One Loop Mass Renormalization of Massive States Using Pure Spinor Formalism

Sitender Pratap Kashyapa, Mritunjay Vermab

a The Institute of Mathematical Sciences, IV Cross Street Road,

CIT Campus, Taramani, Chennai 600113, India

b Mathematical Sciences and STAG Research Centre, University of Southampton,

Highfield, Southampton SO17 1BJ, UK

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

As a check of the first massive integrated vertex operator in the pure spinor
formalism constructed in arXiv:1802.04486, we compute the one loop 2-point
function of the stable non BPS massive states in $SO(32)$ heterotic string
theory. This allows us to compute the one loop renormalized mass of these
states using the pure spinor formalism. Our results are in agreement with the
corresponding results obtained by Sen using the RNS formalism. This provides
an instance of the equivalence between the RNS and the pure spinor formalism
for the massive states at loop level.

###### Contents

  1. 1 Introduction
  2. 2 Massive Vertex Operators for Closed Strings
    1. 2.1 Unintegrated Vertex Operator
    2. 2.2 Integrated Vertex Operator
  3. 3 One Loop 2-point Function and mass renormalization
  4. 4 Discussions
  5. A $SO(32)$ Heterotic String in Pure Spinor Formalism
    1. A.1 World-sheet Theory
    2. A.2 Loop Amplitude Prescription
    3. A.3 First Massive States
  6. B  Some details about closed string vertex operators
    1. B.1 Unintegrated vertex operator
    2. B.2 Integrated vertex operator
    3. B.3 Theta Expansion
  7. C  Some useful pure spinor identities

##  1 Introduction

The string scattering amplitudes are the most important ingredients for
probing the perturbative aspects of string theory. Due to this reason, the
calculation of scattering amplitudes received a lot of attention since the
early days of string theory. There are numerous results at tree and loop level
for the massless scattering amplitudes. In contrast, the scattering of massive
states has not received similar attention due to the complexities associated
with these states. However, even at the perturbative level, the massive states
encode a lot of interesting physics which require the knowledge of their
scattering amplitudes [1, 2, 3]. Moreover, the massive states also become
important at low energy scale in the case of large extra dimensions [4, 5].
Some black hole states can also be thought as massive string excitations (see
e.g., [6, 7]) and their scattering can be analysed from the scattering of the
elementary strings. Thus, the computation of the scattering amplitudes of
massive string states is a crucial requirement.

A lot of success in computing the massless string scattering has come from the
development of the pure spinor approach to superstring theory [8]. The RNS
formalism becomes very complicated for the loop amplitudes and the higher
point functions (due to complexities such as requirement of sum over spin
structures). On the other hand, the pure spinor formalism has been used very
successfully to compute the massless string amplitudes which were very
difficult in the RNS approach [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] (see
[19, 20] for recent reviews). However, a similar success for the massive state
scatterings has been missing even in the pure spinor formalism. One reason for
this in the pure spinor formalism is that the vertex operators of massive
states were not known until very recently.

In [21], the first massive unintegrated vertex operator in pure spinor
formalism was constructed in open string theory. However, the equations
satisfied by the superfields appearing in the vertex operator were worked out
only in the rest frame. Due to this, the theta expansion needed for computing
the scattering amplitudes could not be worked out. This problem was solved in
[22] and a systematic method to construct any higher massive integrated or
unintegrated vertex operator was given in [23] which also constructed the
integrated version of the first massive vertex operator. Using these results,
[24] showed that the pure spinor tree level amplitudes involving the massive
states agree with the corresponding RNS results. This only made use of the
unintegrated vertex operator. So far, there has been no computation involving
the integrated massive vertex.

In this paper, we initiate the computations using the first massive integrated
vertex. For this, we shall consider a simple example, namely, one loop 2-point
function of first massive state in the Heterotic string theory. Our motivation
is two fold. First, we want to show that the loop calculations in pure spinor
formalism involving the massive string states are also consistent with the RNS
results. Secondly, in long term, we want to study the 2-loop mass
renormalization of massive states using the pure spinor formalism.

A generic string state, unless protected by some symmetry, undergoes mass
renormalization. However, one can not compute the renormalised masses using
the standard Polyakov formulation of string theory. The reason for this is
that one needs to go off-shell for computing the renormalised masses. In a
quantum field theory, if we are interested in computing the renormalised mass,
we need to reorganise the perturbative expansion for 2-point Green’s function
in terms of 1PI and 1PR diagrams. This perturbative series can be resummed and
the location of pole in this gives the renormalised mass. However, the
Polyakov’s approach to string perturbation theory computes a single diagram at
each loop order without distinguishing between the 1PI and 1PR contributions.
This situation was remedied in [1, 2] which gave a prescription to go off-
shell in the Polyakov’s approach (also see [25]). The most efficient way to do
this is to use the string field theory which neatly separates the 1PI and 1PR
contributions of a given amplitude (see [26] for a review). The off-shell
version of the pure spinor formalism is still not developed. Hence, we can’t
use an off-shell method to compute the mass renormalization using the pure
spinor formalism. However, at one loop, we can compute the mass
renormalization by remaining on-shell since there is no internal free
propagator connecting two 1PI blobs which diverges on-shell. Hence, this
computation can be performed in pure spinor as well.

As mentioned above, we shall work in Heterotic string theory. The first
massive states in SO(32) Heterotic string theory are stable non BPS states and
they undergo mass renormalization due to loop effects [1, 27]. The mass
renormalization of these states can’t be computed by examining the poles in
the S-matrix of the massless states since these massive states do not appear
as single particle intermediate states in the scattering of the massless
particles of the theory [1]. For this reason, one needs to compute the
renormalized masses of these states directly. Another feature of these states
is that they don’t mix with the unphysical states when they undergo mass
renormalization unlike a generic string state [1]. These features make it
ideal for studying the mass renormalization in string theory as well as
comparing the massive one loop calculations in RNS and pure spinor formalisms.
One reason for focussing on pure spinor formalism for this purpose is that as
in the case of massless states, the RNS formalism is, in general, very
difficult to use if we want to compute the higher genus amplitudes involving
the massive states to analyse such effects.

The rest of the paper is organized as follows. In section 2, we give the
expressions of the integrated and unintegrated SO(32) Heterotic first massive
closed string vertex operators, that we shall require in this work. In section
3, we compute the one loop 2-point function of our massive states and their
one loop renormalized mass and show that the results are in agreement with
[27]. We conclude in section 4 with a brief discussion. In the various
appendices we provide additional details required in the main draft. In
appendix A, we review the Heterotic strings in the pure spinor formalism. We
focus on the world-sheet theory, one loop amplitude prescription and the first
massive states of the theory. Appendix B gives some details about the relation
between the open and closed string vertex operators and in appendix C, we note
some useful pure spinor identities involving the massive superfields which are
useful in simplifying our calculations.

##  2 Massive Vertex Operators for Closed Strings

In this section, we construct the integrated and unintegrated vertex operators
for the closed strings using the corresponding open string results. More
details can be found in appendix B. Since we shall describe the supersymmetric
sector of the world-sheet theory in terms of the superfields $B_{mnp},G_{mn}$
and $\Psi_{m\alpha}$, we start by writing down the equations satisfied by
these superfields. As described in appendix B, the closed string vertex
operators can be obtained from the open string vertex operators by rescaling
the momenta as $k\rightarrow\frac{k}{2}$. This is also true for the equations
of motion satisfied by the superfields. Thus, the superspace equations
satisfied by the superfields describing the supersymmetric sector of the
theory are thus obtained from the open string result [22] as

$\displaystyle D_{\alpha}G_{sm}={8}i\ k^{p}(\gamma_{p(s}\Psi_{m)})_{\alpha}$
(2.1) $\displaystyle
D_{\alpha}B_{mnp}=12(\gamma_{[mn}\Psi_{p]})_{\alpha}+{6}i\alpha^{\prime}k^{t}k_{[m}(\gamma_{|t|n}\Psi_{p]})_{\alpha}$
(2.2) $\displaystyle
D_{\alpha}\Psi_{s\beta}=\frac{1}{16}G_{sm}\gamma^{m}_{\alpha\beta}+{\frac{i}{48}}k_{m}B_{nps}(\gamma^{mnp})_{\alpha\beta}-{\frac{i}{288}}k^{m}B^{npq}(\gamma_{smnpq})_{\alpha\beta}$
(2.3)

with the constraints

$\displaystyle(\gamma^{m})^{\alpha\beta}\Psi_{m\beta}=0\quad;\quad\partial^{m}\Psi_{m\beta}=0\quad;\quad\partial^{m}B_{mnp}=0\quad;\quad\partial^{m}G_{mn}=0\;\;\;;\;\;\eta^{mn}G_{mn}=0$
(2.4)

The mutual consistency of these equations is easy to check. For this, we take
the covariant derivative of both sides of these equations and then use the
identity (A.45) for the left hand side and covariant derivative of the
superfield in the right hand side. The both sides then agree as expected.

###  2.1 Unintegrated Vertex Operator

The unintegrated vertex operator $V(z,\bar{z})$ in the closed string theory
satisfy [28]

$\displaystyle QV(z,\bar{z})=0=\bar{Q}V(z,\bar{z})\qquad,\qquad\delta
V=Q\Omega+\bar{Q}\bar{\Omega}$ (2.5)

with the condition $\bar{Q}\Omega=Q\bar{\Omega}=0$.

The equations for both the left and right sectors are essentially same as the
ones arising in the open string case. Thus, the closed string vertex operator
is essentially the tensor product of two copies of the open string operator. A
more detailed analysis shows that if $V(k,z)$ denotes the open string
unintegrated vertex operator with momentum $k$, then the closed string
unintegrated vertex operator is given by (see appendix B for more details)

$\displaystyle V(k,z,\bar{z})=\kappa
V_{{L}}\left(\frac{k}{2},z\right)V_{{R}}\left(\frac{k}{2},\bar{z}\right)e^{ik\cdot
X}$ (2.6)

where $\kappa^{\prime}$ is the overall normalization constant for the vertex
operator. The dependence of $V_{L}$ and $V_{R}$ upon the $X^{m}$ fields is
only through their derivatives $\partial X^{m}$. The fact that we need to take
the tensor product of two copies of the open string vertex operator with only
half of the momenta is also clear from the difference in the OPEs of the open
and closed strings. As discussed in appendix A.1, the closed string OPEs can
be obtained from the corresponding open string OPEs by the replacement
$k\rightarrow\frac{k}{2}$. For the type II theories, both the $V_{L}$ and
$V_{R}$ in the right hand side of (2.6) have same functional form. However,
for the heterotic strings, only the holomorphic factor corresponds to the
supersymmetric open string. The anti-holomorphic factor is constructed
separately as in the RNS formalism. We are interested in the heterotic string
case for which $V_{L}$ can be written down by using the open string pure
spinor result to be111In this paper, we are using the conventions for OPEs
etc. of [16]. The various factors of $\frac{\alpha^{\prime}}{2}$ in the vertex
operators have been inserted as compared to the result given in [22] due to
this change in the conventions. .

$\displaystyle V_{L}\ =\
:\partial\theta^{\beta}\lambda^{\alpha}B_{\alpha\beta}:+\;\frac{\alpha^{\prime}}{2}\;:d_{\beta}\lambda^{\alpha}C^{\beta}_{\;\alpha}:+:\Pi^{m}\lambda^{\alpha}H_{m\alpha}:+\;\frac{\alpha^{\prime}}{2}\;:N^{mn}\lambda^{\alpha}F_{\alpha
mn}:$ (2.7)

where, the superfields are only functions of $\theta^{\alpha}$ (since we have
included the factor of $e^{ik\cdot X}$ separately) and are given in terms of
the basic superfields $B_{mnp}$ and $\Psi_{m\alpha}$ to be [21]

$\displaystyle
H_{s\alpha}=\frac{3}{7}(\gamma^{mn})_{\alpha}^{\;\;\beta}D_{\beta}B_{mns}=-72\Psi_{s\alpha}\;,\quad
C_{mnpq}={\frac{i}{4}}k_{[m}B_{npq]}\;,$ $\displaystyle\hskip
14.45377ptF_{\alpha
mn}={\frac{i}{16}}\biggl{(}7k_{[m}H_{n]\alpha}+k^{q}(\gamma_{q[m})_{\alpha}^{\;\;\beta}H_{n]\beta}\biggl{)}$
(2.8)

Note that in the above equations, we have replaced $k_{m}$ of open string
solution by $\frac{1}{2}k_{m}$.

