# Extended Massive Ambitwistor String

Christian Kunz
E-mail: mailto:kunz.christian.321<EMAIL_ADDRESS>
###### Abstract

This work considers a variation of the massive ambitwistor string model
presented in arXiv:2301.11227 that describes supergravity and super-Yang-Mills
on the Coulomb branch simultaneously with a single Lagrangian. All-
multiplicity tree and one-loop amplitudes are evaluated and shown to have
proper unitary factorization. The massless limit provides Einstein Yang-Mills
Amplitudes including multiple gluon traces. It is argued that the cosmological
constant vanishes at all orders of perturbation theory. As application results
for Compton scattering are obtained and compared with ones in the literature.

###### Contents

  1. 1 Introduction
  2. 2 Extended Massive Ambitwistor String
    1. 2.1 Consistency Checks: Einstein-Yang-Mills Amplitudes and Massless Limit
    2. 2.2 Factorization of Tree Amplitudes
  3. 3 Loop Amplitudes and Factorization
    1. 3.1 Nonseparating Degeneration
    2. 3.2 Separating Degeneration
    3. 3.3 Higher Loops and Cosmological Constant
  4. 4 Compton Scattering Using Massive Twistor Strings
  5. 5 Summary and Outlook

##  1 Introduction

There has been a lot of progress in calculating on-shell scattering amplitudes
for massless and massive particles using the spinor-helicity formalism and
generalized unitarity [1, 2, 3, 4, 5]. There also has been some progress in
deriving such amplitudes directly from a massive ambitwistor string theory
centered on polarized scattering equations [6, 7, 8, 9, 10]. What has been
still missing, was a consolidation of supergravity and super-Yang-Mills (SYM)
on the Coulomb branch into a unified theory that exhibits proper unitary
factorization justifying the use of generalized unitarity. This article is a
step forward in this direction.

The paper is organized as follows:

In section 2 the supergravity version of the massive ambitwistor string
presented in [9, 10] is extended to include additional gauge symmetries. It
has some similarity with the supergravity model of Skinner [11] and is shown
to be anomaly-free in the same manner as the original massive ambitwistor
string. It also has the same $\mathcal{N}\\!=\\!8$ supergravity spectrum. The
all-multiplicity tree amplitude is calculated.

In subsection 2.1 some consistency checks are performed. First a massive
Einstein-Yang-Mills (EYM) amplitude is derived which includes SYM on the
Coulomb branch as special case and has the proper massless limit [12]. Also
the massless limit of the general tree amplitude is shown to encompass known
supergravity amplitudes [13, 7].

In subsection 2.2 proper factorization of tree amplitudes is established.

In section 3 one-loop amplitudes are examined. They vanish on even spin
structures. Their form on the odd spin structure is evaluated.

In subsection 3.1 unitary factorization of the amplitude is demonstrated for
nonseparating degeneration of genus 1 Riemann surfaces.

In subsection 3.2 the same is done for separating degeneration.

In subsection 3.3 some remarks concerning higher loop amplitudes are made. It
is argued that the cosmological constant is zero at all orders of perturbation
theory, and that all higher loop amplitudes vanish on even spin structures.
Further, not even all odd spin structures can contribute to loop amplitudes,
only a particular subset. It is also mentioned that the original ambitwistor
model shares many of the features of the extended model, in particular with
regards to even/odd spin structures and the cosmological constant.

Section 4 serves as another consistency check by applying the model to the
Compton scattering of a massless particle hitting a massive target. A
difference to the original massive ambitwistor string for supergravity is that
the same-helicity amplitude does not vanish for ’gravitinos’, and the result
for ’gluons’ is consistent with the one arising from SYM on the Coulomb
branch.

Section 5 contains summary and discussion.

##  2 Extended Massive Ambitwistor String

In this section the massive ambitwistor string of [10] for maximal
supergravity is extended with an enlarged supersymmetric gauge symmetry. It
can also be viewed as generalization of a massless ambitwistor string model in
[14] to include massive particles.

The twistor action used here is, in similar notation to [14],

$\displaystyle S\;\,\\!=\\!\int_{\Sigma}$
$\displaystyle\mathcal{Z}^{a}\cdot\bar{\partial}\mathcal{Z}_{a}+A_{ab}\mathcal{Z}^{a}\cdot\mathcal{Z}^{b}+a\\!\braket{\lambda_{A}\lambda_{B}}\\!\Omega^{AB}+\tilde{a}\\!\braket{\eta_{\mathcal{I}}\eta_{\mathcal{J}}}\\!\Omega^{\mathcal{I}\mathcal{J}}+S_{\rho_{1}}+S_{(\rho_{2},\tau)},$
$\displaystyle S_{\rho}\,\\!=\\!\int_{\Sigma}$
$\displaystyle\frac{1}{2}\braket{\rho_{A}\bar{\partial}\rho_{B}}\Omega^{AB}+B_{ab}\lambda^{a}_{A}\rho^{b}_{B}\Omega^{AB}+b\braket{\lambda_{A}\rho_{B}}\Omega^{AB}\,,$
(2.1) $\displaystyle S_{(\rho,\tau)}\\!=\\!\int_{\Sigma}$
$\displaystyle\frac{1}{2}(\braket{\rho_{A}\bar{\partial}\rho_{B}}\\!\Omega^{AB}\\!+\\!\braket{\tau_{\mathcal{I}}\bar{\partial}\tau_{\mathcal{J}}}\\!\Omega^{\mathcal{I}\mathcal{J}})+D_{ab}(\lambda^{a}_{A}\rho^{b}_{B}\Omega^{AB}\\!+\\!\eta^{a}_{\mathcal{I}}\tau^{b}_{\mathcal{J}}\\!\Omega^{\mathcal{I}\mathcal{J}})$
$\displaystyle+d\,(\braket{\lambda_{A}\rho_{B}}\Omega^{AB}+\braket{\eta_{\mathcal{I}}\tau_{\mathcal{J}}}\Omega^{\mathcal{I}\mathcal{J}})\,,$

where $a=1,2$ is the little group index with contraction notation
$\xi_{a}\zeta_{b}\varepsilon^{ab}=\xi_{a}\zeta^{a}\equiv\braket{\xi\zeta}$ and
$\mathcal{Z}_{a}$ is a pair of supertwistor fields that are worldsheet
spinors, repackaged into Dirac supertwistors of the form

$\displaystyle\mathcal{Z}\\!=\\!(\lambda_{A},\mu^{A},\eta_{\iota})\\!:\;$
$\displaystyle\lambda_{A}\\!=\\!(\lambda_{\alpha},\tilde{\lambda}_{\dot{\alpha}}),\mu^{A}\\!=\\!(\mu^{\dot{\alpha}},\tilde{\mu}^{\alpha}),\;\eta_{\iota}\\!=\\!(\eta_{\mathcal{I}},\tilde{\eta}^{\mathcal{I}}),\eta_{\mathcal{I}}\\!=\\!(\eta_{I},\tilde{\eta}_{\dot{I}}),\tilde{\eta}^{\mathcal{I}}\\!=\\!(\tilde{\eta}^{\prime
I},\eta^{\prime\dot{I}}),$ $\displaystyle
I,\dot{I}=1\ldots\frac{\mathcal{N}}{2},\,\mathcal{I}=1,\ldots,\mathcal{N},\,\iota=1,\ldots,2\mathcal{N}\,,\,\mathcal{N}=8\,,$

