fmn
K2-loop =
(λγmnpqrλ)(λγsW )FmnFpqFrs
(1.1)
where the pure spinor superspace integration is denoted by 〈. . . 〉, and Aα(x, θ), Wα(x, θ)
and Fmn(x, θ) are the superfields of ten-dimensional Yang-Mills theory.
The kinematic factors in (1.1) have been explicitly evaluated for Neveu-Schwarz states
at two loops [6] and one loop [7], and were found to match the amplitudes derived in
the RNS formalism [8]. This provided important consistency checks in establishing the
validity of the pure spinor amplitude prescriptions. (Related one-loop calculations had
been reported in [9].)
In this paper, it will be shown how to compute the kinematic factors in (1.1) when
the superfields are allowed to contribute fermionic fields, as is relevant for the scattering
of fermionic closed string states as well as Ramond/Ramond bosons. It turns out that
the calculation of fermionic amplitudes presents no additional difficulties, making (1.1)
a good practical starting point for the computation of four-point loop amplitudes in a
unified fashion. This practical aspect of the supersymmetric pure spinor amplitudes was
also emphasised in [10], where the tree-level amplitudes were used to construct the fermion
and Ramond/Ramond form contributions to the four-point effective action of the type II
theories.
This paper is organised as follows. In section 2, different methods to compute pure
spinor superspace integrals are explored. These methods are then applied to the explicit
evaluation of the kinematic factors of massless four-point amplitudes at the one-loop level
in section 3, and at the two-loop level in section 4. In both these sections, the bosonic calcu-
lations are briefly reviewed before separately considering the cases of two and four Ramond
states. Particular attention will be paid to the constraints imposed by simple exchange
symmetries. An appendix contains algorithms which were used to reduce intermediate
expressions encountered in the amplitude calculations to a canonical form.
2. Zero mode integration
The calculation of scattering amplitudes in the pure spinor formalism leads to integrals over
zero modes of the fermionic worldsheet variables λ and θ. Both θ and λ are 16-component
Weyl spinors, the λ are commuting and the θ anticommuting, and λ is subject to the pure
spinor constraint (λγmλ) = 0. The amplitude prescriptions [1, 2] require three zero modes
of λ and five zero modes of θ to be present, and a Lorentz covariant object
T̄ αβγ,δ1...δ5 ≡
λαλβλγθδ1 . . . θδ5
= T̄ (αβγ),[δ1...δ5] (2.1)
was constructed such that the Yang-Mills antighost vertex operator
V = (λγmθ)(λγnθ)(λγpθ)(θγmnpθ) has
= 1 . (2.2)
– 2 –
In this section, different methods of computing such “pure superspace integrals” are ex-
plored. As an example, a typical correlator encountered in the two-loop calculations of
section 4 is considered:
F (ki, ui) = k
(λγmnpq[rλ)(λγs]u1)(θγn
abθ)(θγbu2)(θγqu3)(θγsu4)
(2.3)
Here, ki and ui are the momenta and spinor wavefunctions of the four external particles.
2.1 Symmetry considerations and tensorial formulae
One systematic approach to evaluate the zero mode integrals is to find expressions for all
tensors that can be formed from (2.1). By Fierz transformations, one can always write the
product of two θ spinors as (θγ[3]θ), where γ[k] denotes the antisymmetrised product of k
gamma matrices. Due to the pure spinor constraint, the only bilinear in λ is (λγ[5]λ), and
it is thus sufficient to consider the three cases
(λγ[5]λ)(λ{γ[1] or γ[3] or γ[5]}θ)(θγ[3]θ)(θγ[3]θ)
. (2.4)
Lorentz invariance then implies that it must be possible to express these tensors as sums of
suitably symmetrised products of metric tensors, resulting in a parity-even expression, plus
a parity-odd part made up from terms which in addition contain an epsilon tensor. The
parity-even parts may be constructed [6] starting from the most general ansatz compatible
with the symmetries of the correlator and then using spinor identities along with the
normalisation (2.2) to determine all coefficients in the ansatz. Duality properties of the
spinor bilinears can be used to determine the parity-odd part [7]. An extensive (and almost
exhaustive) list of correlators is found in [11], including the (λγ[1]θ) and (λγ[3]θ) cases of
the above list:
(λγmnpqrλ)(λγuθ)(θγfghθ)(θγjklθ)
= − 4
mnpqr
m̄n̄p̄q̄r̄ +
εmnpqrm̄n̄p̄q̄r̄
δm̄n̄fg δ
(δr̄l δ
u + δ
u − δr̄uδhl )
[fgh][jkl]
(2.5)
(λγmnpqrλ)(λγstuθ)(θγfghθ)(θγjklθ)
= −24
mnpqr
m̄n̄p̄q̄r̄ +
εmnpqrm̄n̄p̄q̄r̄
δm̄j δ
δq̄sδ
u − δkhδr̄u)
[fgh][jkl](fgh↔jkl)
(2.6)
(Here, the brackets (fgh↔ jkl) denote symmetrisation under simultaneous interchange of
fgh with ijk, with weight one.) The remaining correlator with the (λγ[5]θ) factor can be
derived in the same way, using an ansatz consisting of six parity-even structures. Taking
a trace between the two γ[5] factors and noting that
(λγmnpqrλ)(λγabcdeθ) . . .
(λγmnpq[bλ)(λγcde]θ) . . .
one finds a relation to (2.6). This is sufficient to determine all coefficients in the ansatz,
and the result is
(λγmnpqrλ)(λγabcdeθ)(θγfghθ)(θγjklθ)
mnpqr
m̄n̄p̄q̄r̄ +
εmnpqrm̄n̄p̄q̄r̄
m̄n̄p̄
(−δehδr̄l + 2δel δr̄h) + δm̄n̄ab δcdfgδ
(δehδ
l − 3δel δr̄h)
[abcde][fgh][jkl](fgh↔jkl)
(2.7)
– 3 –
One may find it surprising that the derivation of these tensorial expressions only made
use of properties of (pure) spinors, and of the normalisation condition (2.2). However, it
can be seen from representation theory that the correlator (2.1) is uniquely characterised,
up to normalisation, by its symmetry. To see this, note that [12] the spinor products λ3
and θ5 transform in
λ(αλβλγ) : Sym3 S+ = [00003] ⊕ [10001]
θ[δ1 . . . θδ5] : Alt5 S+ = [00030] ⊕ [11010] .
(2.8)
(Here, λ and θ are taken to be in the S+ irrep of SO(1,9), with Dynkin label [00001].) The
tensor product of these contains only one copy of the trivial representation. This applies
to any spinors λ, which means that the pure spinor property cannot be essential to the
derivation of the tensorial identities. The use of the pure spinor constraint merely allows
for simpler derivations of the same identities.