Now, for the heterotic strings, the right moving factor $V_{R}(\bar{z})$ is
same as in the RNS formalism and is given by222For the type II theories,
$V_{R}$ has the same form as $V_{L}$ in equation (2.7).

$\displaystyle
V_{R}(\bar{z})=\hat{c}(\bar{z})\hat{\rho}^{A}(\bar{z})\hat{\rho}^{A}(\bar{z})\equiv\hat{c}(\bar{z})\hat{J}(\bar{z})$
(2.9)

Before proceeding further, we note that the conformal weight of $e^{ik\cdot
X}$ for the closed string is given by
$(\frac{\alpha^{\prime}k^{2}}{4},\frac{\alpha^{\prime}k^{2}}{4})$ and the mass
of the state at $n^{th}$ level is given by
$k^{2}=-\frac{4n}{\alpha^{\prime}}$. This means that the conformal weight of
$e^{ik\cdot X}$ at mass level $n$ is $(-n,-n)$. Demanding the conformal weight
of the unintegrated vertex operator to be $(0,0)$, we find that the conformal
weights of $V_{L}$ and $V_{R}$ should be $(n,0)$ and $(0,n)$ respectively. The
above expressions (2.7) and (2.9) are consistent with this for $n=1$.

###  2.2 Integrated Vertex Operator

The integrated vertex operator $U$ can also be expressed in the factorized
form

$\displaystyle U(z,\bar{z})=\kappa U_{L}(z)U_{R}(\bar{z})e^{ik\cdot X}$ (2.10)

The expression for the left moving sector involving $U_{L}$ can be written
down using the known RNS result and is given by

$\displaystyle
U_{R}(\bar{z})=\hat{\rho}^{A}(\bar{z})\hat{\rho}^{A}(\bar{z})=\hat{J}(\bar{z})$
(2.11)

The factor $U_{L}(z)$ can be written down by using the corresponding open
string result of [23] by rescaling $k_{m}\rightarrow\frac{1}{2}k_{m}$ to be
333The various factors of $\frac{\alpha^{\prime}}{2}$ in the definition of
$U_{R}$ is present as a result of the new convention that we are following.

$\displaystyle\frac{2}{\alpha^{\prime}}\ U_{L}(z)$ $\displaystyle=$
$\displaystyle\
:\Pi^{m}\Pi^{n}F_{mn}:\;+\;\frac{\alpha^{\prime}}{2}:\Pi^{m}d_{\alpha}F_{m}^{\;\;\alpha}:\;+\;:\Pi^{m}\partial\theta^{\alpha}G_{m\alpha}:\;+\;\frac{\alpha^{\prime}}{2}:\Pi^{m}N^{pq}F_{mpq}:$
(2.12)
$\displaystyle+\;\;\left(\frac{\alpha^{\prime}}{2}\right)^{2}:d_{\alpha}d_{\beta}K^{\alpha\beta}:\;+\;\frac{\alpha^{\prime}}{2}:d_{\alpha}\partial\theta^{\beta}F^{\alpha}_{\;\;\beta}:\;+\;\left(\frac{\alpha^{\prime}}{2}\right)^{2}:d_{\alpha}N^{mn}G^{\alpha}_{\;\;mn}:\;$
$\displaystyle+\;:\partial\theta^{\alpha}\partial\theta^{\beta}H_{\alpha\beta}:\;+\;\frac{\alpha^{\prime}}{2}:\partial\theta^{\alpha}N^{mn}H_{mn\alpha}:\;+\;\;\left(\frac{\alpha^{\prime}}{2}\right)^{2}:N^{mn}N^{pq}G_{mnpq}:$

where, the superfields appearing in (2.12) are given as

$\displaystyle F_{mn}$ $\displaystyle=$
$\displaystyle-\frac{18}{\alpha^{\prime}}G_{mn}\quad\;,\qquad
F_{m}^{\;\alpha}\ =\
{\frac{144i}{\alpha^{\prime}}}(\gamma^{r})^{\alpha\beta}k_{r}\Psi_{m\beta}\quad\;\;,\qquad
G_{m\alpha}\ =\ -\frac{432}{\alpha^{\prime}}\Psi_{m\alpha}$ $\displaystyle
F_{mpq}$ $\displaystyle=$
$\displaystyle\frac{12}{(\alpha^{\prime})^{2}}B_{mpq}-{\frac{18i}{\alpha^{\prime}}}k_{[p}G_{q]m}\quad,\quad\;\;K^{\alpha\beta}\
=\ -\frac{1}{(\alpha^{\prime})^{2}}\;\gamma_{mnp}^{\alpha\beta}B^{mnp}$
$\displaystyle F^{\alpha}_{\;\;\;\beta}$ $\displaystyle=$
$\displaystyle-{\frac{2i}{\alpha^{\prime}}}(\gamma^{mnpq})^{\alpha}_{\;\;\;\beta}k_{m}B_{npq}\qquad,\quad\;\;G^{\alpha}_{mn}\
=\
\frac{48}{(\alpha^{\prime})^{2}}\gamma_{[m}^{\alpha\sigma}\Psi_{n]\sigma}-{\frac{48}{\alpha^{\prime}}}\gamma_{r}^{\alpha\sigma}k^{r}k_{[m}\Psi_{n]\sigma}$
$\displaystyle H_{\alpha\beta}\ $ $\displaystyle=$ $\displaystyle\
\frac{2}{\alpha^{\prime}}\gamma^{mnp}_{\alpha\beta}B_{mnp}\quad\qquad\qquad\;\;,\quad
H_{mn\alpha}\ =\
-{\frac{288i}{\alpha^{\prime}}}\;k_{[m}\Psi_{n]\alpha}-{\frac{72i}{\alpha^{\prime}}}k^{q}(\gamma_{q[m})_{\alpha}^{\;\;\sigma}\Psi_{n]\sigma}$
$\displaystyle G_{mnpq}$ $\displaystyle=$
$\displaystyle{\frac{2i}{(\alpha^{\prime})^{2}}}k_{[m}B_{n]pq}+{\frac{2i}{(\alpha^{\prime})^{2}}}k_{[p}B_{q]mn}+{\frac{3}{\alpha^{\prime}}}k_{[p}k_{[m}G_{n]q]}$
(2.13)

For amplitude computations, we also require the theta expansions of the
superfields which are given in appendix B.3.

##  3 One Loop 2-point Function and mass renormalization

In this section, we compute the one loop 2 point function of the first massive
states in the heterotic string theory using the pure spinor formalism. As
reviewed in appendix A.2, the one loop two point function is given by

$\displaystyle\mathcal{A}=\frac{\kappa^{2}}{2}\int d^{2}\tau
d^{2}z\langle\langle\mathcal{N}B\hat{B}:V_{L}(0)V_{R}(0)e^{ik_{1}\cdot
X(0)}:\;:U_{L}(z)U_{R}(\bar{z})e^{ik_{2}\cdot X(z,\bar{z})}:\rangle\rangle$
(3.14)

where, the integrated and unintegrated vertex operators are given in the
previous section, $B$ and $\hat{B}$ are constructed using the beltrami
differentials (see equation (A.51)), $\mathcal{N}$ is the regulator given in
(A.54) and the double bracket $\langle\langle\cdots\rangle\rangle$ denotes the
fact that we have gotten rid of the non-zero modes by making use of the OPEs.
We denote the external momenta as $k_{1}$ and $k_{2}$ and take them to be
incoming so that the on-shell momentum conservation implies

$\displaystyle k_{1}+k_{2}=0$ (3.15)

We first consider the pure spinor part of the calculation in (3.14). As
discussed in appendix A.2, all the world-sheet fields can be expanded in a
complete set of basis of $\bar{\partial}$ operator. The zero and the non zero
modes in this expansion behave differently. The non zero modes of the various
world-sheet fields have OPEs between them as given in section A. However, the
zero modes of different fields do not have OPEs between them. As the first
step in calculation, one uses the OPEs between the non zero modes of the
world-sheet fields present in the vertex operators and $b$ ghosts to eliminate
them from the integrand in (3.14).

Once we get rid of non zero modes of all the fields from the integrand using
their OPEs, we shall have to perform the path integration over non zero modes
with weight factor $e^{-S}$. The result of this integration is given by

$\displaystyle[\mbox{det}^{\prime}{\bar{\partial}}]^{16}[\mbox{det}^{\prime}{\bar{\partial}}]^{-11}[\mbox{det}^{\prime}{\bar{\partial}}]^{-11}[\mbox{det}^{\prime}{\bar{\partial}}]^{11}=[\mbox{det}^{\prime}{\bar{\partial}}]^{5}$
(3.16)

where, the various factors in the left hand side come from the integrations
over the $(p_{\alpha},\theta^{\beta})$, $(\lambda^{\alpha},w_{\beta})$,
$(\bar{\lambda}^{\alpha},\bar{w}_{\beta})$ and $(r_{\alpha},s^{\beta})$
systems respectively. The primes in (3.16) denote that the zero modes are
excluded from the determinant. Note that the non-zero mode measures are such
that the overall coefficient after integration comes out to be 1 [29].

After this, we need to perform the integration over the zero modes. Our task
is simplified by the fact that for the Grassman odd fields, there are not many
ways to saturate the zero modes and hence, most of the terms in the integrand
do not contribute. We shall denote the zero modes of a world-sheet field
$\phi$ by $\phi^{1}$. We start with the field $s^{\alpha}$. This field only
appears in the regulator $\mathcal{N}$ and the $b$ ghost. However, in the $b$
ghost, it appears in the combination
$s^{\alpha}\partial\bar{\lambda}_{\alpha}$ and hence it can only contribute if
there is $\bar{w}^{\alpha}$ in the integrand (otherwise it vanishes since it
does not provide any zero mode of $\bar{\lambda}$). We have chosen our vertex
operators to be independent of the non-minimal variables. Thus, this term will
never contribute. Noting that $s^{\alpha}$ has 11 zero modes on torus, this
means that these zero modes can only be saturated by the regulator through the
factor $\mathcal{N}\rightarrow(s^{1}d^{1})^{11}$.

Next, we consider the 16 zero modes of $d_{\alpha}$. The regulator
$\mathcal{N}$ has already supplied $11$ zero modes. The remaining 5 must come
from the $b$ ghost and the vertex operators. The $b$ ghost can supply either
one or two zero modes of $d_{\alpha}$. For the two point function we are
interested in, the vertex operators can supply a maximum of 3 zero modes.
Thus, it follows that we must choose 2 zero modes from the $b$ ghost and 3
zero modes from the vertex operator. The 3 zero modes from the vertex
operators can only come in a unique way, namely, we must pick only the
$d_{\alpha}$ containing term in the unintegrated vertex and the
$d_{\alpha}d_{\beta}$ containing term in the integrated vertex operator. Using
the expressions of the vertex operators given in the previous section, the two
point function can, thus, be written as

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}}{2}{\left(\frac{\alpha^{\prime}}{2}\right)^{3}}\int
d^{2}\tau\;\;\biggl{\langle}\mathcal{N}(y)\;B\hat{B}\;:d_{\alpha}\lambda^{\beta}C^{\alpha}_{\;\beta}:V_{R}(z)e^{ik_{1}\cdot
X(z)}$ (3.17) $\displaystyle\hskip 108.405pt\int
d^{2}w\;:d_{\sigma}d_{\tau}K^{\sigma\tau}:U_{R}(w)e^{ik_{2}\cdot
X(w)}\biggl{\rangle}$

Next, we compute the beltrami differential $B$. For this, we write
$d_{\alpha}$ as

$\displaystyle d_{\alpha}(x)=\hat{d}_{\alpha}(x)+d^{1}_{\alpha}\omega_{1}(x)$
(3.18)

The $\omega_{1}$ are the holomorphic 1-form given by $\omega_{1}=1$ on torus.
The non zero mode $\hat{d}_{\alpha}$ will have OPE with the other
$\hat{d}_{\alpha}$ as well as the non zero modes of $\theta^{\alpha}$ and
$X^{m}$ fields. However, these OPE terms will not contribute since they will
not provide the required number of zero mode $d_{\alpha}^{1}$. Thus, in the
expression (3.17), we can replace $d_{\alpha}$ by $d^{1}_{\alpha}$ everywhere.
Noting that we need to pick the term containing two $d_{\alpha}$ from the
$b$-ghost, namely

$\displaystyle\frac{\alpha^{\prime}}{2}\frac{(\bar{\lambda}\gamma^{mnp}r)(d\gamma_{mnp}d)}{192(\lambda\bar{\lambda})^{2}}$
(3.19)

the left moving Beltrami differential term becomes

$\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int d^{2}x\
b(x)\mu(x)$ (3.20) $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int
d^{2}x\frac{\alpha^{\prime}}{2}\frac{(\bar{\lambda}\gamma^{mnp}r)(d^{1}\gamma^{mnp}d^{1})}{192(\lambda\bar{\lambda})^{2}}\omega_{1}\omega_{1}\mu$
$\displaystyle=$
$\displaystyle\frac{\alpha^{\prime}}{2}\frac{(\bar{\lambda}\gamma^{mnp}r)(d^{1}\gamma^{mnp}d^{1})}{2\pi\times
192(\lambda\bar{\lambda})^{2}}$

where we used $\int d^{2}zw_{1}w_{1}\mu=1$ in going to the last line.