where $\lambda_{A}$ and $\mu^{A}$ are Dirac spinors made up of the homogeneous
chiral and antichiral components of the twistor
$Z=(\lambda_{\alpha},\mu^{\dot{\alpha}})$ and dual twistor
$\tilde{Z}=(\tilde{\lambda}_{\dot{\alpha}},\tilde{\mu}^{\alpha})$, and
$\eta_{\iota}\\!=\\!(\eta_{\mathcal{I}},\tilde{\eta}^{\mathcal{I}})$ are
fermionic components with $\mathcal{I}=1,\ldots,\mathcal{N}$ as the R-symmetry
index, with $\mathcal{N}=8$ for maximal supergravity. Indices are raised and
lowered with skew 4 $\\!\times\\!$ 4 forms
$\Omega^{EF},\Omega_{EF},\Omega^{EG}\Omega_{GF}=\delta^{E}_{F}$. The
$\mathcal{N}\times\mathcal{N}$ form $\Omega_{\mathcal{I}\mathcal{J}}$
decomposes into two $\frac{\mathcal{N}}{2}\times\frac{\mathcal{N}}{2}$ forms:
$\Omega_{\mathcal{I}\mathcal{J}}=\Omega_{IJ}\otimes\Omega_{\dot{I}\dot{J}}$.
The inner product is defined as
$\mathcal{Z}_{a}\cdot\mathcal{Z}_{b}=\frac{1}{2}(\tilde{Z_{a}}\cdot
Z_{b}+\tilde{Z_{b}}\cdot
Z_{a}+\tilde{\eta}_{a\mathcal{I}}\eta^{\mathcal{I}}_{b}+\tilde{\eta}_{b\mathcal{I}}\eta^{\mathcal{I}}_{a}),\;\;\tilde{Z_{a}}\cdot
Z_{b}=\tilde{\mu}_{a}^{\alpha}\lambda_{b\alpha}+\tilde{\lambda}_{a\dot{\alpha}}\mu_{b}^{\dot{\alpha}}$
,
with special treatment of $\bar{\partial}Z$ when taking the dual:
$\;\widetilde{\bar{\partial}Z}=-\bar{\partial}\tilde{Z}\,$.
$\rho^{a}_{rA}(r\\!=\\!1,2)$ and $\tau^{a}_{\mathcal{I}}$ are auxiliary
fermionic and bosonic worldsheet spinors, respectively, and
$(B_{ab},b,D_{ab},d)$ are fermionic Lagrange multipliers for the constraints
$\lambda^{(a}_{A}\rho^{b)}_{1B}\Omega^{AB}=0=\braket{\lambda_{A}\rho_{1B}},\lambda^{(a}_{A}\rho^{b)}_{2B}\Omega^{AB}\\!+\\!\eta^{(a}_{\mathcal{I}}\tau^{b)}_{\mathcal{J}}\Omega^{\mathcal{I}\mathcal{J}}=0=\braket{\lambda_{A}\rho_{2B}}\Omega^{AB}+\braket{\eta_{\mathcal{I}}\tau_{\mathcal{J}}}\Omega^{\mathcal{I}\mathcal{J}}\\!$
on the supersymmetric gauge currents.
Little group transformations for the twistors are gauged by the fields
$A_{ab}=A_{(ab)}$, and $a$ and $\tilde{a}$ are worldsheet $(0,1)$-forms that
also act as Lagrange multipliers, required in order to have a closed current
algebra.

It can be noted that the action is somewhat similar to the
$\mathcal{N}\\!=\\!8$ supergravity model of Skinner [11] which has the same
kind of auxiliary pair of supertwistors with reverse statistics as in
$S_{(\rho,\tau)}$, but only a single main supertwistor. The action is even
more similar to the massive ambitwistor model of [9, 10] and can be viewed as
an extension of it. The difference consists in doubling the fermionic
components in the main supertwistor, gauging them with auxiliary $\tau$ fields
using the same worldsheet supersymmetry as for $\rho_{2}$, and replacing the
constraints on mass operators with constraints on currents that close the
current algebra.

BRST quantization can be performed similarly to [8]. In addition to the
familiar fermionic ($b,c$) ghost system related to worldsheet gravity (the
action (2) is already written in conformal gauge), the Lagrange multipliers in
(2) are associated with corresponding (antighost, ghost) pairs:
the bosonic fields $\\{A_{ab},a,\tilde{a}\\}$ with fermionic ghosts
$\\{(M^{ab},N_{ab}),(m,n),(\tilde{m},\tilde{n})\\}$ and
the fermionic fields $\\{B_{ab},b,D_{ab},d\\}$ with bosonic ghosts
$\\{(\beta^{ab},\\!\gamma_{ab}),(\beta,\\!\gamma),(\tilde{\beta}^{ab},\\!\tilde{\gamma}_{ab}),(\tilde{\beta},\tilde{\gamma})\\}$.
All the Lagrange multipliers can be gauged to zero and the BRST operator
becomes $Q=\oint\mathrm{d}zJ_{BRST}(z)$, where the BRST current $J_{BRST}(z)$
is (based on the formula $Q=c^{i}K_{i}-\frac{1}{2}f^{ij}_{k}c^{i}c^{j}b_{k}$
for the current algebra $K_{i}$ with structure constants $f^{ij}_{k}$, ghosts
$c^{i}$, and antighosts $b_{k}$) :

$\displaystyle J_{BRST}\\!=$
$\displaystyle\,c\,T\\!+\\!N_{ab}(J^{ab}\\!+\\!M^{a}_{c}N^{bc})\\!+\\!n\\!\braket{\lambda_{A}\lambda^{A}}\\!+\\!\tilde{n}\\!\braket{\eta_{\mathcal{I}}\eta^{\mathcal{I}}}\\!+\\!\gamma_{ab}\lambda^{aA}\rho^{b}_{1A}\\!+\\!\gamma\\!\braket{\lambda^{A}\rho_{1A}}\\!+\\!m\gamma\gamma$
$\displaystyle+\\!\tilde{\gamma}_{ab}(\lambda^{aA}\rho^{b}_{2A}\\!+\\!\eta^{a\mathcal{I}}\tau^{b}_{\mathcal{I}})\\!+\\!\tilde{\gamma}(\braket{\lambda^{A}\rho_{2A}}\\!+\\!\braket{\eta^{\mathcal{I}}\tau_{\mathcal{I}}})\\!+\\!(m\\!+\\!\tilde{m})\tilde{\gamma}\tilde{\gamma}\,,$
$\displaystyle T\\!=$
$\displaystyle\mathcal{Z}^{a}\\!\\!\cdot\\!\partial\mathcal{Z}_{a}\\!+\\!\frac{1}{2}\\!\sum_{r=1,2}\\!\\!\braket{\rho_{rA}\partial\rho_{r}^{A}}\\!+\\!\beta^{ab}\partial\gamma_{ab}\\!+\\!\beta\partial\gamma\\!+\\!\frac{1}{2}\\!\braket{\tau_{\mathcal{I}}\partial\tau^{\mathcal{I}}}\\!+\\!\tilde{\beta}^{ab}\partial\tilde{\gamma}_{ab}\\!+\\!\tilde{\beta}\partial\tilde{\gamma}$
$\displaystyle+\\!M^{ab}\partial N_{ab}+m\partial
n\\!+\\!\tilde{m}\partial\tilde{n}\\!+\\!\frac{1}{2}(b\partial
c+\partial(bc))\,,$ $\displaystyle J^{ab}\\!=$
$\displaystyle\mathcal{Z}^{a}\cdot\mathcal{Z}^{b}\\!+\\!\frac{1}{2}\\!\sum_{r=1,2}\rho^{(a}_{rA}\rho_{r}^{b)A}\\!+\\!\beta^{c(a}\gamma^{b)}_{c}\\!+\\!\frac{1}{2}\tau^{(a}_{\mathcal{I}}\tau^{b)\mathcal{I}}\\!+\\!\tilde{\beta}^{c(a}\tilde{\gamma}_{c}^{b)}\,,$

where $T$ is the energy-momentum current and $J^{ab}$ the sl(2, $\mathbb{C}$)
current. Obstructions to the nilpotency of the BRST charge $Q$ can only come
from the sl(2,$\mathbb{C}$) algebra and the Virasoro algebra. The sl(2,
$\mathbb{C}$) anomaly coefficient is, using
$\text{tr}_{\text{adj}}(t^{k}t^{k})=6,\text{trf}_{\text{F}}(t^{k}t^{k})=3/2$,

$a_{sl2}=\frac{1}{2}\text{trf}_{\text{F}}(t^{k}t^{k})((8\\!-\\!2\mathcal{N})_{\\!\mathcal{Z}}\\!-\\!4_{\rho_{1}}\\!-\\!4_{\rho_{2}}\\!+\\!\mathcal{N}_{\tau})+\text{tr}_{\text{adj}}(t^{k}t^{k})(\\!-1_{M\\!N}+1_{\beta\gamma}+1_{\tilde{\beta}\tilde{\gamma}})\\!=\\!\frac{3}{4}(8-\mathcal{N})=0\,.$