As an illustration of this approach, consider the correlator of eq. (2.3). Leaving the
momenta aside for the moment by setting F = k2ak
r F̃ , the task is to compute
(λγmnpq[rλ)(λγs]u1)(θγn
abθ)(θγbu2)(θγqu3)(θγsu4)
After applying two Fierz transformations,
(λγmnpq[r|λ)(λγcθ)(θγn
abθ)(θγjklθ)
|s]γcγbu2)
3!·16
(λγmnpq[r|λ)(λγcdeθ)(θγn
abθ)(θγjklθ)
|s]γcdeγbu2)
2·5!·16
(λγmnpq[r|λ)(λγcdefgθ)(θγn
abθ)(θγjklθ)
|s]γcdefgγbu2)
3!·16(u3γqγjklγsu4) ,
one obtains a combination of the fundamental correlators listed in (2.5), (2.6) and (2.7).
A reliable evaluation of the numerous index symmetrisations is made possible by the use of
a computer algebra program. In doing these calculations with Mathematica, an essential
tool is the GAMMA package [13], expanding the products of gamma matrices in a γ[k] basis.
The result consists of two parts, F̃ = F̃ (δ) + F̃ (ε), with
F̃ (δ) = 1
mpru2)(u3γ
au4) +
r (u1γ
iu2)(u3γiu4) + . . .
ai1i2u2)(u3γ
i1i2u4) (92 terms) (2.9)
F̃ (ε) = − 1
1209600
εi1...i7
mpr(u1γ
i1...i7u2)(u3γ
au4) + . . .
604800
εampri1...i6(u1γ
i3...i9u2)(u3γ
i7i8i9u4) (34 terms) (2.10)
The epsilon tensors in the second part can be eliminated using the fact that the ui are
chiral spinors: If all the indices on γ[k]ui are contracted into an epsilon tensor, one uses
εi1...ik′j1...jkγ
j1...jkγ11 = (−)
k(k+1)
k! γi1...ik′ , (2.11)
where γ11 = 1
εi0...i9γ
i0...i9 . More generally, if all but r indices of γ[k]ui are contracted,
εi1...ik′ j1...jkγ
p1...prj1...jkγ11 = (−)
k(k+1)
(k′ − r)!
pr ...p1
[i1...ir
γir+1...i′k]
. (2.12)
– 4 –
The result of these manipulations is
F̃ (ε) =− 1
mpru2)(u3γ
au4)− 1280δ
r (u1γ
amiu2)(u3γiu4) + . . .
11200
i1i2i3u2)(u3γ
i1i2i3u4) (53 terms) (2.13)
(Note that while the epsilon terms in the basic correlator formulae were easily obtained from
the delta terms by using Poincaré duality, this cannot be done here in any obvious way.)
The last step in the evaluation of (2.3) is to contract with the momenta, F = k2ak
r F̃ ,
and to simplify the expressions using the on-shell identities
i ki = 0, k
i = 0, /kiui = 0. It
is shown in appendix A.2 that there are only ten independent scalars, denoted by B1 . . . B10,
that can be formed from four momenta and the four spinors u1 . . . u4. With respect to this
basis, the result is
F (δ) = 1
48·10080
695s12(u1/k3u2)(u3/k1u4) + · · ·+ 233s213(u1γau2)(u3γau4)
(7 terms)
48·10080 (695, 775, 0,−80, 356, 356, 0, 233, 233, 0)B1 ...B10 ,
F (ε) = 1
48·10080 (−23,−7, 0,−16, 28, 28, 0, 7, 7, 0)B1 ...B10 ,
F = 1
10080
(14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.14)
where sij = ki · kj .
2.2 A spinorial formula
While the derivation of tensorial identities for correlators of the form (2.4) is relatively
straightforward and elegant, it may be a tedious task to transform the expressions encoun-
tered in amplitude calculations to match this pattern. As seen in the example calculated
above, this is particularly true if additional spinors are involved, making it necessary to ap-
ply Fierz transformations. It is therefore desirable to use a covariant correlator expression
with open spinor indices. Such an expression was given in [1, 2]:
T̄αβγ,δ1...δ5 = N−1
(γm)αδ1(γn)βδ2(γp)γδ3(γmnp)
(αβγ)[δ1...δ5]
, (2.15)
where N is a normalisation constant and the brackets ()[] denote (anti-)symmetrisation
with weight one. (Note that the right hand side is automatically gamma-matrix traceless:
any gamma-trace
(γr)αβ × (γm)α[δ1|(γn)β|δ2|(γp)γ|δ3(γmnp)δ4δ5] = −(γmnr)[δ1δ2(γmnp)δ3δ4(γp)δ5]γ = 0
vanishes due to the double-trace identity (γabθ)
α(θγabcθ) = 0, which follows from the
fact that the tensor product (Alt3 S+)⊗ S− does not contain a vector representation and
therefore the vector (ψγabθ)(θγ
abcθ) has to vanish for all spinors ψ, and can also be shown
by applying a Fierz transformation.) This prescription was originally motivated [2] by the
fermionic expansion of the Yang-Mills antighost vertex operator V ,
V = Tαβγ,δ1...δ5λ
αλβλγθδ1 . . . θδ5 (2.16)
Tαβγ,δ1...δ5 =
(γm)αδ1(γ
n)βδ2(γ
p)γδ3(γmnp)δ4δ5
(αβγ)[δ1...δ5]
– 5 –
where T is related to T̄ by a parity transformation, up to the overall constant N . (Since
T̄ is uniquely determined by its symmetries, any covariant expression will be proportional
to T̄ , after symmetrisation of the spinor indices, and this is merely the simplest choice.)
Equation (2.15) immediately yields an algorithm to convert any correlator into traces
of gamma matrices or, if additional spinors are involved, bilinears in those spinors. It
is, however, already very tiresome to determine the normalisation constant N by hand.
The main advantage of this approach is that it clearly lends itself to implementation on
a computer algebra system, which can easily carry out the spinor index symmetrisations,
simplify the gamma products (again using the GAMMA package), and compute the traces.
For example,
N〈V 〉 =
(γm)αδ1(γn)βδ2(γp)γδ3(γmnp)
(αβγ)[δ1...δ5]
(γx)αδ1(γy)βδ2(γz)γδ3(γ
xyz)δ4δ5
= − 1
Tr(γxγ
m)Tr(γyγ
n)Tr(γzγ
p)Tr(γxyzγpnm) + . . .
Tr(γzγpnmγ
zyxγnγxγ
p) (60 terms)
= 5160960 .
The correct normalisation is therefore obtained by setting N = 5160960.
Returning to the example correlator (2.3), one finds that the calculation is by far
simpler than with the previous method. After carrying out the symmetrisations (αβγ)[δi],
one obtains
NF̃ = 1
Tr(γxγ
mnpq[r|)(u3γqγ
xyzγsu4)(u1γ
|s]γzγbu2) + . . .
(u2γbγ
xyzγqu3)(u1γsγyγ
mnpq[rγzγ
s]u4) , (24 terms)
where elementary index re-sorting has reduced the number of terms from 60 to 24. Ex-
panding the gamma products leads to
NF̃ = 476
δpr (u1γ
mu4)(u2γ
au3) + · · ·+ 815(u1γ
ai1i2i3i4u2)(u3γ
i1i2i3i4u4) , (294 terms)
which, in contrast to (2.10), contains no epsilon terms as there are not enough free indices
present. Note that this intermediate result contains terms with with u1 paired with u3
or u4, so it is not possible to directly compare to eqs. (2.9) and (2.13). However, after
contracting with the momenta k2ak
r and decomposing the result in the basis B1 . . . B10,
one again obtains
F = 1
10080
(14, 16, 0,−2, 8, 8, 0, 5, 5, 0)B1 ...B10 , (2.17)
in agreement with (2.14).