Using the above results, the 2-point function can be written as

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha^{\prime}}{2}\right)^{4}\frac{\kappa^{2}}{4\pi\times
192}\int
d^{2}\tau\Big{\langle}\mathcal{N}(y)\hat{B}\;\frac{(\bar{\lambda}\gamma_{mnp}r)(d^{1}\gamma^{mnp}d^{1})}{(\lambda\bar{\lambda})^{2}}d^{1}_{\alpha}\lambda^{\beta}C^{\alpha}_{\;\beta}(z)V_{R}e^{ik_{1}\cdot
X(z)}$ (3.21) $\displaystyle\times\int
d^{2}w\;d^{1}_{\sigma}d^{1}_{\tau}K^{\sigma\tau}(w)U_{R}e^{ik_{2}\cdot
X(w)}\Big{\rangle}$

Next, we use the trick that $r_{\alpha}$ present in the $b$ ghost term in
(3.21) can be replaced by the covariant derivative which acts on all the
superfields in the integrand [11]. To see this, we note that $r_{\alpha}$
inside the integrand can be manipulated as (denoting all the other terms in
the integrand by $T^{\alpha}$ and recalling that the regulator contains the
term $e^{-r\theta}$)

$\displaystyle\left(r_{\alpha}e^{-r_{\beta}\theta^{\beta}}\right)T^{\alpha}$
$\displaystyle=$
$\displaystyle\left(\frac{\partial}{\partial\theta^{\alpha}}e^{-r_{\beta}\theta^{\beta}}\right)T^{\alpha}$
(3.22) $\displaystyle=$
$\displaystyle-e^{-r\theta}(D_{\alpha}-\frac{1}{2}\gamma^{m}_{\alpha\beta}\partial_{m})T^{\alpha}$
$\displaystyle=$
$\displaystyle-e^{-r\theta}D_{\alpha}T^{\alpha}-e^{-r\theta}\frac{i}{2}\gamma^{m}_{\alpha\beta}(k_{1}+k_{2})_{m}T^{\alpha}$
$\displaystyle=$ $\displaystyle-e^{-r\theta}D_{\alpha}T^{\alpha}$

In going to the second line, we used integration by parts and and the
definition of covariant derivative. In going to the 3rd line, we used the fact
that the integrand $T^{\alpha}$ contains the $X^{m}$ fields only through
$e^{ik\cdot X}$. In going to the final line, we used the momentum conservation
$k_{1}+k_{2}=0$.

We now use the zero mode measure for $s^{\alpha}$ given in (A.56) to write

$\displaystyle[ds]e^{sd}$ $\displaystyle=$ $\displaystyle
c_{s}(\lambda\bar{\lambda})^{-3}T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}\epsilon^{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}\partial^{s}_{\rho_{1}}\cdots\partial^{s}_{\rho_{11}}e^{sd}$
(3.23) $\displaystyle=$
$\displaystyle{\left(\frac{\alpha^{\prime}}{2}\right)^{2}\frac{(2\pi)^{11/2}}{2^{6}11!5!}\frac{Z_{1}^{11}}{R(\lambda\bar{\lambda})^{3}}}T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}\epsilon^{\alpha_{1}\cdots\alpha_{5}\beta_{1}\cdots\beta_{11}}d_{\beta_{1}}\cdots
d_{\beta_{11}}$

Using these results, the expression in (3.21) can be written as

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-{\left(\frac{\alpha^{\prime}}{2}\right)^{6}\frac{\kappa^{2}}{4\pi\times
192}\frac{(2\pi)^{11/2}}{2^{6}11!5!}\frac{Z_{1}^{11}}{R}}\int d^{2}\tau\ \int
d^{2}w\int[d\lambda][d\bar{\lambda}][dw^{1}][d\bar{w}^{1}][dd^{1}][dr][d\theta]$
(3.24) $\displaystyle
T_{\sigma_{1}\cdots\sigma_{5}}\epsilon^{\sigma_{1}\cdots\sigma_{5}\rho_{1}\cdots\rho_{11}}d^{1}_{\rho_{1}}\cdots
d^{1}_{\rho_{11}}\;e^{-\lambda\bar{\lambda}-w\bar{w}-r\theta}\frac{(\bar{\lambda}\gamma_{mnp}D)(d^{1}\gamma^{mnp}d^{1})}{(\lambda\bar{\lambda})^{(2+3)}}$
$\displaystyle
d^{1}_{\alpha}\lambda^{\beta}C^{\alpha}_{\;\beta}(z)\;d^{1}_{\sigma}d^{1}_{\tau}K^{\sigma\tau}(w)e^{ik_{1}\cdot
X(z)}e^{ik_{2}\cdot X(w)}\Bigl{\langle}\hat{B}V_{R}(z)U_{R}(w)\Bigl{\rangle}$

We can now perform the integration over the $d_{\alpha},w_{\alpha}$ and
$\bar{w}^{\alpha}$ zero modes. The measures for the zero modes in (A.56) give
at one loop (see, e.g., [18])

$\displaystyle
T_{\sigma_{1}\cdots\sigma_{5}}\epsilon^{\sigma_{1}\cdots\sigma_{5}\rho_{1}\cdots\rho_{11}}\int[dd^{1}]d^{1}_{\rho_{1}}\cdots
d^{1}_{\rho_{11}}(d^{1}\gamma^{mnp}d^{1})d^{1}_{\alpha}d^{1}_{\sigma}d^{1}_{\tau}=11!\
5!\ 96\
c_{d}(\lambda\gamma^{[m})_{\alpha}(\lambda\gamma^{n})_{\sigma}(\lambda\gamma^{p]})_{\tau}$
$\displaystyle\int[dw^{1}][d\bar{w}^{1}]\exp\left(-w^{I}_{\alpha}\bar{w}_{I}^{\alpha}\right)=c_{w}c_{\bar{w}}(11!)^{2}5!(2\pi)^{11}(\lambda\bar{\lambda})^{3}=\frac{(\lambda\bar{\lambda})^{3}}{(2\pi)^{11}z_{1}^{22}}$
(3.25)

Thus, after substituting $c_{d}$ and making use of the above results, the two
point function becomes

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-{\left(\frac{\alpha^{\prime}}{2}\right)^{2}\frac{(2\pi)^{3/2}Z_{1}^{5}\kappa^{2}}{2^{8}R}}\int
d^{2}\tau\ \int
d^{2}w\int[d\lambda][d\bar{\lambda}][dr][d\theta]e^{-\lambda\bar{\lambda}-r\theta}\;(\lambda\gamma^{m})_{\alpha}(\lambda\gamma^{n})_{\sigma}(\lambda\gamma^{p})_{\tau}$
(3.26)
$\displaystyle\times\frac{(\bar{\lambda}_{\rho}\gamma_{mnp}^{\rho\delta}D_{\delta})}{(\lambda\bar{\lambda})^{2}}\lambda^{\beta}C^{\alpha}_{\;\beta}(z)\;K^{\sigma\tau}(w)e^{ik_{1}\cdot
X(z)}e^{ik_{2}\cdot X(w)}\Bigl{\langle}\hat{B}V_{R}(z)U_{R}(w)\Bigl{\rangle}$

In principle, one can now perform the zero mode integrals over the remaining
variables. However, as explained in [18], these integrals are precisely the
integrals which one needs to perform at the tree level. Hence, after doing the
theta expansion of the superfields, one can make use of the pure spinor
superspace identities listed in [11, 13]. For this, one makes use of the
bracket (A.62) described in appendix A.2. Thus, using the definition (A.62)
and equations (2.8) and (2.13), we can write the two point function in the
form

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle{i\left(\frac{\alpha^{\prime}}{2}\right)\frac{(2\pi)^{3/2}Z_{1}^{5}\kappa^{2}}{2^{11}R}}\int
d^{2}\tau\int d^{2}w\ \Bigl{\langle}\hat{B}V_{R}(z)U_{R}(w)\Bigl{\rangle}$
$\displaystyle\;\Bigl{\langle}(\gamma^{abcd})^{\alpha}_{\beta}\gamma_{qrs}^{\sigma\tau}(k_{1})_{a}(\lambda\gamma^{m})_{\alpha}(\lambda\gamma^{n})_{\sigma}(\lambda\gamma^{p})_{\tau}\lambda^{\beta}(\bar{\lambda}_{\rho}\gamma_{mnp}^{\rho\delta}D_{\delta})B^{1}_{bcd}(z)\;B_{2}^{qrs}(w)e^{ik_{1}\cdot
X(z)}e^{ik_{2}\cdot X(w)}\Bigl{\rangle}_{(1,1)}$

We now simplify the pure spinor part of the correlator as follows

$\displaystyle\langle f\rangle_{(1,1)}$ $\displaystyle\equiv$
$\displaystyle\Bigl{\langle}(\gamma^{abcd})^{\alpha}_{\beta}\gamma_{qrs}^{\sigma\tau}(k_{1})_{a}(\lambda\gamma^{m})_{\alpha}(\lambda\gamma^{n})_{\sigma}(\lambda\gamma^{p})_{\tau}(\bar{\lambda}_{\rho}\gamma_{mnp}^{\rho\delta}D_{\delta})\lambda^{\beta}B^{1}_{bcd}\;B_{2}^{qrs}\Bigl{\rangle}_{(1,1)}$
(3.28) $\displaystyle=$
$\displaystyle\Bigl{\langle}\lambda^{\alpha_{1}}\lambda^{\alpha_{2}}\lambda^{\alpha_{3}}\lambda^{\alpha_{4}}\bar{\lambda}_{\beta_{1}}\Bigl{[}(k_{1})_{a}\gamma^{mabcd}_{\alpha_{1}\alpha_{4}}\gamma^{nqrsp}_{\alpha_{2}\alpha_{3}}\gamma_{mnp}^{\beta_{1}\delta}D_{\delta}(B^{1}_{bcd}\;B_{2}^{qrs})\Bigl{]}\Bigl{\rangle}_{(1,1)}$
$\displaystyle\equiv$
$\displaystyle\Bigl{\langle}\lambda^{\alpha_{1}}\lambda^{\alpha_{2}}\lambda^{\alpha_{3}}\lambda^{\alpha_{4}}\bar{\lambda}_{\beta_{1}}f_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{\beta_{1}}(\theta)\Bigl{\rangle}_{(1,1)}$
$\displaystyle=$
$\displaystyle\Bigl{\langle}672\lambda^{\sigma_{1}}\lambda^{\sigma_{2}}\lambda^{\sigma_{3}}\mathcal{T}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{\sigma_{1}\sigma_{2}\sigma_{3}\beta_{1}}f^{\beta_{1}}_{\alpha_{1}\cdots\alpha_{4}}(\theta)\Bigl{\rangle}_{(2,1)}$

In going from 3rd to 4th line, we made use of the identity (A.64) and the
tensor $\mathcal{T}$ is defined by