For the central charge it is assumed that there is a contribution of 12 from
compactifying 6 dimensions like for the massive ambitwistor model of [10],
e.g. with help of 6 bosonic scalars as in appendix A of [10]. Then it can be
seen to vanish for $\mathcal{N}=8$:

$c\\!=\\!(\\!-8+2\mathcal{N}\\!)_{\\!\mathcal{Z}}\\!-26_{bc}\\!-6_{M\\!N}\\!-2_{mn}\\!-2_{\tilde{m}\tilde{n}}\\!+4_{\rho_{1}}\\!+8_{\beta\gamma}\\!+4_{\rho_{2}}\\!-\\!\mathcal{N}_{\\!\tau}+8_{\tilde{\beta}\tilde{\gamma}}\\!+12_{\text{comp}}\\!=\\!-8+\\!\mathcal{N}\\!=\\!0\,.$

Therefore, the model can be viewed as conditionally anomaly-free. The
additional contribution to the central charge can also be obtained by gauging
6 additional worldsheet supersymmetries between 3 or 4 additional auxiliary
supertwistors $(\rho_{i}^{A},\tau_{i}^{A}),i=3,4,5(,6)$ that do not affect the
spectrum and vertex operators. In this manner the model can become anomaly-
free without resorting to extra spacetime dimensions. Of course, one could
argue that the auxiliary spinors are remnants from compactifying extra
dimensions in twistor space, and there might be many more ways to come up with
an auxiliary action that achieves the same outcome.

To find the spectrum one needs to determine the Virasoro $L_{0}$ constant $a$
[14]. In the NS sector $a$ is calculated by changing in the calculation of the
central charge the $bc$ contribution from $-26_{bc}$ to $-2_{bc}$ and dividing
the result by 24, such that $a=1$ in the NS sector. In the R sector $a$ is:

$24a\\!=\\!\\!(\\!16-\\!4\mathcal{N}\\!)_{\\!\mathcal{Z}}\\!-\\!2_{bc}\\!-6_{M\\!N}\\!-2_{mn}\\!-2_{\tilde{m}\tilde{n}}\\!-8_{\rho_{1}}\\!+8_{\beta\gamma}\\!-(8_{\rho_{2}}\\!-\\!2\mathcal{N}_{\\!\tau})+8_{\tilde{\beta}\tilde{\gamma}}\\!+12_{\text{comp}}\\!=\\!2(8-\\!\mathcal{N})\\!=\\!0\,.$

i.e. $a=0$ in the R sector.
The situation is parallel to the massive ambitwistor model of [10], and based
on BRST cohomology the spectrum is again the one of $\mathcal{N}\\!=\\!8$
supergravity with SU(4) $\\!\times\\!$ SU(4) $R$-symmetry. Although the
fermionic modes in the supertwistors have been doubled, similarly to the
exclusion of $\mu^{aA}_{n}$ modes from the spectrum because of BRST
cohomology, now the $\tilde{\eta}^{a\mathcal{I}}_{n}$ cannot contribute either
to the spectrum because of the requirement that the non-negative modes of the
$\eta^{a\mathcal{I}}\tau^{b}_{\mathcal{I}}$ and
$\braket{\eta^{\mathcal{I}}\tau_{\mathcal{I}}}$ currents annihilate physical
states. One other difference is that the action in [10] contains mass
operators that ensure that the supersymmetric multiplet has a particular mass
whereas the current model leaves the value of the mass open.

Moving on to vertex operators, following [8, 9, 10] vertex operators are
sought in the form

$\genfrac{(}{)}{0.0pt}{}{\mathcal{V}}{V}\\!=\\!\\!\int\\!\mathrm{d}^{2}\\!u\mathrm{d}^{2}\\!v\genfrac{(}{)}{0.0pt}{}{\mathcal{W}(u)}{w(u)}\bar{\delta}(\\!\braket{v\epsilon}\\!-\\!1)\bar{\delta}^{4}(\\!\braket{u\lambda_{A}}\\!-\\!\braket{v\kappa_{A}})\bar{\delta}^{\mathcal{N}}\\!(\\!\braket{u\eta_{\mathcal{I}}}\\!-\\!\braket{v\zeta_{\mathcal{I}}})\;e^{\braket{u\mu^{A}}\epsilon_{A}+\braket{u\tilde{\eta}^{\mathcal{I}}}q_{\mathcal{I}}}.\\\
$ (2.2)

Here $\epsilon_{a}$ are little group spinors,
$\kappa^{A}_{a}=(\kappa^{\alpha}_{a},\tilde{\kappa}^{\dot{\alpha}}_{a})$ are
momentum spinors with associated polarization spinors
$\epsilon^{A}\\!=\\!\braket{\epsilon\kappa^{A}}$,$\;\zeta^{\mathcal{I}}_{a}$
are supermomenta with superpolarization spinors
$q^{\mathcal{I}}\\!=\\!\braket{\epsilon\zeta^{\mathcal{I}}}$, and
$\mathcal{V}$ and $V$ stand for fixed and integrated vertex operators,
respectively, which differ only by an operator $\mathcal{W}$ or $w$ with
$\mathcal{W}$ typically being just a product of fermionic ghost fields and
delta functions of bosonic ghost fields. As the handling of fixed and
integrated vertex operators concerning the $(b,c)$ ghosts is standard, the
difference being just an integration over the worldsheet, the $(b,c)$ ghosts
and worldsheet integrations will be omitted in the formulae for $\mathcal{W}$
and $w$, but then added later in the correlators. Also the effect of $(m,n)$
is just to fix
$\braket{\lambda_{\alpha}\lambda^{\alpha}}=\braket{\lambda_{\dot{\alpha}}\lambda^{\dot{\alpha}}}$
which can be viewed as a little group transformation outside of sl(2,
$\mathbb{C}$) [2, 15] and will be skipped as well, and similarly for
$(\tilde{m},\tilde{n})$. Finally it is assumed that the bosonic fields from
compactifying extra dimensions or additional supersymmetric gauge symmetries
do not affect the spectrum or vertex operators111They will be crucial for loop
amplitudes in the next section.. The remainder can be simplified even further
because the form (2.2) is such that it is sufficient that the delta functions
of $\gamma_{ab}$ and $\tilde{\gamma}_{ab}$ in $\mathcal{W}$ force only the
ghost components orthogonal to $u$ to vanish [8]. Consequently

$\displaystyle\mathcal{W}(u)\\!=$
$\displaystyle\,\delta(\gamma)\,\delta(u^{a}u^{b}\gamma_{ab})\;\delta(\tilde{\gamma})\,\delta(u^{a}u^{b}\tilde{\gamma}_{ab}),$
$\displaystyle w(u)\\!=$
$\displaystyle\\!\left(\frac{\braket{\hat{u}\lambda_{A}}\epsilon^{A}}{\braket{u\hat{u}}}\\!+\\!\frac{\\!\braket{\rho_{1A}\rho_{1B}}}{2}\epsilon^{A}\epsilon^{B}\\!\right)\\!\left(\frac{\braket{\hat{u}\lambda_{A}}\\!\epsilon^{A}}{\braket{u\hat{u}}}\\!+\\!\frac{\\!\braket{\rho_{2A}\rho_{2B}}}{2}\epsilon^{A}\epsilon^{B}\\!+\\!\frac{\braket{\hat{u}\eta_{\mathcal{I}}}\\!q^{\mathcal{I}}}{\braket{u\hat{u}}}\\!+\\!\frac{\\!\braket{\tau_{\mathcal{I}}\tau_{\mathcal{J}}}}{2}q^{\mathcal{I}}\\!q^{\mathcal{J}}\\!\right).$

Here $\hat{u}$ is a reference little group spinor that forms a basis with $u$
such that $\braket{u\hat{u}}\neq 0$. In amplitudes it will fall out based on
solutions of the scattering equations [8].

The all-multiplicity tree amplitude of this model can now be evaluated:

$\mathcal{A}_{n}^{\text{tree}}=\left<c\mathcal{V}_{1}^{(-1,-1)}c\mathcal{V}_{2}^{(-1,0)}c\mathcal{V}_{3}^{(-1,0)}\,\prod_{j=4}^{n}\int_{\Sigma}\\!V_{j}\right>\,,$

where $\mathcal{V}^{(-1,-1)}$ denotes a fixed vertex operator with regards to
all
$(\beta^{ab},\\!\gamma_{ab}),(\tilde{\beta}^{ab},\\!\tilde{\gamma}_{ab}),(\beta,\\!\gamma),$
$(\tilde{\beta},\\!\tilde{\gamma})$ ghosts and $\mathcal{V}^{(-1,0)}$ only
fixed with regards to
$(\beta^{ab},\gamma_{ab}),(\tilde{\beta}^{ab},\tilde{\gamma}_{ab})$.