The algorithm just outlined will be the method of choice for all correlator calculations
in the later sections of this paper and can easily be applied to a wider range of problems.
The only limitation is that the larger the number of gamma matrices and open indices
of the correlator, the slower the computer evaluation will be. For example, the correlator
considered in eq. (5.2) of [11],
mnm1n1...m4n4
(λγpγm1n1θ)(λγqγm2n2θ)(λγrγm3n3θ)(θγmγnγpqrγ
m4n4θ)
= − 2
m1n1...m4n4
εmnm1n1...m4n4
, (2.18)
can still be verified with this method but this already requires substantial runtime.
– 6 –
2.3 Component-based approach
A third method to evaluate the zero mode integrals consists of choosing a gamma matrix
representation, expanding the integrand as a polynomial in spinor components, and then
applying (2.15) to the individual monomials. This procedure seems particularly appealing
if at some stage of the calculation one works with a matrix representation anyhow, in
order to reduce the results to a canonical form (e.g. as outlined in appendix A). An
efficient decomposition algorithm (of k4u1u2u3u4 scalars, say) only needs a few non-zero
momentum and spinor wavefunction components to distinguish all independent scalars, and
therefore k and u can be replaced by sparse vectors. Furthermore, a trivial observation
allows for a much quicker numeric evaluation of correlator components than a naive use
of (2.15): In view of (2.16), one can equivalently compute the components of the parity-
transformed expression V̄ = (λ̄γmθ̄)(λ̄γnθ̄)(λ̄γpθ̄)(θ̄γmnpθ̄), where λ̄ and θ̄ are spinors of
chirality opposite to that of λ, θ. In the representation given in appendix B, V̄ coincides
with V |λ→λ̄,θ→θ̄, and
V = 192λ9λ9λ9θ1θ2θ3θ4θ9 + · · ·+ 480λ1λ2λ3θ1θ9θ10θ13θ15 + . . . (100352 terms)
The monomials in the fermionic expansion of V̄ then correspond to the arguments of
non-zero correlators, and the coefficients of those monomials are, up to normalisation and
symmetry factors, the correlator values.
Unfortunately, it turns out that the complexity of typical correlators (e.g. the one
given in (2.3)) makes it difficult to carry out the expansion in fermionic components in
any straightforward way and limits this method to special applications. For example, the
coefficients in (2.18) can be checked relatively easily by choosing particular index values,
such as
(λγpγ12θ)(λγqγ21θ)(λγrγ34θ)(θγ0γ0γpqrγ
12λ1λ1λ1θ1θ9θ10θ11θ12 + · · ·+ 12λ16λ16λ16θ5θ6θ7θ8θ16
(For fixed values of pqr, one gets no more than about 105 monomials of the form λ3θ5).
This approach may thus still be helpful in situations where the result has been narrowed
down to a simple ansatz.
3. One-loop amplitudes
The amplitude for the scattering of four massless states of the type IIB superstring was
computed [2] in the pure spinor formalism as
A = KK̄
(Im τ)5
G(zi, zj)
ki·kj , (3.1)
where G(zi, zj) is the scalar Green’s function, and the kinematic factor is given by the
product KK̄ of left- and right-moving open superstring expressions,
K1-loop =
(λA1)(λγ
mW2)(λγ
nW3)F4,mn
cycl(234)
. (3.2)
– 7 –
Here the indices 1 . . . 4 label the external states and “· · ·+
cycl(234)
” denotes the addition
of two other terms obtained by cyclic permutation of the indices 234. The spinor super-
field Aα and its supercovariant derivatives, the vector gauge superfield Am =
m DαAβ
as well as the spinor and vector field strengths Wα = 1
(γm)αβ(DβAm − ∂mAβ) and
Fmn = 18(γmn)
β = 2∂[mAn], describe ten-dimensional super-Yang-Mills theory.
The physical fields of this theory, a gauge boson and a gaugino, are found in the leading
components Am| = ζm and Wα| = ûα and correspond to the Neveu-Schwarz and Ramond
superstring states.
The superfields Aα and W
α as well as the gaugino field ûα are anticommuting.1 To
facilitate computer calculations involving polynomials in the spinor components, and for
easier comparison with the literature, it will be more convenient to work with commuting
fermion wavefunctions uα. Fortunately, as the kinematic factors with fermionic external
states are multilinear functions of the distinctly labelled spinors ûi, it is straightforward to
translate between the two conventions: Any monomial expression in û1 . . . û4 (and possibly
fermionic coordinates θ) corresponds to the same expression in u1 . . . u4, multiplied by the
signature of the permutation sorting the ûi (and any θ variables) into some fixed order,
such as (θ · · · θ)ûα11 û
Choosing a gauge where θαAα = 0, the on-shell identities
2D(αAβ) = γ
αβAm , DαW
β = 1
(γmn)α
have been used to derive recursive relations [10, 14, 15] for the fermionic expansion
A(n)α =
(γmθ)αA
(n−1)
m , A
(θγmW
(n−1)) , Wα(n) = − 1
(γmnθ)α∂mA
(n−1)
where f (n) = 1
θαn · · · θα1(Dα1 · · ·Dαnf)|. These recursion relations were explicitly solved
in [10], reducing the fermionic expansion to a simple repeated application of the derivative
operator Omq = 12 (θγm
qpθ)∂p:
A(2k)m =
(2k)!
[Ok]mqζq ,
A(2k+1)m =
(2k+1)!
[Ok]mq(θγqû) .
(3.3)
With this solution at hand, one has all ingredients to evaluate the kinematic factor (3.2)
for the three cases of zero, two, or four fermionic states.
3.1 Review: four bosons
The kinematic factor involving four bosons was considered in [7] and this calculation will
now be reviewed briefly. First, note that the outcome is not fixed by symmetry: The result
must be gauge invariant [2] and therefore expressible in terms of the field strengths F1 . . . F4.
The cyclic symmetrisation in (3.2) yields expressions symmetric in F2, F3, F4, and acting
on scalars constructed from the Fi only, the (234) symmetrisation is equivalent to complete
symmetrisation in all labels (1234). Thus the result must be a linear combination of the
1Thanks to Carlos Mafra for pointing this out.
– 8 –
two gauge invariant symmetric F 4 scalars, namely the single trace Tr(F(1F2F3F4)) and
double trace Tr(F(1F2)Tr(F3F4)), leaving one relative coefficient to be determined.
Since all four states are of the same kind, one may first evaluate the correlator for one
labelling and then carry out the cyclic symmetrisation:
1-loop =
(λA1)(λγ
mW2)(λγ
nW3)F4,mn
cycl (234)
The different ways to saturate θ5 result in a sum of terms of the form
XABCD =
1 )(λγ
2 )(λγ
(3.4)
with A+B +C +D = 5 and A, B, C odd, D even:
(λA1)(λγ
mW2)(λγ
nW3)F4,mn
= X3110 +X1310 +X1130 +X1112 .