$\displaystyle\mathcal{T}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}}=\frac{1}{2772}\left[\delta^{(\alpha_{1}}_{\sigma_{1}}\cdots\delta^{\alpha_{4})}_{\sigma_{4}}-\frac{1}{4}\delta^{(\alpha_{1}}_{(\sigma_{1}}\delta^{\alpha_{2}}_{\sigma_{2}}\gamma_{z}^{\alpha_{3}\alpha_{4})}\gamma^{z}_{\sigma_{3}\sigma_{4})}+\frac{1}{160}\gamma_{z}^{(\alpha_{1}\alpha_{2}}\gamma^{z}_{(\sigma_{1}\sigma_{2}}\gamma_{x}^{\alpha_{3}\alpha_{4})}\gamma^{x}_{\sigma_{3}\sigma_{4})}\right]$
(3.29)

By making use of the pure spinor superspace identities and the equation of
motion (2.2), the quantity in (3.28) can be evaluated with the help of Cadabra
[30, 31] or Mathematica package GAMMA [32] to be

$\displaystyle\langle{f}\rangle_{(1,1)}$ $\displaystyle=$
$\displaystyle-{4608}\,\Bigl{\langle}{B^{2}}_{abc}({k_{1}})_{d}(\lambda{\Psi^{1}}_{e})(\lambda{\gamma}_{abcde}\lambda)\Bigl{\rangle}_{(2,1)}+{6912}\,\Bigl{\langle}{B^{1}}_{abc}{(k_{1})}_{d}(\lambda{\Psi^{2}}_{e})(\lambda{\gamma}_{abcde}\lambda)\Bigl{\rangle}_{(2,1)}$
(3.30) $\displaystyle=$
$\displaystyle{-\frac{18i}{\alpha^{\prime}}}N_{(2,1)}e_{mn}e^{mn}$

Note that the correlator $\langle{f}\rangle$ does not have any $\tau$
dependence coming from the OPEs since we did not need to take any OPEs in our
calculation. If we consider one loop higher point amplitudes or higher genus
amplitudes, then there would be more ways to saturate the $d_{\alpha}$ zero
modes and the modular dependence will arise from the OPEs.

Using the above result (3.30) and equation (A.63), the two point function
(LABEL:t2point) thus becomes

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle
162(A_{1})^{-5/2}\pi^{4}Z_{1}^{5}\alpha^{\prime 2}e_{mn}e^{mn}\int d^{2}\tau\
\int d^{2}w\
\Bigl{\langle}\hat{B}V_{R}(\bar{z})U_{R}(\bar{w})\Bigl{\rangle}\Bigl{\langle}e^{ik_{1}\cdot
X(z)}e^{ik_{2}\cdot X(w)}\Bigl{\rangle}$ (3.31)

The first correlator in the second line of (3.31) essentially factorizes in
terms of the correlator involving the ghost field of the non supersymmetric
sector, namely $\langle\hat{b}\hat{c}\rangle$, and a correlator involving the
SO(32) fields $\hat{\rho}^{A}$. These and the last correlator of (3.31) are
essentially same as the ones which also arise in the heterotic string in the
RNS formalism. Thus, we can use the results of [27, 33, 34]

$\displaystyle\langle\hat{b}\hat{c}\rangle$ $\displaystyle=$
$\displaystyle\bigl{[}\overline{\eta(\tau)}\bigl{]}^{2}$ (3.32)
$\displaystyle\langle\hat{J}(\bar{z})\hat{J}(\bar{w})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}\overline{\eta(\tau)}\bigl{)}^{-16}\left(\frac{\overline{\vartheta^{\prime}_{1}(0,\tau)}}{\overline{\vartheta_{1}(w-z)}}\right)^{4}\sum_{\nu}\overline{\vartheta_{\nu}\left((w-z)/2\right)}^{16}$
(3.33) $\displaystyle\Bigl{\langle}e^{ik_{1}\cdot X(z)}e^{ik_{2}\cdot
X(w)}\Bigl{\rangle}$ $\displaystyle=$
$\displaystyle(2\pi)^{10}\delta^{10}\left(k_{1}+k_{2}\right)(A_{1})^{5}\left(-2\pi^{2}\alpha^{\prime}\mbox{det}^{\prime}\partial\bar{\partial}\right)^{-5}\left|\frac{\vartheta_{1}(z-w,\tau)}{\vartheta^{\prime}_{1}(0,\tau)}\right|^{4}$
(3.34) $\displaystyle\exp\left[-4\pi\frac{(Im(z-w))^{2}}{\tau_{2}}\right]$

The correlators in (3.32) and (3.33) are given only upto the overall numerical
factor (see footnote 4). Using these results, we can now complete the
calculation of 2-point function. However, it is instructive to first count the
$\tau$ dependence hidden in the normalization factors. The $\tau$ dependence
comes from the factor of $Z_{g}$ and $A_{g}$ present in the pure spinor path
integral measures and the zero mode integral of $X^{m}$ fields. The $Z_{g}$
appears in the measure of conformal weight one fields and its total power at
one loop is $Z_{1}^{5}$ due to the factor $c_{d}c_{w}c_{\bar{w}}c_{s}$. This
contributes $\tau_{2}^{-5/2}$. The $A_{g}$ appears in the measure of conformal
weight zero fields with the factor $A_{1}^{-5/2}$ and due to zero mode
integral of $X^{m}$ fields with the factor $A_{1}^{5}$. These contribute
$(\tau_{2})^{5/2}$. Thus, the $\tau$ dependence arising from the measures and
zero mode integrals mutually cancel.

Thus, using (A.57), (A.58), (3.16), (3.34), the theta function identity
$\vartheta^{\prime}_{1}(0,\tau)=-2\pi\eta^{3}(\tau)$ and noting that the
determinant of the operator $\partial$ can be expressed in terms of eta
function as $\mbox{det}^{\prime}(\partial)=\tau_{2}\overline{\eta(\tau)}^{2}$,
we finally obtain

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{N}{\alpha^{\prime}}e_{mn}e^{mn}\int\frac{d^{2}\tau}{\tau_{2}^{5}}\int
d^{2}w\
\left(\overline{\eta(\tau)}\right)^{-18}\eta^{-6}(\tau)\left(\frac{{\vartheta_{1}(w-z,\tau)}}{\overline{\vartheta_{1}(w-z)}}\right)^{2}\sum_{\nu}\left\\{\overline{\vartheta_{\nu}\left((w-z)/2\right)}^{16}\right\\}$
(3.35) $\displaystyle\hskip
28.90755pt\exp\left[-4\pi\frac{(Im(z-w))^{2}}{\tau_{2}}\right]$

The $N$ is the overall numerical factor. This is undetermined since the
normalization of $\langle bc\rangle$ correlator in the non supersymmetric
world-sheet sector is not completely fixed.444As discussed in [35], the
normalization of the ghost correlators in RNS is still an unsolved problem and
there is no concensus even on the normalization of the simple correlators such
as $\langle bc\rangle$ [36, 37, 38, 39]. Apart from $N$, the expression
matches with the corresponding expression (4.17) in [27]. The [27] fixed the
overall numerical factor by comparing the above result with the expected
result in the low energy effective theory. Since the functional dependence
coming from the integral is same in two cases, the numerical factor is
guaranteed to be identical in RNS and pure spinor.

It would be interesting to compare the numerical factors in the RNS and pure
spinor. In the supersymmetric side treated using the pure spinor, we have kept
track of the numerical factors. However, in the calculation using the RNS
formalism in [27], this is not completely fixed due to the ghost correlators.
Equating the RNS and pure spinor results, the correlators coming from the
bosonic world-sheet side cancel each other and we get a prediction for the
numerical constant in the product of two ghost correlators
$\langle\hat{b}\hat{c}\rangle$ and $\langle\hat{\beta}\hat{\gamma}\rangle$ in
the RNS formalism.

The one loop renormalized mass of the heterotic massive states can be
expressed in terms of the two point function computed above. The shift in the
mass due to the one loop effect is given by [27]

$\displaystyle\delta M=MK_{w}g^{2}$ (3.36)

where, $M$ is the tree level mass, $g^{2}$ is the string coupling constant and
$K_{w}$ is the integral expression in (3.35) including the numerical constant
for $N=-1/(64\pi)$ [27].

##  4 Discussions

In this paper, we have computed the one loop 2-point function involving the
first massive string states in the SO(32) Heterotic string theory using the
pure spinor formalism. This allows us to compute the mass renormalization of
these states in the pure spinor formalism. Our results are consistent with the
results obtained in [27]. This shows that loop amplitudes involving the
massive states in the pure spinor formalism also agree with the corresponding
RNS results.

An important feature of the pure spinor calculation is that we didn’t need to
sum over the spin structure unlike RNS calculation. In [27], same calculation
using the RNS formalism required the use of Riemann identity to simplify the
expressions. In the pure spinor calculation, we directly got the simplified
expression without making use of the Riemann identity. This demonstrates a
generic feature of the pure spinor formalism that it naturally avoids some of
the mathematical complications present in RNS formalism. However, we still
encounter some complications of RNS formalism in heterotic theory. We needed
to use the expression of determinant of the world-sheet operator $\partial$.
In the pure spinor calculations in type II theories, the various determinant
factors coming from left and right moving sectors cancel each other avoiding
the necessity to compute the determinants [29]. This is a simplifying feature
over RNS formalism. However, for the heterotic theories, this feature is
absent since the left moving sector is not described by pure spinors.

Since the pure spinor formalism provides a more efficient way to compute the
higher genus amplitudes, it will be promising for the calculation of higher
genus mass corrections to first massive non BPS heterotic states. The 2-loop
correlator can be efficiently computed using the pure spinor formalism. The
divergences will appear when we try to perform the integration over the moduli
space. This happens due to Riemann surfaces developing long handles in some
corners of moduli space. One needs to separate these contributions for
calculating the renormalized mass. As mentioned in the introduction, the
proper way to compute the higher genus mass corrections is to go off-shell. We
don’t have the pure spinor version of the string field theory at the moment.
Part of this is related to the issue of pure spinor $b$ ghost and its
divergences associated with the poles in $\lambda\bar{\lambda}$. However, the
issue of $\lambda\bar{\lambda}$ pole is not problematic at the level of two
loop we are interested in. Further, even though we don’t have a working string
field theory with pure spinors 555See [29] for an attempt in this direction. ,
we can try to go off-shell in the world-sheet approach as described in [1, 2].
This will require, e.g., making the vertex operators dependent on the local
coordinates. In the pure spinor formalism, this might also involve relaxing
the pure spinor constraint. We hope to report on it in future [40].

Acknowledgments: We thank Subhroneel Chakrabarti for collaboration during
initial stages of this work. We are also thankful to Carlos Mafra and Luis
Alberto Ypanaque Rocha for discussion and Subhroneel Chakrabarti, Carlos
Mafra, Rafaelle Marotta and Ashoke Sen for comments on an earlier version of
this draft. SK is thankful to IOP, Bhubaneshwar and MV is thankful to INFN,
Napoli where part of this work were done. The work of MV is supported in part
by the STFC consolidated grant ST/T000775/1 “New Frontiers in Particle
Physics, Cosmology and Gravity”.

##  Appendix A $SO(32)$ Heterotic String in Pure Spinor Formalism

In this appendix, we briefly review the results regarding the Heterotic
strings in the pure spinor formalism and its first massive states. For more
details on this section, see e.g., [9, 14, 16, 41]. We shall follow the
conventions of [16] about the space-time dimensions and the path integral
measures of the various world-sheet fields.

###  A.1 World-sheet Theory

In the heterotic closed string theory, the world-sheet supersymmetry is kept
only in the left moving sector. The right moving sector has no supersymmetry.
Since the left moving sector has supersymmetry, it can be described by the
pure spinor formalism. On the other hand, the right moving sector is described
by an internal CFT with central charge 16 as in RNS formalism.