Integrating out the $\mu^{A}$ and $\tilde{\eta}^{\mathcal{I}}$ leads, on the
support of the delta functions, to the polarized scattering equations (using
the notation $\sigma_{ij}=\sigma_{i}-\sigma_{j}$):

$\displaystyle\braket{v_{r}\kappa_{r}^{A}}\\!=\\!\braket{u_{r}\lambda^{A}(\sigma_{r})},$
$\displaystyle\lambda^{a}_{A}(\sigma_{r})=\sum_{l\neq
r}\frac{u^{a}_{l}\epsilon_{lA}}{\sigma_{rl}},$ (2.3)
$\displaystyle\braket{v_{r}\zeta_{r}^{\mathcal{I}}}=\\!\braket{u_{r}\eta^{\mathcal{I}}(\sigma_{r})},$
$\displaystyle\eta^{a}_{\mathcal{I}}(\sigma_{r})=\sum_{l\neq
r}\frac{u^{a}_{l}q_{l\mathcal{I}}}{\sigma_{rl}}.$

Based on these scattering equations and because of the delta functions
enforcing $\braket{v_{r}\epsilon_{r}}=1$ it follows that
$\braket{u_{r}\lambda^{A}(\sigma_{r})}\epsilon_{rA}=\braket{v_{r}\kappa_{r}^{A}}\epsilon_{rA}=0$
and
$\braket{u_{r}\eta^{\mathcal{I}}(\sigma_{r})}q_{r\mathcal{I}}=\braket{v_{r}\zeta_{r}^{\mathcal{I}}}q_{r\mathcal{I}}=0$
such that the dependence on $\hat{u}$ falls out of the correlator.

Then the amplitude is evaluated to

$\mathcal{A}_{n}^{\text{tree}}=\int\mathrm{d}\mu_{n}^{\text{pol}|\mathcal{N}}\;\mathcal{I}_{n}\,,$
(2.4)

where $\mathrm{d}\mu_{n}^{\text{pol}|\mathcal{N}}$ is the measure which
localizes on the scattering equations (2.3) and also accounts for the quotient
by the volume of
$\text{SL}(2,\mathbb{C})_{\sigma}\times\text{SL}(2,\mathbb{C})_{u}$ from the
path integral over the $(b,c)$ and $(M^{ab},N_{ab})$ ghost systems:

$\mathrm{d}\mu_{n}^{\text{pol}|\mathcal{N}}\\!:=\\!\frac{\prod_{l}\mathrm{d}\sigma_{l}\mathrm{d}^{2}u_{l}\mathrm{d}^{2}v_{l}}{\text{vol}(\text{SL}(2,\mathbb{C})_{\sigma}\\!\times\\!\text{SL}(2,\mathbb{C})_{u})}\\!\\!\prod_{r=1}^{n}\\!\bar{\delta}(\\!\braket{v_{r}\epsilon_{r}}\\!-\\!1)\bar{\delta}^{4}(\\!\braket{u_{r}\lambda^{A}}\\!-\\!\braket{v_{r}\kappa_{r}^{A}})\bar{\delta}^{\mathcal{N}}(\\!\braket{u_{r}\eta^{\mathcal{I}}}\\!-\\!\braket{v_{r}\zeta_{r}^{\mathcal{I}}})$
(2.5)

and where the integrand $\mathcal{I}_{n}$ is the product of two reduced
determinants,

$\displaystyle\mathcal{I}_{n}=$
$\displaystyle\quad\text{det}^{\prime}H\text{det}^{\prime}G\quad=$
$\displaystyle\frac{\text{det}H^{[ij]}_{[kl]}}{\braket{u_{i}u_{j}}\braket{u_{k}u_{l}}}\;\frac{\text{det}G^{[pr]}_{[st]}}{\braket{u_{p}u_{r}}\braket{u_{s}u_{t}}},$
(2.6) $\displaystyle
H_{ij}=\frac{\epsilon_{iA}\epsilon_{j}^{A}}{\sigma_{ij}},$ $\displaystyle
u_{i}^{a}H_{ii}=-\lambda_{A}^{a}\epsilon_{i}^{A},$ $\displaystyle
G_{ij}=\frac{\epsilon_{iA}\epsilon_{j}^{A}+q_{i\mathcal{I}}q_{j}^{\mathcal{I}}}{\sigma_{ij}},$
$\displaystyle
u_{i}^{a}G_{ii}=-\lambda_{A}^{a}\epsilon_{i}^{A}-\eta^{a}_{\mathcal{I}}q_{i}^{\mathcal{I}},$

where $H$ and $G$ are $n\\!\times\\!n$ matrices with co-rank 2 and
$H^{[ij]}_{[kl]}$ means that rows $i$ and $j$ and columns $k$ and $l$ have
been removed. The reduced determinants are independent of the choice of rows
and columns to remove.

The amplitude (2.4) has been worked out in the so-called little group (LG)
representation [7], mainly because the derivation required ’scattering
equations’ (2.3) for the $\eta$ fields, but in the following it is more
convenient to go over to the R-symmetry preserving representation, by Fourier
transforming both halves of the
$q_{\mathcal{I}}=(q_{I},\tilde{q}_{\dot{I}})=(q_{I},\\!\braket{\epsilon\zeta_{I}})=(\\!\braket{\epsilon\tilde{\zeta}_{\dot{I}}},\tilde{q}_{\dot{I}})$
variables, using $N\\!=\\!\tilde{N}\\!=\\!\frac{\mathcal{N}}{2}$ and
$v_{a}\\!=\\!\xi_{a}\\!+\\!\braket{\xi v}\epsilon_{a}$ where $(\epsilon,\xi)$
form a spinor basis with $\braket{\xi\epsilon}=1$[7]:

$\displaystyle 2^{-N}\\!\int\prod_{i=1}^{n}\mathrm{d}^{N}\\!q_{i}^{I}$
$\displaystyle\prod_{j=1}^{n}e^{-\frac{1}{2}q_{i}^{I}\\!\braket{\xi_{i}\zeta_{iI}}}e^{F_{N}}=\prod_{i=1}^{n}\delta^{N}\\!\left(\sum_{j=1}^{n}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}\\!\braket{\epsilon_{j}\zeta_{jI}}-\braket{v_{i}\zeta_{jI}}\right),$
$\displaystyle
F_{N}=\sum_{i<j}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}q_{iI}q_{j}^{I}-\frac{1}{2}\sum_{i=1}^{n}\braket{\xi_{i}v_{i}}q_{i}^{2},$
$\displaystyle
2^{-\tilde{N}}\\!\int\prod_{i=1}^{n}\mathrm{d}^{\tilde{N}}\\!\tilde{q}_{i}^{\dot{I}}$
$\displaystyle\prod_{j=1}^{n}e^{-\frac{1}{2}\tilde{q}_{i}^{\dot{I}}\\!\braket{\xi_{i}\tilde{\zeta}_{i\dot{I}}}}e^{F_{\tilde{N}}}=\prod_{i=1}^{n}\delta^{\tilde{N}}\\!\left(\sum_{j=1}^{n}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}\\!\braket{\epsilon_{j}\tilde{\zeta}_{j\dot{I}}}-\braket{v_{i}\tilde{\zeta}_{j\dot{I}}}\right),$
$\displaystyle\tilde{F}_{\tilde{N}}=\sum_{i<j}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}\tilde{q}_{i\dot{I}}\tilde{q}_{j}^{\dot{I}}-\frac{1}{2}\sum_{i=1}^{n}\braket{\xi_{i}v_{i}}\tilde{q}_{i}^{2}.$

When doing this, the $q$ variables in the matrix $G$ and in
$F_{\mathcal{N}}=F_{N}+\tilde{F}_{\tilde{N}}$ will have transposed halves, but
because the inner products are the same one can just transpose one set of
these $q$ variables, e.g. the ones in $F_{\mathcal{N}}$. One gets