Note that X1310 and X1130 are related by exchange of the labels 2 and 3. This exchange
can be carried out after computing the correlator, an operation which will in the following
be denoted by π23. Using (3.3) for the superfield expansions and replacing ∂m → ikm, one
obtains
X3110 = − 1512F
tuX̃3110 , X̃3110 =
(λγ[t|γpqθ)(λγ|u]γrsθ)(λγaθ)(θγ
amnθ)
X1112 = − 1128 ik
tuX̃1112 , X̃1112 =
(λγ[m|γpqθ)(λγ|a]γrsθ)(λγnθ)(θγa
X1310 = − 1384 ik
tuX̃1310 , X̃1310 =
(λγ[t|γmaθ)(λγ|u]γrsθ)(λγnθ)(θγa
The method outlined in section 2.2 is readily applicable to these correlators. For example,
for X3111, the trace evaluation yields
X̃3110 = N
Tr(γaγ
z)Tr(γxyzγ
anm)Tr(γxγqpγ
[t|)Tr(γyγsrγ
|u]) + · · ·
· · ·+ 1
Tr(γ[u|γrsγzyxγqpγ
|t]γxγaγ
yγmnaγz)
(60 terms)
δmprs δ
tu − 1315δ
rs − 145δ
δmnpr δ
[mn][pq][rs][tu](pq↔rs)
Upon contracting with the field strengths, momenta and polarisations, and symmetrising
over the cyclic permutations (234) (with weight 3), one finds that all three contributions
are separately gauge invariant:
X3110 +
cycl(234)
= − 11
13440
Tr(F(1F2F3F4)) +
Tr(F(1F2)Tr(F3F4))
X1112 +
cycl(234)
= − 19
53760
Tr(F(1F2F3F4)) +
215040
Tr(F(1F2)Tr(F3F4))
(1 + π23)X1310 +
cycl(234)
= − 1
10240
4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4))
The sum X3110 +X1112 has the right ratio of single- and double-trace terms to be propor-
tional to the well-known result t8F
4, and the last line exhibits the right ratio by itself. The
overall kinematic factor is therefore
K4B1-loop = − 12560
4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4))
= − 1
15360
4 , (3.5)
in agreement with the expressions derived in the RNS [16] and Green-Schwarz [17] for-
malisms.
– 9 –
3.2 Four fermions
The four-fermion kinematic factor could be evaluated in the same way as in the four-boson
case by summing up all terms XABCD, A + B + C + D = 5, now with A, B, C even
and D odd. Note however that this time, the outcome is fixed by symmetry: The cyclic
symmetrisation in (3.2) leads to a completely symmetric dependence on û2, û3, û4, and
therefore to a completely antisymmetric dependence on u2, u3, u4. Acting on scalars of
the form k2u1u2u3u4, antisymmetrising over [234] is equivalent to antisymmetrising over
[1234], and there is only one completely antisymmetric k2u1u2u3u4 scalar. Without further
calculation, one can infer that the kinematic factor is proportional to that scalar,
K4F1-loop = const ·
(u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3)
which of course agrees with the RNS amplitude (see e.g. [16], eq. (3.67)).
3.3 Two bosons, two fermions
In evaluating (3.2) for two bosons and two fermions, the cyclic symmetrisations affect
whether the W and F superfields contribute bosons or fermions. Only the label of the Aα
superfield stays unaffected, and one has to choose whether it should contribute a boson
or a fermion. Since its fermionic expansion starts with the bosonic polarisation vector,
A1,α ∼ (/ζ1θ)α, the calculation can be simplified by choosing a labelling where particle 1 is
a fermion. (Of course, the final result must be independent of this choice.) The assignment
of the other three labels is then irrelevant and will be chosen as f1f2b3b4. Writing out
the cyclic permutations, two of the three terms are essentially the same because they are
related by interchange of the labels 3 and 4. The kinematic factor is then
K2B2F1-loop(f1f2b3b4) = (1 + π34)
(even)
1 )(λγ
(even)
2 )(λγ
(odd)
(even)
(even)
1 )(λγ
(odd)
3 )(λγ
(odd)
(odd)
Unlike in the four-fermion calculation, the result is not fixed by symmetry. There are five
independent ku1u2F3F4 scalars (see appendix A, eq. (A.6)), denoted by C1 . . . C5, and there
are two independent combinations of these scalars with the required [12](34) symmetry.
Expanding the superfields and collecting terms with θ5, the first line yields a combination
of terms XABCD with A, B, D odd and C even. There is only one θ
5 combination coming
from the second line, which will be denoted by X ′2111 ≡ (−π24)X2111:
K2B2F1-loop = (1 + π34) (X4010 +X2210 +X2030 +X2012) +X
2111 ,
with the correlators
X4010 =
ζ3c k
nX̃4010 , X̃4010 =
(λγaθ)(θγa
pqθ)(θγpu1)(λγ
[mu2)(λγ
n]γbcθ)
X2210 = − i12k
nX̃2210 , X̃2210 =
(λγaθ)(θγau1)(λγ
[m|γbcθ)(θγcu2)(λγ
|n]γdeθ)
X2030 = − i36k
nX̃2030 , X̃2030 =
(λγaθ)(θγau1)(λγ
[mu2)(λγ
n]γbcθ)(θγc
X2012 = − i12k
ζ3c k
ζ4e X̃2012 , X̃2012 =
(λγaθ)(θγau1)(λγ
[mu2)(λγ
n]γbcθ)(θγn
X ′2111 =
ζ3c k
2111 , X̃
2111 =
(λγaθ)(θγau1)(λγ
[m|γbcθ)(λγ|n]γdeθ)(θγnu2)
– 10 –
(The numerical coefficient in X ′2111 includes a sign coming from the θ, û ordering: there
is an odd number of θs between u1 and u2.) Evaluating these expressions as outlined in
section 2.2, the spinor wavefunctions ui present no complication. The last part takes the
simplest form: One finds
(λγaθ)(θγau1)(λγ
mγbcθ)(λγnγdeθ)(θγnu2)
= − 1
(2δbcm[d(u1γe]u2) + δ
m(u1γ
c]deu2))
and therefore
X̃ ′2111 = − 1480
δ[bm(u1γ
c]γdeu2) + δ
m(u1γ
e]γbcu2)
The result for X̃4010 is
X̃4010 =
δbqmn(u1γ
cu2)− 190δ
mq(u1γ
nu2) +
δbcmn(u1γ
qu2)− 12520δ
q (u1γ
bcnu2)
δbq(u1γ
cmnu2) +
δbm(u1γ
cnqu2) +
bcmnqu2)
[bc][mn]
For the evaluation of X̃2210, it is useful to consider the more general correlator
(λγaθ)(θγau1)(λγ
[m|γbcθ)(λγ|n]γdeθ)(θγxu2)
mn(u1γ
cu2) + . . .