We shall be making use of the non-minimal version of the pure spinor formalism
for the loop computations [29]. The world-sheet action describing the
heterotic string theory in the non-minimal pure spinor formalism is given (in
the conventions followed in [16] )

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\alpha^{\prime}}\int d^{2}z\ \Bigl{(}\partial
X^{m}\bar{\partial}X_{m}+\alpha^{\prime}p_{\alpha}\bar{\partial}\theta^{\alpha}-\alpha^{\prime}w_{\alpha}\bar{\partial}\lambda^{\alpha}-\alpha^{\prime}\bar{w}^{\alpha}\partial\bar{\lambda}_{\alpha}+\alpha^{\prime}s^{\alpha}\bar{\partial}r_{\alpha}\Bigl{)}$
(A.37) $\displaystyle\hskip 36.135pt+\;\;\frac{1}{4\pi}\int
d^{2}z\Bigl{(}\hat{\rho}^{A}\partial\hat{\rho}^{A}+2\hat{b}\partial\hat{c}\Bigl{)}\
$

In the supersymmetric sector, the fields
$p_{\alpha},\theta^{\alpha},r_{\alpha}$ and $s^{\alpha}$ are fermionic in
nature while the rest of the fields are bosonic in nature. The fields
$w_{\alpha},\bar{w}^{\alpha},p_{\alpha}$ and $s^{\alpha}$ have conformal
weight one while $\lambda^{\alpha},\bar{\lambda}_{\alpha},\theta^{\alpha}$ and
$r_{\alpha}$ have conformal weight zero. For the purpose of writing down the
path integral measures, it is instructive to note the target space length
dimensions of the pure spinor world-sheet fields. The length dimensions of
various quantities appearing in the action are

$\displaystyle[\theta^{\alpha}]=[\lambda^{\alpha}]=[\bar{w}^{\alpha}]=[s^{\alpha}]={\frac{1}{2}}\;,\;[p_{\alpha}]=[w_{\alpha}]=[{\bar{\lambda}}_{\alpha}]=[r_{\alpha}]={-\frac{1}{2}}\;,\;[\alpha^{\prime}]=2,\;\;[X^{m}]=1$
(A.38)

On the Riemann surface, an arbitrary conformal weight +1 field $\phi$ can be
expanded as a linear combination of the eigenfunctions of the operator
$\bar{\partial}$, i.e.,

$\displaystyle\phi=\sum_{i}\phi^{i}f_{i}(z,\bar{z})\qquad,\qquad\bar{\partial}f_{i}(z.\bar{z})=\gamma_{i}f_{i}(z,\bar{z})$
(A.39)

Now, it is a result from the theory of Riemann surfaces that the object of
conformal weight +1 have $g$ zero modes on a genus $g$ Riemann surface
satisfying $\bar{\partial}f_{i}=0$. Separating these zero modes, we can thus
write

$\displaystyle\phi(z)=\hat{\phi}(z)+\sum_{I=1}^{g}\phi^{I}\omega_{I}(z)$
(A.40)

where, $\omega_{I}(z)$ are the $g$ holomorphic one-forms satisfying
$\bar{\partial}\omega_{I}(z,\bar{z})=0$. The $\hat{\phi}$ have no zero modes.
These modes satisfy

$\displaystyle\int_{a_{I}}\omega_{J}=\delta_{IJ}\qquad,\qquad\int_{a_{I}}dz\
\hat{\phi}_{I}(z)=0$ (A.41)

The objects of conformal weight 0 have one zero mode on every genus $g$
Riemann surface. Hence, the number of zero modes of the various pure spinor
world-sheet fields on a genus $g$ Riemann surface can be tabulated as

$X^{m}$ | $\lambda^{\alpha}$ | $w_{\alpha}$ | $\bar{\lambda}_{\alpha}$ | $\bar{w}^{\alpha}$ | $\theta^{\alpha}$ | $p_{\alpha}$ | $r_{\alpha}$ | $s^{\alpha}$
---|---|---|---|---|---|---|---|---
10 | 11 | 11g | 11 | 11g | 16 | 16g | 11 | 11g

Since, we do not have fundamental world-sheet fields with higher conformal
weights, this is all we need. At one loop, $g=1$, so that all the fields have
one zero mode.

As in open strings, it is convenient to define the SUSY invariant combinations

$\displaystyle d_{\alpha}$ $\displaystyle=$ $\displaystyle
p_{\alpha}-\frac{1}{\alpha^{\prime}}\gamma^{m}_{\;\;\alpha\beta}\theta^{\beta}\partial
X_{m}-\frac{1}{4\alpha^{\prime}}\gamma^{m}_{\alpha\beta}\gamma_{m\sigma\delta}\theta^{\beta}\theta^{\sigma}\partial\theta^{\delta}$
$\displaystyle\Pi^{m}$ $\displaystyle=$ $\displaystyle\partial
X^{m}+\frac{1}{2}\gamma^{m}_{\alpha\beta}\theta^{\alpha}\partial\theta^{\beta}$
(A.42)

The various OPEs are given by

$\displaystyle
d_{\alpha}(z)d_{\beta}(w)=-\frac{2}{\alpha^{\prime}}\frac{\gamma^{m}_{\alpha\beta}}{(z-w)}\Pi_{m}(w)+\cdots\quad,\qquad
d_{\alpha}(z)\Pi^{m}(w)=\frac{\gamma^{m}_{\alpha\beta}}{(z-w)}\partial\theta^{\beta}(w)+\cdots$
$\displaystyle
d_{\alpha}(z)V(w)=\frac{D_{\alpha}V(w)}{(z-w)}+\cdots\quad,\qquad\Pi^{m}(z)V(w)=-{\frac{\alpha^{\prime}}{2}}\frac{\partial^{m}V(w)}{(z-w)}+\cdots$
$\displaystyle\Pi^{m}(z)\Pi^{n}(w)=-\frac{\alpha^{\prime}\eta^{mn}}{2(z-w)^{2}}+\cdots\quad,\qquad
N^{mn}(z)\lambda^{\alpha}(w)=\frac{(\gamma^{mn})^{\alpha}_{\;\;\beta}}{2(z-w)}\
\lambda^{\beta}(w)+\cdots$ $\displaystyle
J(z)J(w)=-\frac{4}{(z-w)^{2}}+\cdots\quad,\qquad
J(z)\lambda^{\alpha}(w)=\frac{1}{(z-w)}\lambda^{\alpha}(w)+\cdots$
$\displaystyle
N^{mn}(z)N^{pq}(w)=-\frac{6}{(z-w)^{2}}\eta^{m[q}\eta^{p]n}-\frac{2}{(z-w)}\Bigl{(}\eta^{p[n}N^{m]q}-\eta^{q[n}N^{m]p}\Bigl{)}+\cdots$
(A.43)

In the above OPEs, $\partial_{m}$ is the derivative with respect to the
spacetime coordinate $X^{m}$ and $\partial$ is the derivative with respect to
the world-sheet coordinate. The $V(w,\bar{w})$ denotes an arbitrary superfield
which depends upon the world-sheet fields $X^{m}$ through the plane wave
factor $e^{ik\cdot X}$. The $D_{\alpha}$ is the supercovariant derivative
whose definition is taken to be different from that for the open string case

$\displaystyle
D_{\alpha}\equiv\partial_{\alpha}+{\frac{1}{2}}\gamma^{m}_{\alpha\beta}\theta^{\beta}\partial_{m}$
(A.44)

This supercovariant derivative satisfies the identity

$\displaystyle\\{D_{\alpha},D_{\beta}\\}=(\gamma^{m})_{\alpha\beta}\partial_{m}\quad\implies\qquad{\frac{1}{8}}(\gamma_{m})^{\alpha\beta}D_{\alpha}D_{\beta}=\partial_{m}$
(A.45)

The above OPEs and the results in (A.44) and (A.45) can be derived from the
corresponding open string results by making the rescaling
$k^{m}\rightarrow\frac{1}{2}k^{m}$ or equivalently
$\partial_{m}\rightarrow\frac{1}{2}\partial_{m}$ (see [23] for the
corresponding results for open strings in our conventions). In fact, this
trick is far more general than just obtaining the OPEs. E.g., for a
holomorphic function, using chain rule, we have

$\displaystyle\partial f(z)=\partial X^{m}\partial_{m}f\ +\
\partial\theta^{\alpha}\partial_{\alpha}f\ =\ \Pi^{m}\partial_{m}f\ +\
\partial\theta^{\alpha}D_{\alpha}f$ (A.46)

where, in going to the second equality, we have made use of equations (A.42)
and (A.44). The corresponding open string result which was derived in [23]
includes a factor of $2$ in front of the term containing
$\Pi^{m}\partial_{m}$. This shows that the above result (A.46) can be obtained
from the corresponding open string result by rescaling
$\partial_{m}\rightarrow\frac{1}{2}\partial_{m}$.

For the amplitude computations, one introduces a composite $b$ ghost in the
supersymmetric pure spinor sector which is given by [9]

$\displaystyle b$ $\displaystyle=$ $\displaystyle
s^{\alpha}\partial\bar{\lambda}_{\alpha}+\frac{2\Pi^{m}(\bar{\lambda}\gamma_{m}d)-N_{mn}(\bar{\lambda}\gamma^{mn}\partial\theta)-J_{\lambda}(\bar{\lambda}\partial\theta)-(\bar{\lambda}\partial^{2}\theta)}{4(\bar{\lambda}\lambda)}$
$\displaystyle+\frac{(\bar{\lambda}\gamma^{mnp}r)(\left(\frac{\alpha^{\prime}}{2}\right)d\gamma_{mnp}d+24N_{mn}\Pi_{p})}{192(\bar{\lambda}\lambda)^{2}}-\left(\frac{\alpha^{\prime}}{2}\right)\frac{(r\gamma_{mnp}r)(\bar{\lambda}\gamma_{m}d)N_{np}}{16(\bar{\lambda}\lambda)^{3}}$
$\displaystyle+\left(\frac{\alpha^{\prime}}{2}\right)\frac{(r\gamma_{mnp}r)(\bar{\lambda}\gamma^{pqr}r)N^{mn}N_{qr}}{128(\bar{\lambda}\lambda)^{4}}$

Finally, the BRST operators in the left and right moving sectors are given by

$\displaystyle Q$ $\displaystyle=$ $\displaystyle\oint\frac{dz}{2\pi i}\
\Bigl{(}\lambda^{\alpha}d_{\alpha}+\bar{w}^{\alpha}r_{\alpha}\Bigl{)}$
$\displaystyle\bar{Q}$ $\displaystyle=$
$\displaystyle\oint\frac{d\bar{z}}{2\pi i}\
\biggl{[}\hat{c}(\bar{z})\hat{T}(\bar{z})+\hat{b}(\bar{z})\hat{c}(\bar{z})\bar{\partial}\hat{c}(\bar{z})\biggl{]}$
(A.47)

where, the right moving component of the matter stress energy tensor is given
by

$\displaystyle\hat{T}=-\frac{1}{\alpha^{\prime}}\bar{\partial}X^{m}\bar{\partial}X_{m}-\frac{1}{2}\hat{\rho}^{A}\bar{\partial}\hat{\rho}^{A}$
(A.48)

The total BRST operator is $Q+\bar{Q}$.

###  A.2 Loop Amplitude Prescription

In this appendix, we briefly review the loop amplitude prescription in the
pure spinor heterotic theory following [9, 16, 18, 29]. The multiloop
$n$-point closed string amplitudes are given by [29]

$\displaystyle\mathcal{A}^{(0)}_{n}$ $\displaystyle=$
$\displaystyle\int\prod_{j=4}^{n}d^{2}z_{i}\langle\langle\mathcal{N}^{(0)}\;V^{1}(0)V^{2}(1)V^{3}(\infty)U^{i}(z_{i},\bar{z}_{i})\rangle\rangle$
(A.49) $\displaystyle\mathcal{A}^{(1)}_{n}$ $\displaystyle=$ $\displaystyle
S_{1}\int_{\mathcal{M}_{1}}\;d^{2}\tau\int\prod_{j=i}^{n}d^{2}z_{i}\langle\langle\mathcal{N}^{(1)}\;B\hat{B}V^{1}(0)U^{2}(z_{i},\bar{z}_{i})\rangle\rangle$
(A.50) $\displaystyle\mathcal{A}^{(g)}_{n}$ $\displaystyle=$ $\displaystyle
S_{g}e^{(2g-2)\lambda}\;\int_{\mathcal{M}_{g}}\;\prod_{i=1}^{3g-3}\;d^{2}\tau_{i}\int\prod_{j=1}^{n}d^{2}z_{j}\langle\langle\mathcal{N}^{(g)}\;B_{j}\hat{B}_{j}\;U^{j}(z_{j},\bar{z}_{i}))\rangle\rangle\quad,\;\;g\geq
2$

In the above expressions, $S_{g}$ are the symmetry factors for a genus $g$
surface. To be precise $S_{1}=1/2$ [42, 43], $S_{2}=1/2$[44] and $S_{g}=1$ for
$g>2$ . Also, $\langle\langle\cdots\rangle\rangle$ stands for the correlator
in which we have gotten rid of the non-zero modes by the OPE computation. $B$
and $\hat{B}$ involve the beltrami differential

$\displaystyle B=\frac{1}{2\pi}\int
d^{2}w\mu(w)b(w)\qquad,\qquad\hat{B}=\frac{1}{2\pi}\int
d^{2}w\hat{\mu}(\bar{w})\hat{b}(\bar{w})$ (A.51)

where $b(w)$ is the composite pure spinor “$b$-ghost”defined in (A.1) and
$\hat{b}(\bar{w})$ is the elemenatary “$b$-ghost”given in (A.37). The $\mu$
and $\hat{\mu}$ are the beltrami differentials given on the torus by
$\mu=\hat{\mu}=1/2\tau_{2}$. $\mathcal{M}_{1}$ denotes the fundamental domain
of the moduli space of torus.