$\displaystyle\mathcal{A}_{n}^{\text{tree}}$
$\displaystyle=\int\mathrm{d}\mu_{n}^{\text{pol}}\;\mathcal{I}_{n}e^{F_{\mathcal{N}}},\qquad
F_{\mathcal{N}}=\sum_{i<j}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}q_{i\mathcal{I}}q_{j}^{\mathcal{I}}-\frac{1}{2}\sum_{i=1}^{n}\braket{\xi_{i}v_{i}}q_{i\mathcal{I}}q_{i}^{\mathcal{I}}\,,$
(2.7)
$\displaystyle\mathrm{d}\mu_{n}^{\text{pol}}\\!:=\\!\frac{\prod_{l}\mathrm{d}\sigma_{l}\mathrm{d}^{2}u_{l}\mathrm{d}^{2}v_{l}}{\text{vol}(\text{SL}(2,\mathbb{C})_{\sigma}\\!\times\\!\text{SL}(2,\mathbb{C})_{u})}\\!\\!\prod_{r=1}^{n}\\!\bar{\delta}(\\!\braket{v_{r}\epsilon_{r}}\\!-\\!1)\bar{\delta}^{4}(\\!\braket{u_{r}\lambda^{A}}\\!-\\!\braket{v_{r}\kappa_{r}^{A}}),$

where compared to (2.5) the measure has lost the delta function in the
fermionic fields, replaced by the exponential factor exp($F_{\mathcal{N}}$).

Now the only difference in the tree amplitude to the original one [7, 5] is
that a second occurrence of det${}^{\prime}H$ is replaced by
det${}^{\prime}G$. The elements in $G$ might look surprising as the terms
containing $\epsilon^{A}$ and the ones containing $q^{\mathcal{I}}$ seem to
refer to incompatible quantities, but one has to keep in mind that the
amplitude in (2.7) is a sum of component amplitudes describing different
physical situations, and the sum of quantities in elements from the matrix $G$
are always split up among component amplitudes. In other words, (2.7) should
be viewed as a power series in the $q$ variables with the coefficients being
component amplitudes describing different physics, although some of them are
just permutations of each other. In terms of natural units the elements in $G$
containing $\epsilon^{A}$ should be viewed as multiplied with the
gravitational coupling constant $\kappa\\!=\\!\sqrt{G_{N}}$, i.e. the
$q^{\mathcal{I}}$-terms in $G$ effectively lower the power in gravitational
couplings, in contrast to the terms in exp($F_{\mathcal{N}})$. One important
benefit of (2.7) is that component amplitudes exist that contain a term
$q_{i}^{\mathcal{I}}q_{j\mathcal{I}}$ even when $\braket{u_{i}u_{j}}\\!=\\!0$,
allowing for more realistic situations, particularly between massless
particles of the same helicity when the exponential factor
exp($F_{\mathcal{N}}$) cannot contribute because of
$\braket{u_{i}u_{j}}\\!=\\!0$.

###  2.1 Consistency Checks: Einstein-Yang-Mills Amplitudes and Massless Limit

The purpose of this subsection is to check whether some component amplitudes
are compatible with known amplitudes that describe interesting physics. One
example are Einstein-Yang-Mills (EYM) amplitudes.

First it is to be noted that all off-diagonal elements in the $G$ matrix come
from the auxiliary spinor fields and only the diagonal elements reflect
contractions of the main supertwistors that are relevant for physical
interpretation. Collect all diagonal elements with non-zero $q$-dependence and
consider all component amplitudes that contain the product of these diagonal
elements with the contributions from $\lambda_{A}$ removed and all other
component amplitudes with the same (maximum) power in $q$ variables from
det${}^{\prime}G$. Consider the case where all $q$-products are connected to
each other to form a single trace (loop) of length $m$. Then the component
amplitude arises from the determinant of the product of two matrices $\hat{H}$
and $\hat{G}$, with $(n\\!-\\!m)\\!\times\\!(n\\!-\\!m)$ matrix $\hat{H}$
having all rows and columns containing a $q$ dependence removed and
$m\\!\times\\!m$ matrix $\hat{G}$ containing only the $q$ dependent elements
that have been removed from $\hat{H}$. Obviously $\hat{G}$ inherits co-rank 2
from the original matrix $G$, and $\hat{H}$ is no longer singular. Therefore,
for a single trace amplitude:

$\displaystyle\mathcal{A}_{n,1\text{tr}}^{\text{tree}}$
$\displaystyle=\int\mathrm{d}\mu_{n}^{\text{pol}}\frac{\text{det}H^{[ij]}_{[kl]}}{\braket{u_{i}u_{j}}\braket{u_{k}u_{l}}}\;\text{det}\hat{H}\frac{\text{det}\hat{G}^{[pr]}_{[st]}}{\braket{u_{p}u_{r}}\braket{u_{s}u_{t}}}e^{F_{\mathcal{N}}}.$
(2.8)

By pulling down
$\frac{\braket{u_{p}u_{r}}}{\sigma_{pr}}q_{p\mathcal{I}}q_{r}^{\mathcal{I}}$
and
$\frac{\braket{u_{s}u_{t}}}{\sigma_{st}}q_{s\mathcal{I}}q_{t}^{\mathcal{I}}$
from the exponential factor $F_{\mathcal{N}}$ to close the trace (also
recognizing that the term
$\braket{\xi_{i}v_{i}}q_{i\mathcal{I}}q_{i}^{\mathcal{I}}$ in
$F_{\mathcal{N}}$ vanishes for a particle referred to by a singly indexed $q$
and does not contribute) one gets a Parke-Taylor(PT) factor with $q$-dependent
numerator $N(q)$:

$\displaystyle\mathcal{A}_{n,1\text{tr}}^{\text{tree}}$
$\displaystyle=\int\mathrm{d}\mu_{n}^{\text{pol}}\text{det}^{\prime}H\;\text{det}\hat{H}\,\text{PT}\,e^{F_{\mathcal{N}}},\quad\text{PT}\\!=\\!\sum_{\text{Perm}}\frac{N(q)}{\sigma_{m1}\prod_{i=1}^{m-1}\sigma_{i\,i+1}}.$
(2.9)

This is the single trace ’massive EYM’ amplitude with no restrictions on the
individual masses. When $m=n$, $\hat{H}$ is empty and (2.9) is then a SYM
amplitude on the Coulomb branch [7, 10], i.e. the current model covers both
supergravity and SYM. But it should be noted that by being embedded in
$\mathcal{N}=8$ supergravity this SYM has more supersymmetry than the
conventional SYM.

Concerning multiple traces the same procedure of removing all non-zero
$q$-dependent elements in the matrix $G$ still would result in a degeneration
of $G$ into a product of two matrices $\hat{H}$ and $\hat{G}$, but $\hat{G}$
would have a co-rank of twice the number of traces. Therefore, instead of
removing all non-zero $q$-dependent elements in $G$ one should keep 2 rows and
2 columns for all traces except one and sum over all ways of doing this. In
each summand $\hat{H}$ gets all $q$-dependence removed and is then a non-
singular $t\\!\times\\!t$ matrix with
$t\\!=\\!n\\!-\\!2\\!-\\!\sum_{k}(l_{k}\\!-\\!2)$, where $l_{k}$ is the length
of trace $k$, whereas $\hat{G}$ again has co-rank 2. To get a PT factor for
each trace one needs to pull down two elements from the exponential factor
$F_{\mathcal{N}}$ for each trace corresponding either to the two rows and
columns kept in $\hat{H}$ or to the two rows and columns removed from
$\hat{G}$ like for the single trace.