201600
δmx (u1γ
bcdenu2) + · · · − 11403200 (u1γ
bcdemnxu2)
[mn][bc][de]
(27 terms)
9676800
εbcdemni1i2i3i4(u1γ
i1i2i3i4xu2)− 12419200εbcdemnxi1i2i3(u1γ
i1i2i3u2) .
This time, even using the method of section 2.2, there are sufficiently many open indices
and long enough traces for epsilon tensors to appear. Using eqs. (2.11) and (2.12), they
can be re-written into γ[5,7] terms:
(λγaθ)(θγau1)(λγ
[m|γbcθ)(λγ|n]γdeθ)(θγxu2)
mn(u1γ
cu2) + . . .
16800
δmx (u1γ
bcdenu2) + · · · − 133600 (u1γ
bcdemnxu2)
[mn][bc][de]
(27 terms)
A good check on the sign of the epsilon contributions is that X̃ ′2111 is recovered when
contracting with ηnx, involving a cancellation of all γ
[5] terms. To obtain X̃2210, one
multiplies by −ηcx:
X̃2210 =
δdemn(u1γ
bu2) +
δbdmn(u1γ
eu2) +
δbmde (u1γ
nu2) +
20160
δdm(u1γ
benu2)
δbm(u1γ
denu2) +
20160
δbd(u1γ
emnu2) +
bdemnu2)
[de][mn]
For the calculation of X2030 and X2012, one may first evaluate a more general correlator
〈(λγaθ)(θγau1)(λγ[mu2)(λγn]γbcθ)(θγxγdeθ)〉 and then contract with ηcx and ηnx, respec-
tively. The results are
X̃2030 =
δdemn(u1γ
bu2) +
δbdmn(u1γ
eu2)− 11440δ
de (u1γ
nu2)− 1710080δ
m(u1γ
benu2)
10080
δbm(u1γ
denu2)− 11440δ
d(u1γ
emnu2) +
bdemnu2)
[mn][de]
X̃2012 =
δdebm(u1γ
cu2) +
δbcdm(u1γ
eu2)− 11440δ
de(u1γ
mu2) +
δdm(u1γ
bceu2)
10080
δbm(u1γ
cdeu2) +
10080
δbd(u1γ
cemu2)− 13360 (u1γ
bcdemu2)
[bc][de]
– 11 –
After multiplication with the momenta and polarisations, all individual contributions are
gauge invariant and can be expanded in the basis C1 . . . C5 listed in (A.6):
(1 + π34)X4010 =
483840
(−6,−16,−40, 6, 0)C1 ...C5
(1 + π34)X2210 =
483840
(−18,−104,−176, 18, 0)C1 ...C5
(1 + π34)X2030 =
483840
(−21, 42,−42, 21, 0)C1 ...C5
(1 + π34)X2012 =
483840
(−39, 78,−78, 39, 0)C1 ...C5
X ′2111 = − i11520 (1, 0, 4,−1, 0)C1 ...C5
The sum can be written as
K2B2F1-loop = X
2111 = − i3840 (1, 0, 4,−1, 0)C1 ...C5
= − i
s13(u2/ζ3(/k2 + /k3)/ζ4u1) + s23(u2/ζ4(/k2 + /k4)/ζ3u1)
(3.6)
and again agrees with the amplitude computed in the RNS result, see [16] eq. (3.37).
4. Two-loop amplitudes
The pure spinor formalism was used in [4, 2] to compute the two-loop type-IIB amplitude
involving four massless states,
d2Ω11d
2Ω12d
i,j ki · kj G(zi, zj)
(det ImΩ)5
K2-loop(ki, zi) ,
where Ω is the genus-two period matrix, and the integration over fermionic zero modes is
encapsulated in
K2-loop = ∆12∆34
(λγmnpqrλ)(λγsW1)F2,mnF3,pqF4,rs
perm(1234)
(4.1)
≡ ∆12∆34K12 +∆13∆24K13 +∆14∆23K14 . (4.2)
The kinematic factors K12, K13, K14 are accompanied by the basic antisymmetric biholo-
morphic 1-form ∆, which is related to a canonical basis ω1, ω2 of holomorphic differentials
via ∆ij = ∆(zi, zj) = ω1(zi)ω2(zj) − ω2(zi)ω1(zj). The superfields Wαi and Fi,mn are the
spinor and vector field strengths of the i-th external state, as in section 3. One encounters
superspace integrals of the form
Y (abcd) =
(λγmnpqrλ)(λγsWa)Fb,mnFc,pqFd,rs
. (4.3)
The symmetries of the λ3 combination [4] in this correlator include the obvious symmetry
under mn↔ pq, and also (λγ[mnpqrλ)(λγs])α = 0 (this holds for pure spinors λ and can be
seen by dualising, and holds for unconstrained spinors λ as part of a λ3θ5 scalar, as seen
from the representation content (2.8)), and allow one to shuffle the F factors:
Y (abcd) = Y (acbd) , Y (abcd) + Y (acdb) + Y (adbc) = 0 . (4.4)
– 12 –
4.1 Review: four bosons
The case of four Neveu-Schwarz states was considered in [6] and will be briefly reviewed
here. As all three kinematic factors K12, K13 and K14 are equivalent, it is sufficient to
consider K12 in detail. With all external states being identical, the symmetrisations of
(4.1) can be carried out at the end of the calculation:
K4B12 = 4
W[1F2]F[3F4]
W[3F4]F[1F2]
= (1− π12)(1− π34)(1 + π13π24)
W1F2F3F4
Expanding the superfields and adopting the notation
YABCD(abcd) =
(λγmnpqrλ)(λγsW (A)a )F
F (C)c,pqF
the Neveu-Schwarz states come from terms of the form YABCD ≡ YABCD(1234) with A odd
and B, C, D even. Using the shuffling identities (4.4) to simplify, one obtains
W1F2F3F4
= Y5000 + Y1400 + Y1040 + Y1004 + Y3200 + Y3020 + Y3002 + Y1220 + Y1202 + Y1022
= (1 + π23)(1− π24)
Y5000 + Y1400 + Y3200 + Y1022
and therefore K4B12 can be written as the image of a symmetrisation operator S4B:
K4B12 = S4B
Y5000 + Y1400 + Y3200 + Y1022
S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24)
It is worth noting at this point that, on the sixteen-dimensional space of Lorentz scalars
built from the four field strengths Fi and two momenta, the symmetriser S4B has rank four.
The correlators were computed in [6], using the method outlined in section 2.1. Two are
zero, Y5000 = Y1400 = 0, and the remaining ones are
Y3200 =
(λγmnpqrλ)(λγsγabθ)(θγb
cdθ)(θγn
Y1022 =
F 1abF
(λγmnpq[rλ)(λγs]γabθ)(θγq
cdθ)(θγs
In reducing those two contributions to a set of independent scalars, one finds that they
both are not just sums of (k · k)F 4 terms but also contain terms of the form k · F terms.
The latter are projected out by the symmetriser S4B, and the result is
K4B12 = S4B(Y3200 + Y1022) = 1120 (s13 − s23)
4Tr(F(1F2F3F4))− Tr(F(1F2)Tr(F3F4))
(s13 − s23)t8F 4 .