On the torus, there is only one globally defined analytic vector field (hence
one conformal killing vector) which allows us to fix the position of one of
the vertex operator in (A.50). The regulator $\mathcal{N}$ is needed to
regularize the zero mode integrals of the pure spinor world-sheet fields. To
see what kind of terms are allowed for this purpose, one uses the fact that
the expression inside the correlator apart from $\mathcal{N}$ is BRST
invariant. We don’t want to break the BRST invariance. Hence, the allowed form
of the regulator should take the form [9]

$\displaystyle\mathcal{N}(y)=e^{\\{Q,\chi(y)\\}}=1+\cdots$ (A.52)

where, $\\{Q,\chi(y)\\}=\oint
dz(\lambda^{\alpha}d_{\alpha}+\bar{w}^{\alpha}r_{\alpha})\chi(y)$ and $\cdots$
terms are BRST invariant. A convenient choice for $\chi$ which works upto two
loop is given by [15]

$\displaystyle\chi=-(\bar{\lambda}\theta)-(w^{I}s^{I})\qquad,\qquad
I=1,\cdots,g$ (A.53)

which gives

$\displaystyle\mathcal{N}(y)=\exp\left[-(\lambda\bar{\lambda})-(r\theta)-(w^{I}\bar{w}^{I})+(s^{I}d^{I})\right]$
(A.54)

The integration over the non zero modes of pure spinor world-sheet fields in
the correlators is done using the OPEs given in the previous subsection. On
the other hand, the integration over the zero modes is performed using the
following measure

$\displaystyle\int[d\lambda][d\bar{\lambda}][dw^{I}][d\bar{w}^{I}][dd^{I}][dr][ds][d\theta]$
(A.55)

where [16],

$\displaystyle[d\lambda]T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}$
$\displaystyle=$ $\displaystyle
c_{\lambda}\epsilon_{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}d\lambda^{\rho_{1}}\cdots
d\lambda^{\rho_{11}}$
$\displaystyle[d\bar{\lambda}]\bar{T}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}$
$\displaystyle=$ $\displaystyle
c_{\bar{\lambda}}\epsilon^{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}d\bar{\lambda}_{\rho_{1}}\cdots
d\bar{\lambda}_{\rho_{11}}$
$\displaystyle[d\bar{\omega}]T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}$
$\displaystyle=$ $\displaystyle
c_{\bar{\omega}}(\lambda\bar{\lambda})^{3}\epsilon_{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}d\bar{\omega}^{\rho_{1}}\cdots
d\bar{\omega}^{\rho_{11}}$ $\displaystyle[d\omega]$ $\displaystyle=$
$\displaystyle
c_{\omega}T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}\epsilon^{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}d\omega_{\rho_{1}}\cdots
d\omega_{\rho_{11}}$ $\displaystyle[dr]$ $\displaystyle=$ $\displaystyle
c_{r}\bar{T}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}\epsilon_{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}\partial_{r}^{\rho_{1}}\cdots\partial_{r}^{\rho_{11}}$
$\displaystyle[ds]$ $\displaystyle=$
$\displaystyle\frac{c_{s}}{(\lambda\bar{\lambda})^{3}}T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}\epsilon^{\alpha_{1}\cdots\alpha_{5}\rho_{1}\cdots\rho_{11}}\partial^{s}_{\rho_{1}}\cdots\partial^{s}_{\rho_{11}}$
$\displaystyle[d\theta]$ $\displaystyle=$ $\displaystyle
c_{\theta}d^{16}\theta\qquad,\quad[dd^{I}]=c_{d}d^{16}d^{I}$ (A.56)

where, the various coefficients are given by

$\displaystyle c_{\lambda}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha^{\prime}}{2}\right)^{-2}\frac{1}{11!}\left(\frac{A_{g}}{4\pi^{2}}\right)^{11/2}\qquad,\quad
c_{\bar{\lambda}}=\left(\frac{\alpha^{\prime}}{2}\right)^{2}\frac{2^{6}}{11!}\left(\frac{A_{g}}{4\pi^{2}}\right)^{11/2}$
$\displaystyle c_{w}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha^{\prime}}{2}\right)^{2}\frac{(4\pi^{2})^{-11/2}}{11!5!}\frac{1}{Z_{g}^{11/g}}\qquad,\quad
c_{\bar{w}}=\left(\frac{\alpha^{\prime}}{2}\right)^{-2}\frac{(4\pi^{2})^{-11/2}}{11!}\frac{(\lambda\bar{\lambda})^{3}}{Z_{g}^{11/g}}$
$\displaystyle c_{r}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha^{\prime}}{2}\right)^{-2}\frac{R}{11!5!}\left(\frac{2\pi}{A_{g}}\right)^{11/2}\qquad,\quad
c_{s}=\left(\frac{\alpha^{\prime}}{2}\right)^{2}\frac{(2\pi)^{11/2}}{2^{6}11!5!}\frac{Z_{g}^{11/g}}{R(\lambda\bar{\lambda})^{3}}$
$\displaystyle c_{\theta}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha^{\prime}}{2}\right)^{4}\left(\frac{2\pi}{A_{g}}\right)^{16/2}\qquad,\quad
c_{d}=\left(\frac{\alpha^{\prime}}{2}\right)^{-4}(2\pi)^{16/2}Z_{g}^{16/g}$
(A.57)

The parameter $R$ in the above coefficients is fixed, by demanding the tree
amplitudes to be same in the RNS and pure spinor calculations, to be
$R^{2}=\frac{\sqrt{2}}{2^{16}\pi}$ [16] and the quantities $A_{g}$ and $Z_{g}$
are given by (denoting the period matrix by $\Omega_{IJ}$)

$\displaystyle A_{g}=\int d^{2}z\sqrt{g}\qquad,\qquad
Z_{g}=\frac{1}{\sqrt{\mbox{det}(2\mbox{Im}(\Omega_{IJ}))}}$ (A.58)

The advantage of including these factors in the measure is that the norm of
the zero mode basis becomes independent of the modular parameters [16] (See
[14] for calculation in which measure has been kept independent of the modular
parameters)

The tensors $T$ and $\bar{T}$ appearing in the measures are defined as

$\displaystyle
T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}=(\lambda\gamma^{m})_{\alpha_{1}}(\lambda\gamma^{n})_{\alpha_{2}}(\lambda\gamma^{p})_{\alpha_{3}}(\gamma_{mnp})_{\alpha_{4}\alpha_{5}}$
(A.59)
$\displaystyle\bar{T}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}=(\bar{\lambda}\gamma^{m})^{\alpha_{1}}(\bar{\lambda}\gamma^{n})^{\alpha_{2}}(\bar{\lambda}\gamma^{p})^{\alpha_{3}}(\gamma_{mnp})^{\alpha_{4}\alpha_{5}}$
(A.60)

and they satisfy

$\displaystyle
T_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}\bar{T}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}\alpha_{5}}=5!2^{6}(\lambda\bar{\lambda})^{3}$
(A.61)

After integrating over the non zero modes, one is left with the integrations
over the zero modes of
$\lambda^{\alpha},\bar{\lambda}_{\beta},\theta^{\gamma}$ and $r_{\delta}$. To
make use of the pure spinor superspace identities which appear at tree level,
one defines an arbitrary function of pure spinor variables
$(\lambda,\bar{\lambda},r,\theta)$ as [16]

$\displaystyle\Bigl{\langle}M(\lambda,\bar{\lambda},\theta,r)\Bigl{\rangle}_{(n,g)}\equiv\int[d\lambda][d\bar{\lambda}][dr][d\theta]\
\frac{e^{-\lambda\bar{\lambda}-r\theta}}{(\lambda\bar{\lambda})^{3-n}}\
M(\lambda,\bar{\lambda},\theta,r)$ (A.62)

The normalization for this bracket is fixed by setting
$M=(\lambda^{3}\theta^{5})$ which gives

$\displaystyle\bigl{\langle}(\lambda^{3}\theta^{5})\bigl{\rangle}_{(n,g)}$
$\displaystyle=$
$\displaystyle{N}_{n,g}\bigl{\langle}(\lambda^{3}\theta^{5})\bigl{\rangle}\qquad,\qquad
N_{(n,g)}\equiv
2^{7}R\left(\frac{2\pi}{A_{g}}\right)^{5/2}\left(\frac{\alpha^{\prime}}{2}\right)^{2}\frac{(7+n)!}{7!}$
(A.63)

where $A_{g}$ is the area of the Riemann surface and $R$ is the arbitrary
constant which is fixed by the normalization of tree level amplitudes as
mentioned above and the bracket $\langle\cdots\rangle$ can be evaluated by
using the same pure spinor superspace identities which are used at tree level
[11, 13].

The introduction of the bracket $\langle\cdots\rangle_{(n,g)}$ is quite useful
because it satisfies the following identity [17]

$\displaystyle\bigl{\langle}f(\lambda^{n+3},\bar{\lambda}^{n},\theta)\bigl{\rangle}_{(n,g)}=\bigl{\langle}(\lambda\bar{\lambda})^{n}\hat{f}(\lambda^{3},\theta^{5})\bigl{\rangle}_{(n,g)}$
(A.64)

where, the functions $f$ can be expressed as

$\displaystyle
f(\lambda^{n+3},\bar{\lambda}^{n},\theta)=\lambda^{\alpha_{1}}\cdots\lambda^{\alpha_{n+3}}\bar{\lambda}_{\beta_{1}}\cdots\bar{\lambda}_{\beta_{n}}f^{\beta_{1}\cdots\beta_{n}}_{\alpha_{1}\cdots\alpha_{n+3}}(\theta)$
(A.65)

and $\hat{f}$ is given by

$\displaystyle\hat{f}(\lambda^{3},\theta^{5})=672\lambda^{\beta_{1}}\lambda^{\beta_{2}}\lambda^{\beta_{3}}\mathcal{T}^{\sigma_{1}\cdots\sigma_{n+3}}_{\beta_{1}\cdots\beta_{n+3}}f^{\beta_{4}\cdots\beta_{n+3}}_{\sigma_{1}\cdots\sigma_{n+3};\delta_{1}\cdots\delta_{5}}\theta^{\delta_{1}}\cdots\theta^{\delta_{5}}$
(A.66)

where $\mathcal{T}$ is a symmetric and gamma traceless $SO(10)$ invariant
tensor whose explicit form can be found in [17].

###  A.3 First Massive States

The heterotic closed string spectrum can be analyzed by considering the tensor
product of the left and right moving states. For a detailed analysis of the
first massive states, see e.g., [41]. At the first mass level, the
supersymmetric sector has the same number of states as in the case of open
strings, namely 128 bosonic and 128 fermionic states. The bosonic states
represent a symmetric traceless second rank tensor with $44$ degrees of
freedom and an anti-symmetric 3-form field with $84$ degrees of freedom. The
128 fermionic states represent a tensor spinor field. The Lorentz group
transformation properties of these states can be represented as

$\displaystyle(\textbf{44},\textbf{1})\oplus(\textbf{84},\textbf{1})\oplus(\textbf{128},\textbf{1})$
(A.67)

The first label corresponds to SO(9) and the second label corresponds to
SO(32) group666More precisely, these are spin(9) and spin$(32)/Z_{2}$ [41].