Another consistency check is to take the massless limit. This is a trivial
generalization of section 6 in [7]. The amplitude for $k$ negative helicity
and $n\\!-\\!k$ positive helicity massless particles becomes, using the
standard notation
$\braket{\epsilon_{i}\epsilon_{j}}\\!=\\!\varepsilon^{\alpha\beta}\epsilon_{i\alpha}\epsilon_{j\beta}\\!=\\!-\\!\braket{\epsilon_{j}\epsilon_{i}},[\epsilon_{i}\epsilon_{j}]\\!=\\!\varepsilon^{\dot{\alpha}\dot{\beta}}\epsilon_{i\dot{\alpha}}\epsilon_{j\dot{\beta}}\\!=\\!-[\epsilon_{j}\epsilon_{i}]$
(not to be confused with similar usage of $\braket{}$ for little group
contractions)

$\displaystyle\mathcal{A}_{n,k}^{4d,\text{tree}}$
$\displaystyle=\int\mathrm{d}\mu^{4d}_{n,k}\;\text{det}^{\prime}G_{-}\;\text{det}^{\prime}G_{+}\;e^{F^{k}_{\mathcal{N}}},\qquad\text{det}^{\prime}G_{\pm}=\frac{\text{det}G_{\pm}^{[st]}}{u_{s}u_{t}}\,,$
(2.10)
$\displaystyle\mathrm{d}\mu^{4d}_{n,k}\\!:=\\!\frac{\prod_{l=1}^{n}\mathrm{d}\sigma_{l}\mathrm{d}u_{l}/u_{l}}{\text{vol}(\text{GL}(2,\mathbb{C}))}\\!\\!\prod_{i=1}^{k}\bar{\delta}^{2}(u_{i}\tilde{\lambda}_{\dot{\alpha}}(\sigma_{i})\\!-\\!\frac{\tilde{\kappa}_{i\dot{\alpha}}}{\epsilon_{i}})\\!\prod_{p=k+1}^{n}\bar{\delta}^{2}(u_{p}\lambda_{\alpha}(\sigma_{p})\\!-\\!\frac{\kappa_{p\alpha}}{\tilde{\epsilon}_{p}}),$
$\displaystyle
F^{k}_{\mathcal{N}}=\sum^{k}_{i=1}\sum^{n}_{j=k+1}\frac{u_{i}u_{j}}{\sigma_{ij}}q_{i\mathcal{I}}q_{j}^{\mathcal{I}},\quad\lambda_{\alpha}(\sigma)\\!=\\!\sum_{i=1}^{k}\frac{u_{i}\epsilon_{i\alpha}}{\sigma\\!-\\!\sigma_{i}},\quad\tilde{\lambda}_{\dot{\alpha}}(\sigma)\\!=\\!\sum_{i=k+1}^{n}\frac{u_{i}\epsilon_{i\dot{\alpha}}}{\sigma\\!-\\!\sigma_{i}},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\;\;\,\eta_{I}(\sigma)\\!=\\!\sum_{i=1}^{k}\frac{u_{i}q_{iI}}{\sigma\\!-\\!\sigma_{i}},\quad\tilde{\eta}_{\dot{I}}(\sigma)\\!=\\!\sum_{i=k+1}^{n}\frac{u_{i}q_{i\dot{I}}}{\sigma\\!-\\!\sigma_{i}},$
$\displaystyle
G_{-}^{ij}=\frac{\braket{\epsilon_{i}\epsilon_{j}}\\!+\\!q_{iI}q_{j}^{I}}{\sigma_{ij}},\quad
G_{-}^{ii}=-\frac{\braket{\epsilon_{i}\lambda(\sigma_{i})}+q_{i}^{I}\eta_{I}(\sigma_{i})}{u_{i}},\quad
1\leq i,j\leq k,$ $\displaystyle
G_{+}^{ij}=\frac{[\tilde{\epsilon}_{i}\tilde{\epsilon}_{j}]\\!+\\!q_{i\dot{I}}q_{j}^{\dot{I}}}{\sigma_{ij}},\,\quad
G_{+}^{ii}=-\frac{[\tilde{\epsilon}_{i}\tilde{\lambda}(\sigma_{i})]\,+q_{i}^{\dot{I}}\tilde{\eta}_{\dot{I}}(\sigma_{i})}{u_{i}},\quad
k+1\leq i,j\leq n,$

where $G^{[st]}$ denotes a matrix $G$ with row $s$ and column $t$ removed. Up
to the additional entries in the matrices coming from fermionic fields, this
is the tree amplitude of [13, 7]. The direct massless limit of the single
trace EYM amplitude (2.8) is obtained similarly:

$\mathcal{A}_{n,k,1\text{tr}}^{4d,\text{tree}}=\int\mathrm{d}\mu^{4d}_{n,k}\;\text{det}\hat{H}_{-}\text{det}\hat{H}_{+}\;\text{PT}\;e^{F^{k}_{\mathcal{N}}},$

where $\hat{H}_{\pm}$ is the matrix obtained from $G_{\pm}$ by removing all
rows and columns containing non-vanishing fermionic entries. This is exactly
the single trace EYM amplitude of [12] if there the reduced determinants are
calculated by removing the last row and column representing ’squeezed’ gluons.
It is interesting to notice that trying to get EYM amplitudes as component
amplitudes directly from (2.10) runs into the limitation that all positive
helicity gluons must be adjacent to each other. The same obstruction occurred
in the massless models of [14].

###  2.2 Factorization of Tree Amplitudes

Again this becomes a straightforward exercise by building upon work done in
section 7 of [7]. There it has been shown that the measure
$\mathrm{d}\mu_{n}^{\text{pol}}$, the exponential factor
$e^{F_{\mathcal{N}}}$, and the reduced determinants det${}^{\prime}H$ without
fermionic elements factorize correctly. Therefore, what remains to be shown is
proper factorization of the reduced determinant of $G$ with fermionic
contributions included. One observation is that the matrix elements of $G$ can
be rewritten as, when using the ’super’ notation
$\epsilon_{j}^{\mathcal{A}}=(\epsilon_{j}^{A},q_{i}^{\mathcal{I}}),\zeta^{a}_{\mathcal{A}}=(\lambda^{a}_{A},\eta^{a}_{\mathcal{I}})$:

$G_{ij}=\frac{\epsilon_{i\mathcal{A}}\epsilon_{j}^{\mathcal{A}}}{\sigma_{ij}},\quad
u^{a}_{i}G_{ii}=-\zeta^{a}_{\mathcal{A}}\epsilon_{i}^{\mathcal{A}},$

with the ’super’ spinor fulfilling the polarized scattering equations (2.3) in
unified notation

$\braket{v_{i}\kappa_{i}^{\mathcal{A}}}=\braket{u_{i}\zeta^{\mathcal{A}}(\sigma_{i})},\quad\zeta^{a}_{\mathcal{A}}(\sigma_{i})=\sum_{j\neq
i}\frac{u^{a}_{j}\epsilon_{j\mathcal{A}}}{\sigma_{ij}},$

where $\kappa_{i}^{\mathcal{A}}=(\kappa_{i}^{A},\zeta_{i}^{\mathcal{I}})$.
Then the proof of factorization of the reduced determinants in [7] can be
repeated basically word by word by replacing
$(\epsilon^{A},\lambda_{A},\kappa^{A})$ with
$(\epsilon^{\mathcal{A}},\zeta_{\mathcal{A}},\kappa^{\mathcal{A}})$.

More effort will be spent on the proof of factorization at the one-loop level
in the next section.

##  3 Loop Amplitudes and Factorization

When calculating multi-loop amplitudes in QFT, this is typically done assuming
generalized unitarity [4]. On the other hand, in this work using a model with
a Lagrangian, consistency requires that this model exhibits proper
factorization of scattering amplitudes to justify using generalized unitarity.

Before evaluating one-loop amplitudes it is interesting to look at partition
functions. The model has two groups of 16 bosonic and 24 fermionic spinors
each, one representing twistors and one representing dual twistors, 8 bosonic
and 6 fermionic ghost-antighost pairs, and additionally 6 bosonic scalars or
ghost-antighost pairs, so in balance the partition function is the same as for
the RNS ambitwistor string [16] when the spin structures of the two particle
groups are taken independently, but is modular invariant even when they are
selected to be the same, because of the modularity of
$\eta(\tau)^{-24}(\theta_{2}(\tau)^{8}+\theta_{3}(\tau)^{8}+\theta_{4}(\tau)^{8})$
of weight -8. On the other hand, because of Jacobi’s abstruse identity
$\theta_{2}(\tau)^{4}-\theta_{3}(\tau)^{4}+\theta_{4}(\tau)^{4}=0$ in the
former case one obtains a zero cosmological constant at the one-loop level
like for RNS, which is preferable.