By trivial index exchange, one obtains K13 and K14, and the total is
K4B2-loop =
(s13 − s23)∆12∆34 + (s12 − s23)∆13∆24 + (s12 − s13)∆14∆23
4 , (4.5)
a product of the completely symmetric one-loop kinematic factor t8F
4 and a completely
symmetric combination of the momenta and the ∆ij.
– 13 –
4.2 Four fermions
The calculation involving four Ramond states is very similar to the bosonic one. Focussing
on the K12 part, the symmetrisations in (4.1) can again be rewritten as action of sym-
metrisation operators on the correlator of superfields with one particular labelling:
K4F12 (ûi) = (1− π12)(1 − π34)(1 + π13π24)
W1F2F3F4
û1û2û3û4
= 4(1− π12)
W1F2F3F4
û1û2û3û4
The last step follows from the fact that all scalars of the form k4u4 (see appendix A.2),
and therefore all k4û4 scalars, are invariant under π13π24 and have π12 = π34. This time,
on expanding the superfields, one collects the terms YABCD with A even and B, C, D odd.
After using (4.4) to simplify,
W1F2F3F4
û1û2û3û4
= Y2111 + Y0311 + Y0131 + Y0113
= (1 + π23)(1− π24)
Y2111 + Y0311
and after translating to commuting wavefunctions ui, which multiplies every permutation
operator with its signature, one obtains
K4F12 (ui) = S4F
Y2111(ui) + Y0311(ui)
, S4F = 4(1 + π12)(1− π23)(1 + π24) .
This symmetriser has rank three, and the result is again not determined by symmetry.
Two correlators have to be computed:
Y2111(ui) = (−2)k1ak2mk3pk4r
(λγmnpq[rλ)(λγs]γabθ)(θγbu1)(θγnu2)(θγqu3)(θγsu4)
Y0311(ui) = (−23)k
(λγmnpq[rλ)(λγs]u1)(θγn
abθ)(θγbu2)(θγqu3)(θγsu4)
With four fermions present, the method of section 2.2 is preferred as it does not involve re-
arranging the fermions using Fierz identities. The first correlator was covered as an example
in that section, and the second one can be evaluated in the same fashion. Expressed in the
basis listed in (A.5), the results are
Y2111(ui) =
(−19,−21, 21, 19,−17,−17, 0, 0, 0, 0)B1 ...B10 ,
Y0311(ui) =
15120
(−14,−16, 0, 2,−8,−8, 0,−5,−5, 0)B1 ...B10 .
After acting with the symmetriser S4F, one obtains the same u4 scalar encountered in the
one-loop amplitude,
K4F12 (ui) = S4F(13Y2111(ui) + Y0311(ui)) =
(−1,−2, 1, 2,−1,−2, 0, 0, 0, 0)B1 ...B10
(s23 − s13)
(u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3)
The K13 and K14 parts again follow by index exchange, and the total result
K4F2-loop(ui) =
(s23 − s13)∆12∆34 + (s23 − s12)∆13∆24 + (s13 − s12)∆14∆23
(u1/k3u2)(u3/k1u4)− (u1/k2u3)(u2/k1u4) + (u1/k2u4)(u2/k1u3)
(4.6)
is again a simple product of the one-loop kinematic factor and a combination of the ∆ij
and momenta.
– 14 –
4.3 Two bosons, two fermions
As in the one-loop calculation of section 3.3, in the mixed case one has to pay some attention
to the permutations in (4.1) since they affect which superfields contribute fermionic fields.
The complete symmetrisation makes it irrelevant which labels are assigned to the two
fermions, and the convention f1f2b3b4 will be used here. The kinematic factor K
12 is then
distinguished from the other two, K2B2F13 and K
14 . Carrying out the symmetrisations in
(4.1) and using the identities (4.4), one finds
K12(û1, û2, ζ3, ζ4) = (1− π12)(1− π34)K̃ ,
K13(û1, û2, ζ3, ζ4) = (2 · 1+ π12 + π34 + 2π12π34)K̃ ,
K14(û1, û2, ζ3, ζ4) = (1+ 2π12 + 2π34 + π12π34)K̃ ,
where, schematically,
(even)
(odd)
(even)
(even)
(odd)
(even)
(odd)
(odd)
. (4.7)
In translating to commuting variables u1 and u2, the permutation operator π12 changes
sign, and therefore2
K12(u1, u2, ζ3, ζ4) = (1+ π12)(1− π34)K̃ ,
K13(u1, u2, ζ3, ζ4) = (2 · 1− π12 + π34 − 2π12π34)K̃ ,
K14(u1, u2, ζ3, ζ4) = (1− 2π12 + 2π34 − π12π34)K̃ .
Expanding the superfields, the contributions to K̃ are:
Y4100 = − i48k
(λγmnpqrλ)(λγsγabθ)(θγbγ
cdθ)(θγcu1)(θγnu2)
Y0500 =
(λγmnpqrλ)(λγsu1)(θγn
abθ)(θγb
cdθ)(θγdu2)
Y0140 =
(λγmnpqrλ)(λγsu1)(θγnu2)(θγq
abθ)(θγb
Y0104 =
(λγmnpqrλ)(λγsu1)(θγnu2)(θγ|s]
abθ)(θγb
Y2300 =
(λγmnpqrλ)(λγsγabθ)(θγbu1)(θγn
cdθ)(θγeu2)
Y2120 =
(λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγq
Y2102 =
(λγmnpqrλ)(λγsγabθ)(θγbu1)(θγnu2)(θγ|s]
Y0320 =
(λγmnpqrλ)(λγsu1)(θγn
abθ)(θγbu2)(θγq
Y0302 =
(λγmnpqrλ)(λγsu1)(θγn
abθ)(θγbu2)(θγ|s]
Y0122 =
(λγmnpqrλ)(λγsu1)(θγnu2)(θγq
abθ)(θγs]
Y3011 =
(λγmnpqrλ)(λγsγabθ)(θγb
cdθ)(θγcu1)(θγnu2)
Y1211 =
F 3abk
(λγmnpqrλ)(λγsγabθ)(θγn
cdθ)(θγqu1)(θγ|s]u2)
Y1031 =
F 3abF
(λγmnpqrλ)(λγsγabθ)(θγq
cdθ)(θγdu1)(θγ|s]u2)
Y1013 =
F 3abF
(λγmnpqrλ)(λγsγabθ)(θγqu1)(θγ|s]
cdθ)(θγdu2)
2This sign change is crucial to avoid the erroneous conclusion that the two-boson, two-fermion kinematic
factor cannot be of the same product form as in the four-boson or four-fermion cases, which would be in
contradiction to the supersymmetric identities derived in [18].
– 15 –
These correlators can be evaluated exactly as described in section 3.3. One finds that
Y0500 = Y0140 = Y0104 = 0, and the sum of the remaining terms reduces to
K̃ = Y4100 + Y2300 + Y2120 + Y2102 + Y0320 + Y0302 + Y0122 + Y3011 + Y1211 + Y1031 + Y1013
(s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 .