The physical states are obtained by tensoring these with the states from the
non supersymmetric sector. For the SO(32) heterotic string, the number of
states in the non supersymmetric sector is 73764 which transform as

$\displaystyle(\textbf{44},\textbf{1})\oplus(\textbf{9},\textbf{496})\oplus(\textbf{1},\textbf{69256})$
(A.68)

Again, the first and the second labels correspond to the transformation under
the $SO(9)$ and $SO(32)$ groups respectively. These 73764 left moving states
transform as scalars, spinors, 2nd rank anti-symmetric tensors, 4th rank anti-
symmetric tensors and 2nd rank symmetric traceless tensors of SO(32).

Thus, the total physical degrees of freedom at the first massive level of
SO(32) heterotic string is $256\times 73764=18883584$. These states form the
$N=1$ supermultiplet in 10 dimensions. A basic fact about these states is that
they do not satisfy the BPS condition. However, they are still stable and
undergo mass renormalization.

As mentioned earlier, the supersymmetric sector of the world-sheet theory can
be described by the pure spinor formalism. This means that the $256$ states in
the supersymmetric sector can be described in terms of superfields in a
manifestly supersymmetric invariant manner. Since these states are
mathematically same as the ones which arise in the open string theory in 10
dimensions, we can use the results of [21, 22] to describe these 256 states in
terms of the superfields $B_{mnp},G_{mn}$ and $\Psi_{m\alpha}$.

##  Appendix B  Some details about closed string vertex operators

In this appendix, we argue that the vertex operator for the closed strings can
be obtained using the result for the open strings (see also [45]). We consider
the integrated and unintegrated vertex operators separately.

###  B.1 Unintegrated vertex operator

The closed string unintegrated vertex operators satisfy

$\displaystyle QV(z,\bar{z})=0=\bar{Q}V(z,\bar{z})\qquad,\qquad\delta
V=Q\Omega+\bar{Q}\bar{\Omega}$ (B.69)

where, $\bar{Q}\Omega=0=Q\bar{\Omega}$.

The above set of equations can be solved by assuming an ansatz of the form
$V(z,\bar{z})=V_{R}(z)V_{L}(\bar{z})e^{ik\cdot X}$. With this ansatz, it is
easy to see that the BRST equations can be satisfied provided

$\displaystyle Q\left[V_{R}(z)e^{ik\cdot
X}\right]=0\qquad,\quad\bar{Q}\left[V_{L}(\bar{z})e^{ik\cdot X}\right]=0$
(B.70)

Both the above equations are essentially the same equations which arise also
in the open string case. We only need to take care of the difference in the
OPEs of the open and closed strings while setting up these equations.
Focussing on the right sector for simplicity, we recall that the main
difference between the closed and open string expressions are in the
definition of the cvariant derivative $D_{\alpha}$ and the OPE involving
$\Pi^{m}$, namely

$\displaystyle
D^{O}_{\alpha}=\partial_{\alpha}+(\gamma_{m}\theta)_{\alpha}\partial_{m}\quad$
$\displaystyle\rightarrow$ $\displaystyle\quad
D^{C}_{\alpha}=\partial_{\alpha}+\frac{1}{2}(\gamma_{m}\theta)_{\alpha}\partial_{m}$
$\displaystyle\Pi^{O}_{m}(z)V^{O}(w)=-\alpha^{\prime}\frac{\partial_{m}V^{O}(w)}{z-w}+\cdots\quad$
$\displaystyle\rightarrow$
$\displaystyle\quad\Pi^{C}_{m}(z)V^{C}(w)=-{\frac{\alpha^{\prime}}{2}}\frac{\partial_{m}V^{C}(w)}{(z-w)}+\cdots$

where the superfield $V$ include the factor $e^{ik\cdot X}$.

We further note that if we use the expression of $\partial_{m}$ in terms of
$D_{\alpha}$, namely
$\partial_{m}=\frac{1}{16}\gamma_{m}^{\alpha\beta}D^{O}_{\alpha}D^{O}_{\beta}$
for the open string and
$\partial_{m}=\frac{1}{8}\gamma_{m}^{\alpha\beta}D^{C}_{\alpha}D^{C}_{\beta}$
for the closed strings, then the OPE expressions look exactly identical,

$\displaystyle\Pi^{O}_{m}(z)V^{O}(w)=-\frac{\alpha^{\prime}}{16}\gamma_{m}^{\alpha\beta}\frac{D^{O}_{\alpha}D^{O}_{\beta}V^{O}(w)}{z-w}+\cdots\quad\rightarrow\quad\Pi^{C}_{m}(z)V^{C}(w)=-{\frac{\alpha^{\prime}}{16}}\gamma_{m}^{\alpha\beta}\frac{D^{C}_{\alpha}D^{C}_{\beta}V^{C}(w)}{(z-w)}+\cdots$

Since the BRST equation of motion only depend upon the OPEs, the above
analysis shows that both the open and two sectors of the closed string
equations take an identical covariant form when expressed using the covariant
derivatives.

Thus, an arbitrary BRST equation can be schematically written in the form

$\displaystyle\sum_{n}f_{n}(D^{O}_{\alpha})\left[\hat{V}^{O}_{n}e^{ik\cdot
X}\right]=0\quad\rightarrow\quad\sum_{n}f_{n}(D^{C}_{\alpha})\left[\hat{V}^{C}_{n}e^{ik\cdot
X}\right]=0$ (B.71)

In the above equation, $\hat{V}$ do not depend upon $X$ but only on the
derivative of $X$. If we work in momentum space, the effect of $e^{ik\cdot X}$
is in replacing $\partial_{m}$ in $D_{\alpha}$ by $k_{m}$. Hence, we can
factor out $e^{ik\cdot X}$ in momentum space and can write

$\displaystyle\sum_{n}f_{n}(D^{O}_{\alpha})\left[\hat{V}^{O}_{n}\right]=0\quad\rightarrow\quad\sum_{n}f_{n}(D^{C}_{\alpha})\left[\hat{V}^{C}_{n}\right]=0$
(B.72)

If we write $D_{\alpha}$ as
$D_{\alpha}=\partial_{\alpha}+i\tilde{k}_{m}(\gamma_{m}\theta)_{\alpha}$,
where $\tilde{k}=k^{O}$ for open string and $k=\frac{1}{2}k^{C}$ for the
closed string, then it is clear that the functional form of the superfield
$\hat{V}$ for both open and closed strings will be same and we can obtain the
left and/or right sector of the closed string vertex operators by replacing
the momenta $k$ by $\frac{1}{2}k$ in the open string vertex operator

$\displaystyle\hat{V}^{C}=\hat{V}^{O}\left(k\rightarrow\frac{1}{2}k\right)$
(B.73)

and hence, the full pure spinor closed string vertex operator can be expressed
as

$\displaystyle
V(k,z,\bar{z})=V_{R}\left(\frac{k}{2},z\right)V_{L}\left(\frac{k}{2},\bar{z}\right)e^{ik\cdot
X}$ (B.74)

(where both $V_{L}$ and $V_{R}$ have the same functional form as in open
string case) for type II and

$\displaystyle
V(k,z,\bar{z})=V_{R}\left(\frac{k}{2},z\right)V_{L}\left(\bar{z}\right)e^{ik\cdot
X}$ (B.75)

(where $V_{R}$ has same functional form as open string case and $V_{L}$ is
constructed using SO(32) or $E_{8}\times E_{8}$ fields as in RNS) for the
heterotic case.

In the above analysis, we have not yet taken into account the gauge freedom
given in (B.69). However, both the gauge superfields $\Omega$ and
$\bar{\Omega}$ in the case of type II theories (or just $\Omega$ in the case
of heterotic theories) have the same number of degrees of freedom as the
single gauge superfield of open strings. Hence, once we obtain the full closed
string vertex operators, they can be gauge fixed in exactly the same manner as
in the open string case. See [28] for this analysis in explicit detail for the
massless states.

###  B.2 Integrated vertex operator

The integrated vertex operator satisfies

$\displaystyle Q\bar{Q}U(z,\bar{z})=\partial\bar{\partial}V(z,\bar{z})$ (B.76)

Using the form of $V$ described above, the right hand side of (B.76) can be
written as

$\displaystyle\partial\bar{\partial}V$ $\displaystyle=$
$\displaystyle\partial\bar{\partial}\Bigl{(}V_{L}(z)V_{R}(\bar{z})e^{ik\cdot
X(z,\bar{z})}\Bigl{)}$ (B.77) $\displaystyle=$
$\displaystyle\partial\Bigl{(}V_{L}\bar{\partial}V_{R}e^{ik\cdot
X}+V_{L}V_{R}\bar{\partial}e^{ik\cdot X}\Bigl{)}$ $\displaystyle=$
$\displaystyle\partial V_{L}\bar{\partial}V_{R}e^{ik\cdot
X}+V_{L}\bar{\partial}V_{R}\partial e^{ik\cdot X}+\partial
V_{L}V_{R}\bar{\partial}e^{ik\cdot
X}+V_{L}V_{R}\partial\bar{\partial}e^{ik\cdot X}$

To satisfy (B.76), we propose

$\displaystyle U(z,\bar{z})=U_{L}(z)U_{R}(\bar{z})e^{ik\cdot X}$ (B.78)

such that

$\displaystyle Q\Bigl{(}U_{L}e^{ik\cdot
X}\Bigl{)}=\partial\bigl{(}V_{L}e^{ik\cdot
X}\bigl{)}\quad,\qquad\bar{Q}\Bigl{(}U_{R}e^{ik\cdot
X}\Bigl{)}=\bar{\partial}\bigl{(}V_{R}e^{ik\cdot X}\bigl{)}$ (B.79)

With this, the left hand side of (B.76) becomes

$\displaystyle Q\bar{Q}U$ $\displaystyle=$ $\displaystyle
Q\Bigl{[}U_{L}\bar{Q}\bigl{(}U_{R}e^{ik\cdot X}\bigl{)}\Bigl{]}$ (B.80)
$\displaystyle=$ $\displaystyle Q\Bigl{[}U_{L}\bar{\partial}V_{R}e^{ik\cdot
X}+U_{L}V_{R}\bar{\partial}e^{ik\cdot X}\Bigl{]}$ $\displaystyle=$
$\displaystyle Q\bigl{(}U_{L}e^{ik\cdot
X}\bigl{)}\bar{\partial}V_{R}+Q\bigl{(}U_{L}\bar{\partial}e^{ik\cdot
X}\bigl{)}V_{R}$ $\displaystyle=$
$\displaystyle\partial\bigl{(}V_{L}e^{ik\cdot
X}\bigl{)}\bar{\partial}V_{R}+\bar{\partial}\partial\bigl{(}V_{L}e^{ik\cdot
X}\bigl{)}V_{R}$ $\displaystyle=$ $\displaystyle\partial
V_{L}\bar{\partial}V_{R}e^{ik\cdot X}+V_{L}\bar{\partial}V_{R}\partial
e^{ik\cdot X}+\partial V_{L}V_{R}\bar{\partial}e^{ik\cdot
X}+V_{L}V_{R}\partial\bar{\partial}e^{ik\cdot X}$

This is same as the left hand side (B.77). This shows that the ansatz (B.78)
satisfies the BRST equation (B.76) provided (B.79) holds.

Thus, the construction of the integrated vertex has reduced to solving the two
decoupled equations given in (B.79). These equations also depend upon the OPEs
of the theory and have the same form as the corresponding equations in open
string case. Hence, using the same arguments as described above for the
unintegrated vertices, we see that the $U_{L}$ and $U_{R}$ for the type II
theories (or $U_{R}$ for the heterotic theories) have exactly the same
functional form as the open string integrated vertex but with momenta $k^{m}$
replaced by $\frac{1}{2}k^{m}$.