Moving on to the one-loop scattering amplitudes themselves, the scattering
equations (2.3) change to

$\displaystyle\braket{v_{r}\kappa_{r}^{A}}\\!=\\!\braket{u_{r}\lambda^{A}(\sigma_{r})},$
$\displaystyle\lambda^{a}_{A}(\sigma_{r})=\lambda^{a}_{0A}+\sum_{l\neq
r}u^{a}_{l}\epsilon_{lA}\frac{S_{\alpha}(\sigma_{r},\sigma_{l};\tau)}{\sqrt{\mathrm{d}\sigma_{r}}\sqrt{\mathrm{d}\sigma_{l}}},$
(3.1)
$\displaystyle\braket{v_{r}\zeta_{r}^{\mathcal{I}}}=\\!\braket{u_{r}\eta^{\mathcal{I}}(\sigma_{r})},$
$\displaystyle\eta^{a}_{\mathcal{I}}(\sigma_{r})=\eta^{a}_{0\mathcal{I}}+\sum_{l\neq
r}u^{a}_{l}q_{l\mathcal{I}}\frac{S_{\alpha}(\sigma_{r},\sigma_{l};\tau)}{\sqrt{\mathrm{d}\sigma_{r}}\sqrt{\mathrm{d}\sigma_{l}}}.$

where $\lambda^{a}_{0A}$ and $\eta^{a}_{0\mathcal{I}}$ are zero modes on an
odd spin structure $\alpha=1$ but generally do not exist on even spin
structures $\alpha=2,3,4$, and $S_{\alpha}(\sigma,\tau)$ is the Szegö kernel
for spin structure $\alpha$. Zero modes of the auxiliary fields
$\rho^{a}_{1,2}$ and $\tau^{a}$ are not allowed as these fields do not
contribute to the spectrum and, therefore, are only allowed as virtual fields,
so their zero modes are omitted, but all propagators in the amplitudes,
including the ones of the auxiliary fields, are changed to Szegö kernels.

One important change for the one-loop level is that there is only one zero
mode of the $c$ ghost left and none for the ghosts of the supersymmetric gauge
symmetries, i.e. only one fixed vertex operator with regard to the conformal
symmetry.

It follows that for even spin structures the amplitude looks like (2.4) with
SL$(2,\mathbb{C})_{\sigma}\\!\times\\!$ SL$(2,\mathbb{C})_{u}$ replaced with
GL(1)σ, all propagators $1/{\sigma_{rl}}$ with Szegö kernels
$S_{\alpha}(\sigma_{r},\sigma_{l};\tau)\\!/$
$\\!(\sqrt{\mathrm{d}\sigma_{r}}\sqrt{\mathrm{d}\sigma_{l}})$, reduced
determinants with their full versions, and with an additional integration over
the torus modulus $\tau$. Because the partition function and the Szegö kernels
are modular invariant so is the amplitude, like for the RNS ambitwistor
string. On the other hand, it becomes immediately clear that all these
determinants still have co-rank 2 and vanish, i.e. the even spin structures do
not contribute to the one-loop scattering amplitude. Although this seems
surprising, one should keep in mind, that in contrast to the conventional RNS
string the spectrum of the current model is the same for the NS and R sector
[14], so it is understandable that only the odd spin structure contributes a
non-zero amplitude, saved from vanishing by the zero modes, as will be now
shown explicitly.

The one-loop amplitude for odd spin structure becomes

$\displaystyle\mathcal{A}_{n}^{\text{1-loop}}$
$\displaystyle=\delta^{4}\\!\\!\left(\sum_{i=1}^{n}\braket{\kappa_{i}^{\alpha}\tilde{\kappa}_{i}^{\dot{\alpha}}}\right)\int\mathrm{d}\mu^{\text{pol}}_{n|\text{1-loop}}\text{det}H\;\text{det}G\;e^{\frac{1}{2}\\!\braket{\eta_{0\mathcal{I}}\eta_{0}^{\mathcal{I}}}+F^{\text{1-loop}}_{\mathcal{N}}}$
(3.2)
$\displaystyle\mathrm{d}\mu^{\text{pol}}_{n|\text{1-loop}}\\!:=\\!\frac{\prod_{l=1}^{n}\mathrm{d}\sigma_{l}\mathrm{d}^{2}u_{l}\mathrm{d}^{2}v_{l}\;\mathrm{d}\tau}{\text{vol}(\text{GL}(1)_{\sigma})}\mathrm{d}^{8}\\!\lambda_{0A}^{a}\mathrm{d}^{2\mathcal{N}}\\!\eta_{0\mathcal{I}}^{a}\\!\prod_{r=1}^{n}\\!\bar{\delta}(\\!\braket{v_{r}\epsilon_{r}}\\!-\\!1\\!)\bar{\delta}^{4}(\\!\braket{u_{r}\lambda^{A}}\\!-\\!\braket{v_{r}\kappa_{r}^{A}}\\!),$
$\displaystyle
F_{\mathcal{N}}=\sum_{i<j}\braket{u_{i}u_{j}}q_{i\mathcal{I}}q_{j}^{\mathcal{I}}\,\frac{S_{\alpha}(\sigma_{i},\sigma_{j};\tau)}{\sqrt{\mathrm{d}\sigma_{i}}\sqrt{\mathrm{d}\sigma_{j}}}+\sum_{i=1}^{n}\braket{u_{i}\eta_{0\mathcal{I}}}q_{i}^{\mathcal{I}}-\frac{1}{2}\sum_{i=1}^{n}\braket{\xi_{i}v_{i}}q_{i\mathcal{I}}q_{i}^{\mathcal{I}},$
$\displaystyle
H_{ij}=\epsilon_{iA}\epsilon_{j}^{A}\frac{S_{\alpha}(\sigma_{i},\sigma_{j};\tau)}{\sqrt{\mathrm{d}\sigma_{i}}\sqrt{\mathrm{d}\sigma_{j}}},\qquad\qquad\quad\,u_{i}^{a}H_{ii}=-\lambda_{A}^{a}\epsilon_{i}^{A},$
$\displaystyle
G_{ij}=(\epsilon_{iA}\epsilon_{j}^{A}+q_{i\mathcal{I}}q_{j}^{\mathcal{I}})\frac{S_{\alpha}(\sigma_{i},\sigma_{j};\tau)}{\sqrt{\mathrm{d}\sigma_{i}}\sqrt{\mathrm{d}\sigma_{j}}},\quad
u_{i}^{a}G_{ii}=-\\!\lambda_{A}^{a}\epsilon_{i}^{A}-\eta^{a}_{\mathcal{I}}q_{i}^{\mathcal{I}},$

where $\lambda^{a}_{A}$ and $\eta^{a}_{\mathcal{I}}$ are to be inserted from
(3.1) and momentum conservation is expressed explicitly with a delta function,
considered to come from the integration over the zero modes $\mu^{aA}_{0}$. An
additional factor
exp($\frac{1}{2}\\!\braket{\eta_{0\mathcal{I}}\eta_{0}^{\mathcal{I}}}$) has
been added to exp($F_{\mathcal{N}}$) to bring the integration over
$\eta^{a}_{0\mathcal{I}}$ into normal form [17]222A similar exponential factor
appears in [12, 10].. It could also be regarded as substitute for the same
effect as a delta function arising from the integration over the zero modes
$\tilde{\eta}^{a\mathcal{I}}_{0}$ when working in the LG representation, cf.
(2.4).

Including the zero modes of $\mu^{aA}_{0}$ and
$\tilde{\eta}^{a\mathcal{I}}_{0}$ which already have been integrated out,
there are 16 eight bosonic and 32 fermionic spinor zero modes resulting in a
modular weight of -8, making the amplitude modular invariant and UV-finite,
when taking the 8 bosonic contributions from the momentum integrations over
the loop and disregarded zero modes of the $bc$ ghosts into account, like for
the partition function.