After applying the symmetrisation operators,
(1+ π12)(1− π34)K̃ = i180 (s12 + 2s13)× (1, 0, 4,−1, 0)C1 ...C5 ,
(2 · 1− π12 + π34 − 2π12π34)K̃ = i180 (2s12 + s13)× (1, 0, 4,−1, 0)C1 ...C5 ,
(1− 2π12 + 2π34 − π12π34)K̃ = i180 (s12 − s13)× (1, 0, 4,−1, 0)C1 ...C5 ,
the total kinematic factor is seen to be
K2-loop(u1, u2, ζ3, ζ4) = − i180
(s23−s13)∆12∆34+(s23−s12)∆12∆34+(s13−s12)∆12∆34
× (1, 0, 4,−1, 0)C1 ...C5 (4.8)
and displays the same simple product form as in the four-boson and four-fermion case.
5. Discussion
In this paper, different methods were discussed to efficiently evaluate the superspace inte-
grals appearing in multiloop amplitudes derived in the pure spinor formalism. Extending
previous calculations [6, 7] restricted to Neveu-Schwarz states, it was then shown how the
treatment of Ramond states poses no additional difficulties.
While the bosonic calculations of [6, 7] have, in conjunction with supersymmetry,
already established the equivalence of the massless four-point amplitudes derived in the
pure spinor and RNS formalisms, it would be interesting to make contact between the
results of sections 4.2 / 4.3 and two-loop amplitudes involving Ramond states as computed
in the RNS formalism (see for example [19]).
The assistance of a computer algebra system seems indispensible in explicitly evaluat-
ing pure spinor superspace integrals. To avoid excessive use of custom-made algorithms,
it would be desirable to implement these calculations in a wider computational framework
particular adapted to field theory calculations [20].
The methods outlined in this paper should be easily applicable to future higher-loop
amplitude expressions derived from the pure spinor formalism, and, it is hoped, to other
superspace integrals.
Acknowledgements
The author would like to thank Louise Dolan for discussions, and Carlos Mafra for valuable
correspondence. This work is supported by the U.S. Department of Energy, grant no. DE-
FG01-06ER06-01, Task A.
– 16 –
A. Reduction to kinematic bases
In calculating scattering amplitudes one encounters kinematic factors which are Lorentz
invariant polynomials in the momenta, polarisations and/or spinor wavefunctions of the
scattered particles. It can be a non-trivial task to simplify such expressions, taking into
account the on-shell identities
i ki = 0, k
i = 0, ki · ζi = 0, /kiui = 0, and, in the case of
fermions, re-arrangements stemming from Fierz identities.
More generally, one would like to know how many independent combinations of some
given fields (subject to on-shell identitites) there are, and how to reduce an arbitrary expres-
sion with respect to some chosen basis. This appendix outlines methods to address these
problems, with an emphasis on algorithms which can easily be transferred to a computer
algebra system. These methods are not limited to dealing with pure spinor calculations
but the scope will be restricted to amplitudes of four massless vector or spinor particles in
ten dimensions.
A.1 Four bosons
It is not difficult to reduce polynomials in the momenta and polarisations to a canonical
form. The momentum conservation constraint
i ki = 0 is solved by eliminating one
momentum (for example k4), all k
i are set to zero, and one of the two remaining quadratic
combinations of momenta is eliminated (for example s23 → −s12− s13, where sij ≡ ki ·kj).
Then all products ki · ζi are set to zero, and one extra k · ζ product is replaced (when
eliminating k4, the replacement is k3 · ζ4 → (−k1 − k2) · ζ4). The remaining monomials are
then independent. (This is at least the case with the low powers of momenta encountered
in the calculations of sections 3 and 4, where there are enough spatial directions for all
momenta/polarisations to be linearly independent.)
The implementation of these reduction rules on a computer is straightforward. The
easiest way to obtain scalars which are also invariant under the gauge symmetry ki → ζi
is to start with expressions constructed from the field strengths F abi = 2∂
i . For the
one-loop calculations of section 3.1, the relevant basis consists of gauge invariant scalars
containing only the four field strengths F1 . . . F4. One finds six independent combinations,
Tr(F1F2F3F4)
Tr(F1F2F4F3)
Tr(F1F3F2F4)
Tr(F1F2)Tr(F3F4)
Tr(F1F3)Tr(F2F4)
Tr(F1F4)Tr(F2F3)
In the two-loop calculations of section 4.1, all monomials have two more momenta. There
are sixteen independent gauge invariant scalars of the form kkF1F2F3F4, and twelve of
them may be constructed from the previous basis by multiplication with s12 and s13:
A1 = s12 Tr(F1F2F3F4), A2 = s13 Tr(F1F2F3F4), etc. One choice for the additional four is
A13 = k3 · F1 · F2 · k3 Tr(F3F4) A15 = k3 · F1 · F4 · k2 Tr(F2F3)
A14 = k4 · F1 · F3 · k2 Tr(F2F4) A16 = k4 · F2 · F3 · k4 Tr(F1F4) .
– 17 –
As an example application of the computer algorithms, one may check that the symmetri-
sation operator of section 4.1,
S4B = (1− π12)(1− π34)(1 + π13π24)(1 + π23)(1− π24) ,
acts as
S4BA1 = 8A1 + 4A2 − 4A3 + 4A4 + 8A5 + 16A6
. . .
S4BA16 = −6A1 + 6A3 − 6A5 − 12A6 + 32A7 + 3A8 +
A9 + 3A10 +
A11 + 3A12
and has rank four.
A.2 Four fermions
In dealing with the spinor wavefunctions ui one has to face two issues: Fierz identities, and
the Dirac equation. Fierz identities not only allow one to change the order of the spinors
but also give rise to relations between different expressions in one spinor order. The Dirac
equation often simplifies terms with momenta contracted into (uiγ
[n]uj) bilinears.
In this section it is shown how to construct bases for terms of the form (k2 or k4) ×
u1u2u3u4. A significant simplification comes from noting that the Dirac equation allows
one to rewrite (uiγ
[n]uj) bilinears into terms with lower n if more than one momentum is
contracted into the γ[n]. A good first step is therefore to disregard the momenta temporarily
and find all independent scalars and two-index tensors built from u1, . . . , u4. From the
SO(10) representation content,
(S+)⊗4 = 2 · 1+ 6 · + 3 ·˜+ (tensors with rank > 2) ,
one expects two scalars and nine 2-tensors. The scalars are easily found by considering, as
in [21],
T1(1234) = (u1γ
au2)(u3γau4) ,
T3(1234) = (u1γ
abcu2)(u3γabcu4) .
and similarly for the other two inequivalent orders of the four spinors. (Note there is no T5
because of self-duality of the γ[5].) From Fierz transformations, one learns that all T3 terms
can be reduced to T1 by T3(1234) = −12T1(1234)− 24T1(1324) and permutations, and the
identity (γa)(αβ(γ
a)γ)δ = 0 implies that T1(1234) + T1(1324) + T1(1423) = 0, leaving for
example T1(1234) and T1(1324) as independent scalars.