###  B.3 Theta Expansion

From the difference between the open and closed strings BRST equations of
motion satisfied by the superfields, it follows that the theta expansion of
the basic superfields for the closed strings can be obtained from the
corresponding open string theta expansion by replacing
$k^{m}\rightarrow\frac{1}{2}k^{m}$. Thus, using the open string theta
expansion results of [22], we obtain for the closed strings

$\displaystyle\Psi_{s\beta}$ (B.81) $\displaystyle=$
$\displaystyle\psi_{s\beta}+\frac{1}{16}(\gamma^{m}\theta)_{\beta}\
g_{sm}-{\frac{i}{48}}(\gamma^{mnp}\theta)_{\beta}k_{m}b_{nps}-{\frac{i}{288}}(\gamma_{s}^{\;\;npqr}\theta)_{\beta}k_{n}b_{pqr}$
$\displaystyle-{\frac{i}{4}}k^{p}(\gamma^{m}\theta)_{\beta}(\psi_{(m}\gamma_{s)p}\theta)-{\frac{i}{8}}k_{m}(\gamma^{mnp}\theta)_{\beta}(\psi_{[s}\gamma_{np]}\theta)-{\frac{i}{48}}(\gamma_{s}^{\;\;mnpq}\theta)_{\beta}k_{m}(\psi_{q}\gamma_{np}\theta)$
$\displaystyle-{\frac{i}{48}}\alpha^{\prime}k_{m}k^{r}k_{s}(\gamma^{mnp}\theta)_{\beta}(\psi_{p}\gamma_{rn}\theta)+{\frac{i}{2304}}\alpha^{\prime}(\gamma^{mnp}\theta)_{\beta}k_{m}k^{r}k_{s}(\theta\gamma^{q}_{\;\;nr}\theta)\
g_{pq}$
$\displaystyle-{\frac{i}{384}}(\gamma^{mnp}\theta)_{\beta}k_{m}(\theta\gamma^{q}_{\;\;[np}\theta)g_{s]q}-{\frac{i}{2304}}(\gamma_{smnpq}\theta)_{\beta}k^{m}(\theta\gamma_{npt}\theta)\
g^{qt}$
$\displaystyle-{\frac{i}{192}}k^{p}(\gamma^{m}\theta)_{\beta}(\theta\gamma_{pq(s}\theta)\
g_{m)q}-{\frac{1}{6912}}(\gamma^{mnp}\theta)_{\beta}k_{m}(\theta\gamma^{tuvw}_{\;\;\;\;\;\;\;\;\;nps}\theta)k_{t}b_{uvw}$
$\displaystyle-{\frac{1}{864\alpha^{\prime}}}(\gamma_{s}\theta)_{\beta}(\theta\gamma^{npq}\theta)b_{npq}-{\frac{1}{41472}}(\gamma_{s}^{\;\;mnpq}\theta)_{\beta}k_{m}(\theta\gamma_{tuvwnpq}\theta)k^{t}b^{uvw}$
$\displaystyle-{\frac{1}{3456}}(\gamma^{m}\theta)_{\beta}(\theta\gamma^{npq}\theta)b_{npq}k_{m}k_{s}-{\frac{1}{2304}}(\gamma_{smnpq}\theta)_{\beta}k^{m}(\theta\gamma^{tun}\theta)b_{u}^{\;\;pq}k_{t}$
$\displaystyle-\frac{1}{96\alpha^{\prime}}(\gamma^{m}\theta)_{\beta}(\theta\gamma^{qr}_{\;\;\;\;(s}\theta)b_{m)rq}+{\frac{1}{384}}(\gamma^{m}\theta)_{\beta}(\theta\gamma^{nqr}\theta)k_{n}k_{(s}b_{m)qr}$
$\displaystyle+{\frac{1}{384}}(\gamma^{mnp}\theta)_{\beta}k_{m}(\theta\gamma^{r}_{\;\;q[n}\theta)b_{ps]r}k^{q}+O(\theta^{4})$

and

$\displaystyle B_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\gamma^{mnp}_{\alpha\beta}\Biggl{[}b_{mnp}+12(\psi_{p}\gamma_{mn}\theta)+{6}\alpha^{\prime}k^{r}k_{m}(\psi_{p}\gamma_{rn}\theta)+\frac{3}{8}(\theta\gamma_{mn}^{\;\;\;\;\;q}\theta)\
g_{pq}-{\frac{3i}{8}}(\theta\gamma^{tu}_{\;\;\;\;m}\theta)k_{t}b_{unp}$ (B.82)
$\displaystyle\hskip
28.90755pt+{\frac{3}{16}}\alpha^{\prime}k^{r}k_{m}(\theta\gamma_{rn}^{\;\;\;\;\;q}\theta)\
g_{pq}-{\frac{i}{48}}(\theta\gamma_{tuvwmnp}\theta)k^{t}b^{uvw}-{\frac{1}{12}}ik_{{s}}\left(\psi_{{v}}\gamma_{{t}{u}}\theta\right)\left(\theta\gamma_{stuvmnp}\theta\right)$
$\displaystyle\hskip 28.90755pt-{\frac{1}{2}}i\alpha
k_{{s}}k_{{t}}k_{m}\left(\theta\gamma_{{t}{u}n}\theta\right)\left(\psi_{p}\gamma_{{s}{u}}\theta\right)+{\frac{i}{2}}k_{{s}}\left(\theta\gamma_{{t}mn}\theta\right)\left(\psi_{p}\gamma_{{s}{t}}\theta\right)+{\frac{i}{2}}k_{{s}}\left(\theta\gamma_{{t}mn}\theta\right)\left(\psi_{{t}}\gamma_{{s}p}\theta\right)$
$\displaystyle\hskip
28.90755pt+{i}k_{{s}}\left(\theta\gamma_{{s}{t}m}\theta\right)\left(\psi_{n}\gamma_{{t}p}\theta\right)-{\frac{i}{2}}k_{{s}}\left(\theta\gamma_{{s}{t}m}\theta\right)\left(\psi_{{t}}\gamma_{np}\theta\right)+\frac{1}{64\alpha^{\prime}}{(\theta\gamma_{smn}\theta)}{(\theta\gamma_{tup}\theta)}b_{stu}$
$\displaystyle\hskip
28.90755pt-\frac{1}{288\alpha^{\prime}}{(\theta\gamma_{stu}\theta)}{(\theta\gamma_{mnp}\theta)}b_{stu}+\frac{1}{64\alpha^{\prime}}{(\theta\gamma_{stu}\theta)}{(\theta\gamma_{unp}\theta)}b_{stm}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{128}}{(\theta\gamma_{sux}\theta)}{(\theta\gamma_{txp}\theta)}b_{smn}k_{t}k_{u}-{\frac{1}{64}}{(\theta\gamma_{sun}\theta)}{(\theta\gamma_{txp}\theta)}b_{stm}k_{u}k_{x}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{256}}{(\theta\gamma_{stx}\theta)}{(\theta\gamma_{unp}\theta)}b_{stm}k_{u}k_{x}+{\frac{1}{768}}{(\theta\gamma_{xzm}\theta)}{(\theta\gamma_{stuyznp}\theta)}b_{stu}k_{x}k_{y}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{768}}{(\theta\gamma_{uyz}\theta)}{(\theta\gamma_{stxzmnp}\theta)}b_{stu}k_{x}k_{y}+{\frac{1}{13824}}{(\theta\gamma_{stuwxyz}\theta)}{(\theta\gamma_{vxyzmnp}\theta)}b_{stu}k_{v}k_{w}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{128}}{(\theta\gamma_{svn}\theta)}{(\theta\gamma_{tup}\theta)}b_{stu}k_{v}k_{m}+{\frac{1}{256}}{(\theta\gamma_{tuv}\theta)}{(\theta\gamma_{snp}\theta)}b_{stu}k_{v}k_{m}$
$\displaystyle\hskip
28.90755pt-{\frac{1}{384}}{(\theta\gamma_{stu}\theta)}{(\theta\gamma_{vnp}\theta)}b_{stu}k_{v}k_{m}-{\frac{1}{128}}{(\theta\gamma_{stv}\theta)}{(\theta\gamma_{uvp}\theta)}b_{stm}k_{u}k_{n}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{768}}i\left(\theta\gamma_{{t}{v}{w}}\theta\right)\left(\theta\gamma_{{s}{u}{v}{w}mnp}\theta\right)k_{{u}}g_{{s}{t}}+{\frac{1}{64}}i\left(\theta\gamma_{{s}{u}n}\theta\right)\left(\theta\gamma_{{t}{u}p}\theta\right)k_{{t}}g_{{s}m}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{128}}i\left(\theta\gamma_{{s}{t}{u}}\theta\right)\left(\theta\gamma_{{u}np}\theta\right)k_{{t}}g_{{s}m}+{\frac{1}{128}}i\left(\theta\gamma_{{s}mn}\theta\right)\left(\theta\gamma_{{t}{u}p}\theta\right)k_{{u}}g_{{s}{t}}$
$\displaystyle\hskip
28.90755pt+{\frac{1}{128}}i\left(\theta\gamma_{{s}{u}m}\theta\right)\left(\theta\gamma_{{t}np}\theta\right)k_{{u}}g_{{s}{t}}-{\frac{i\alpha^{\prime}}{128}}\left(\theta\gamma_{{s}{u}{v}}\theta\right)\left(\theta\gamma_{{t}{v}p}\theta\right)k_{{t}}k_{{u}}k_{n}g_{{s}m}\;+\
O(\theta^{5})\Biggl{]}$

Using these theta expansion results, the theta expansion of the unintegrated
as well as integrated vertex operators can be easily written down.

##  Appendix C  Some useful pure spinor identities

The following identities turn out to be useful in simplifying the pure spinor
correlators at the superfield level. For an arbitrary tensor-spinor superfield
$\Psi_{m\alpha}$, we have

  1. 1.

$(\lambda\gamma_{mn}{\Psi}_{p})(\lambda{\gamma}_{mqrst}\lambda)=-(\lambda{\Psi}_{p})(\lambda{\gamma}_{nqrst}\lambda)$

  2. 2.

$(\lambda{\gamma}_{mnpq}{\Psi}_{r})(\lambda{\gamma}_{mnstu}\lambda)=-2(\lambda{\Psi}_{r})(\lambda{\gamma}_{pqstu}\lambda)$

  3. 3.

$(\lambda{\gamma}_{mnpqrs}{\Psi}_{t})(\lambda{\gamma}_{mnpuv}\lambda)=6(\lambda{\Psi}_{t})(\lambda{\gamma}_{qrsuv}\lambda)$

  4. 4.

$(\lambda{\gamma}_{mnpqrstu}{\Psi}_{v})(\lambda{\gamma}_{mnpqw}\lambda)=24(\lambda{\Psi}_{v})(\lambda{\gamma}_{rstuw}\lambda)$

These identities can be proved using the gamma matrix properties and the pure
spinor constraint. E.g., to prove the first identity, we start by noting

$\displaystyle(\lambda\gamma_{mn}{\Psi}_{p})(\lambda{\gamma}_{mqrst}\lambda)$
$\displaystyle=$
$\displaystyle\bigl{[}\lambda(\gamma_{m}\gamma_{n}-\eta_{mn}){\Psi}_{p}\bigl{]}(\lambda{\gamma}_{mqrst}\lambda)$

The first term in the right hand side vanishes by pure spinor constraint. To
see this, we use the identity $(\lambda\gamma^{abc}\lambda)=0$ to decompose
the gamma 5-form term as
$(\lambda{\gamma}_{mqrst}\lambda)=(\lambda\gamma_{m}{\gamma}_{qrst}\lambda)$.
With this, the first term of the right hand side vanishes by the identity
$(\lambda\gamma^{m})_{\alpha}(\lambda\gamma_{m})_{\beta}=0$ which follows from
the pure spinor constraint. This proves the first identity. All the other
identities can be proved in similar way.

The following identities also turn out to be useful in simplifying the
calculations after doing the theta expansion

  1. 1.

$(\lambda\gamma^{mnpqr}\lambda)(\lambda\gamma_{mna}\theta)=0$

  2. 2.

$(\lambda\gamma^{mnpqr}\lambda)(\lambda\gamma_{m}\theta)=0$

  3. 3.

$(\lambda\gamma^{mnpqr}\lambda)(\lambda\gamma_{mnpab}\theta)=0$

  4. 4.

$(\theta\gamma^{mnp}\theta)(\theta\gamma_{mnp}\theta)=0$

These can also be proved using the pure spinor constraint by following the
same method as described above.

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