Generalising this approach to two-index tensors, it turns out that it is sufficient to
start with
T11(1234) = (u1γ
mu2)(u3γ
nu4) ,
T31(1234) = (u1γ
aγmγnu2)(u3γau4) ,
T33(1234) = (u1γ
abγmu2)(u3γabγ
nu4) ,
– 18 –
and permutations of the spinor labels. It would be very tiresome to systematically apply
a variety of Fierz transformations by hand and to find an independent set. Fortunately,
by choosing a gamma matrix representation (such as the one listed in appendix B) and
reducing all expressions to polynomials in the independent spinor components u1i , . . . , u
this problem can be solved with computer help. As expected, one finds that the Tij(abcd)
span a nine-dimensional space, and a basis can be chosen as
T11(1234), T11(1324), T11(1423), T11(3412), T11(2413), T11(2314),
T31(1234), T31(1324), T31(2314) . (A.1)
A typical relation reducing the other Tij(abcd) to this basis is
T31(3412) = 2T11(1234) − 2T11(3412) + T31(1324) + T31(2314) + 2ηmnT1(1234) . (A.2)
Having solved the first step, it is now easy to include the two or four momenta, taking
the Dirac equation into account. Consider first the case of two momenta. Starting from
the two-tensors in (A.1), one gets the three independent scalars
(u1/k3u2)(u3/k1u4) , (u1/k2u3)(u2/k1u4) , (u1/k2u4)(u2/k1u3) .
In addition, there are four products of the two independent scalars T1(1234) and T1(1324)
with the two independent momentum invariants s12 and s13. By contracting (A.2) with
momenta, one can show that
s12T1(1324) − s13T1(1234)
= −(u1/k3u2)(u3/k1u4) + (u1/k2u3)(u2/k1u4)− (u1/k2u4)(u2/k1u3) , (A.3)
and this relation can be used to eliminate s12T1(1324). (It will become clear later that
there are no independent other relations like this one.) There are thus six independent
k2u1 · · · u4 scalars:
(u1/k3u2)(u3/k1u4) s12 T1(1234)
(u1/k2u3)(u2/k1u4) s13 T1(1234) (A.4)
(u1/k2u4)(u2/k1u3) s13 T1(1324)
Note that there is only one completely antisymmetric combination of those, given by the
right hand side of (A.3). Similarly, in the case of four momenta, one finds ten independent
k4u1 · · · u4 scalars:
B1 = s12 (u1/k3u2)(u3/k1u4) B2 = s13 (u1/k3u2)(u3/k1u4)
B3 = s12 (u1/k2u3)(u2/k1u4) B4 = s13 (u1/k2u3)(u2/k1u4)
B5 = s12 (u1/k2u4)(u2/k1u3) B6 = s13 (u1/k2u4)(u2/k1u3) (A.5)
B7 = s
12 T1(1234) B8 = s12s13 T1(1234)
B9 = s
13 T1(1234) B10 = s
13 T1(1324)
– 19 –
Working in a gamma matrix representation, it is again simple to construct a computer
algorithm which reduces any given k2u1 · · · u4 or k4u1 · · · u4 scalar into polynomials of the
spinor and momentum components. The Dirac equation can then be solved by breaking
up the sixteen-component spinors ui into eight-dimensional chiral spinors u
i and u
i , as in
eq. (B.1). One obtains polynomials in the momentum components kai and the independent
spinor components (uci )
1...8. However, a great disadvantage of this procedure is that it
breaks manifest Lorentz invariance. For example, one encounters expressions which contain
subsets of terms proportional to the square of a single momentum and are therefore equal
to zero, but it is difficult to recognise this with a simple algorithm. The easiest solution
is to choose several sets of particular vectors ki satisfying k
i = 0 and
i ki = 0 and to
evaluate all expressions on these vectors. (By choosing integer arithmetic, one easily avoids
issues of numerical accuracy.) Substituting these sets of momentum vectors in the bases
(A.4) and (A.5) gives full rank six and ten respectively, showing they are indeed linearly
independent.
Equipped with a computer algorithm for these basis decompositions, one finds, for
example, that the symmetriser S4F of section 4.2,
S4F = 4(1 + π12)(1 − π23)(1 + π24) ,
acts on the B1 . . . B10 basis as
S4FB1 = −12B4 + 12B5 + 12B6 ,
. . .
S4FB10 = 8B1 + 16B2 − 8B3 − 16B4 + 8B5 + 16B6 − 24B7 − 24B8 − 24B9
and has rank three.
A.3 Two bosons, two fermions
The combined methods of the last two sections can easily be extended to the mixed case
of two bosons and two fermions. In the one-loop calculation of section 3.3, one encounters
scalars of the form ku1u2F3F4. A basis of such objects is given by
C1 = (u1γ
au2)k
C2 = (u1γ
au2)F
C3 = (u1γ
au2)F
c (A.6)
C4 = (u1γ
abcu2)F
C5 = (u1γ
abcu2)F
There are two combinations antisymmetric in [12] and symmetric in (34):
−C1 + 4C2 +C4 and C2 + C3 .
Finally, there are ten independent scalars of the form k3u1u2F3F4 (relevant to the two-loop
calculation of section 4.3), and they can all be obtained by multiplication of C1 . . . C5 with
the two momentum invariants s12 and s13.
– 20 –
B. A gamma matrix representation
A convenient representation of the SO(1,9) gamma matrices is given by the 32×32 matrices
0 (γa)αβ
(γa)αβ 0
where
(γ0)αβ = 116 = (γ
0)αβ ,
(γ9)αβ =
−18 0
= −(γ9)αβ ,
and (γa)αβ = −(γa)αβ, a = 1 . . . 8, is a real, symmetric 16×16 representation for the SO(8)
Clifford algebra,
(γa)αβ =
(σa)T 0
, a = 1 . . . 8 ,
as given in appendix 5.B of [21]. The matrices Γa satisfy the SO(1,9) Clifford algebra
relations,
{Γa,Γb} = 2ηab132 , ηab = (+−− · · · −) ,
and bilinears of chiral spinors (with, say, positive chirality) are constructed as
(uΓ[a1...ak]v) = (uγ[a1...ak ]v) = uα(γ[a1)αβ(γ
a2)βγ . . . (γak ])γδv
This representation is particularly suitable for the calculations outlined in appendix A
because it allows a simple decomposition of SO(1,9) spinors into SO(8) spinors due to its
block structure:
Γ0 · · ·Γ9 =
116 0
0 −116
, Γ1 · · ·Γ8 =
18 0 0 0
0 −18 0 0
0 0 18 0
0 0 0 −18
Therefore, the Dirac equation for a chiral 16-component spinor u,
(γa)αβ∂au
α = 0 ,
can be solved by splitting u into two chiral eight-component spinors of SO(8),
with γ1...8
One obtains the coupled equations
(∂0 + ∂9)u
s − (σ · ∂)uc = 0
(∂0 − ∂9)uc − (σT · ∂)us = 0
(with eight-dimensional dot products). These can be solved for us in terms of uc:
(σ · ∂)uc =
(σ · k)uc , (B.1)
where k+ = −i∂+ = −i√
(∂0 + ∂9).
– 21 –
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