###  3.1 Nonseparating Degeneration

This section checks whether the one-loop amplitude (3.2) factorizes properly
for a nonseparating degeneration of the worldsheet. Pinching a nonseparating
cycle occurs in the limit $q=e^{2\pi i\tau}\rightarrow 0$. One obstruction to
dealing with this limit by simply taking the residue at 0 of the integrand as
a function in $q$ is that the integrand is not an analytic function in $q$
because the torus Szegö kernel $S_{1}(\sigma_{1},\sigma_{2};\tau)$ contains an
Im($\tau$)-dependent term:

$S_{1}(\sigma_{1},\sigma_{2};\tau)=\left(\frac{\theta_{1}^{{}^{\prime}}\\!(\sigma_{12},\tau)}{\theta_{1}(\sigma_{12},\tau)}+4\pi\frac{\mathrm{Im}(\sigma_{12})}{\mathrm{Im}(\tau)}\right)\,.$

The behavior of $S_{1}$ can be examined by modeling the torus as a Riemann
sphere with a handle far away from all punctures, with ends located at two
reference points $\sigma_{+}$ and $\sigma_{-}$, i.e. these two points are
added as additional punctures, modeled in terms of local coordinates by

$(\sigma-\sigma_{+})(\sigma-\sigma_{-})=q.$

Then, because according to [18] all Szegö kernels are holomorphic in $q$ for
small enough $q$, the integrand in (3.2) becomes a manifestly meromorphic
function in $q$.

The asymptotic behavior of the Szegö kernel is [19, 20]

$\displaystyle
S_{1}(\sigma_{ij};\tau)\sqrt{\mathrm{d}\sigma_{i}\mathrm{d}\sigma_{j}}\rightarrow$
$\displaystyle\frac{\sqrt{\mathrm{d}z_{i}\mathrm{d}z_{j}}}{2z_{ij}}$
$\displaystyle\left(\sqrt{\frac{z_{i-}z_{j+}}{z_{i+}z_{j-}}}+\sqrt{\frac{z_{i+}z_{j-}}{z_{i-}z_{j+}}}\right)$
$\displaystyle+\,O(q),$ (3.3)

in terms of the coordinates $z=e^{2\pi i(\sigma\\!-\\!\frac{\tau}{2})}$ on the
Riemann sphere. As $q\rightarrow 0$, the handle containing $z_{+}$ and $z_{-}$
stretches and pinches, such that the values of the square roots in (3.3) can
be replaced by 1 up to $O(q)$, i.e. $S_{1}$ becomes just the genus 0
propagator in this limit.

On the Riemann sphere the torus zero modes $\lambda_{0}$ and $\eta_{0}$ are
lifted up to fields $\lambda_{0}/z$ and $\eta_{0}/z$. To proceed, the points
$z_{+}$ and $z_{-}$ are chosen temporarily as 0 and $\infty$. Then the right
hand side of the scattering equations (3.1) for $\lambda$ can be rewritten as,
in abuse of notation changing back from $z$ to $\sigma$,

$\lambda^{aA}(\sigma)=\frac{\lambda^{aA}_{0}}{\sigma}+\sum_{l}\frac{u^{a}_{l}\epsilon^{A}_{l}}{\sigma-\sigma_{l}}+O(q)=\frac{\braket{u^{a}_{+}\epsilon^{A}_{+}+u^{a}_{-}\epsilon^{A}_{-}}}{\sigma}+\sum_{l}\frac{u^{a}_{l}\epsilon^{A}_{l}}{\sigma-\sigma_{l}}+O(q),$

where $\lambda^{aA}_{0}$ has been expanded into a basis of little spinors
$u^{a}_{\pm}$ with $\braket{u_{+}u_{-}}\\!\neq\\!0$. Now reverting to general
$\sigma_{+}$ and $\sigma_{-}$, designating $u^{a}_{\pm}\epsilon^{A}_{\pm}$ to
belong to $\sigma_{\pm}$ by requiring that
$\braket{u_{\pm}\lambda^{A}(\sigma_{\pm})}\\!<\\!\infty$, defining a spinor
basis $\epsilon^{a}_{\pm}$ with $\braket{\epsilon_{+}\epsilon_{-}}\\!=\\!1$
and momenta $\kappa^{a}_{\pm A}$ with
$\kappa^{1}_{+A}=\kappa^{1}_{-A},\kappa^{2}_{+A}=-\kappa^{2}_{-A}$ such that
$\epsilon^{A}_{\pm}\\!=\\!\braket{\epsilon_{\pm}\kappa^{A}_{\pm}}$, and
finally viewing $\sigma_{\pm}$ as additional punctures $\sigma_{n\\!+\\!1}$
and $\sigma_{n\\!+\\!2}$ one gets

$\lambda^{aA}(\sigma)=\sum_{l=1}^{n+2}\frac{u^{a}_{l}\epsilon^{A}_{l}}{\sigma-\sigma_{l}}+O(q).$

A similar procedure for $\eta_{0}$ gives

$\eta^{a\mathcal{I}}(\sigma)\\!=\\!\sum_{l=1}^{n+2}\frac{u^{a}_{l}q^{\mathcal{I}}_{l}}{\sigma-\sigma_{l}}\\!+\\!O(q)\;\;\text{with}\;q^{\mathcal{I}}_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}\\!=\\!\braket{\epsilon_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}\zeta^{\mathcal{I}}_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}},\zeta^{1}_{n\\!+\\!1\mathcal{I}}\\!=\\!\zeta^{1}_{n\\!+\\!2\mathcal{I}},\zeta^{2}_{n\\!+\\!1\mathcal{I}}\\!=\\!-\zeta^{2}_{n\\!+\\!2\mathcal{I}}.$
(3.4)

This way scattering equations for $n\\!+\\!2$ particles have been obtained, so
far fulfilled by the first $n$ particles. Now the measure needs to be
modified. This is done by setting

$\displaystyle\delta^{4}\\!\\!\left(\sum_{i=1}^{n}\braket{\kappa_{i}^{\alpha}\tilde{\kappa}_{i}^{\dot{\alpha}}}\right)$
$\displaystyle\frac{\mathrm{d}^{8}\\!\lambda_{0A}^{a}\,\mathrm{d}^{2\mathcal{N}}\\!\eta_{0\mathcal{I}}^{a}}{\text{vol}(\text{GL}(1)_{\sigma})}\frac{\mathrm{d}q}{q}\\!=\\!\int\\!\\!\frac{\prod_{l=n\\!+\\!1}^{n\\!+\\!2}\\!\mathrm{d}\sigma_{l}\,\mathrm{d}^{2}u_{l}\,\mathrm{d}^{2}v_{l}}{\text{vol}(\text{SL}(2,\mathbb{C})_{\sigma}\\!\times\\!\text{SL}(2,\mathbb{C})_{u})}$
$\displaystyle\frac{\prod_{l=n\\!+\\!1}^{n\\!+\\!2}\mathrm{d}^{4}\epsilon_{l}^{A}\,\mathrm{d}^{\mathcal{N}}\\!q_{l}^{\mathcal{I}}}{\braket{u_{n\\!+\\!1}u_{n\\!+\\!2}}^{4}}\prod_{r=n\\!+\\!1}^{n\\!+\\!2}\\!\bar{\delta}(\\!\braket{v_{r}\epsilon_{r}}\\!-\\!1)\bar{\delta}^{4}(\\!\braket{u_{r}\lambda^{A}}\\!-\\!\braket{v_{r}\kappa_{r}^{A}}).$

That the left hand side is equal to the right hand side can be seen as
follows: on the tree level two more $\sigma_{l}$ values can be fixed (as was
done with $\sigma_{n\\!+\\!1}$ and $\sigma_{n\\!+\\!2}$) and three of the
$(u^{a}_{l},v^{a}_{l})$ variables (e.g. by choosing
$u^{a}_{n\\!+\\!1}\\!=\\!(1,0),u^{a}_{n\\!+\\!2}=(u_{n\\!+\\!2},1)$),
increasing the denominator to
vol(SL(2,$\mathbb{C})_{\sigma}\\!\times$SL(2,$\mathbb{C})_{u}$) and fixing 5
of the 10 delta functions on the right. On the left
$\mathrm{d}^{8}\\!\lambda_{0A}^{a}\mathrm{d}q/q$ is equivalent to
$\mathrm{d}^{4}l_{\mu}\delta(l^{2}-m^{2})\mathrm{d}q/q\sim-(\mathrm{d}^{4}l_{\mu}/(l^{2}-m^{2}))\mathrm{d}q\bar{\delta}(q)$,
and because momentum conservation follows from the scattering equations for
the $n\\!+\\!2$ particles the remaining 5 delta functions on both sides
balance each other out. Finally
$\braket{u_{n\\!+\\!1}u_{n\\!+\\!2}}^{4-\mathcal{N}}$ =
$\braket{u_{n\\!+\\!1}u_{n\\!+\\!2}}^{-4}$ is the Jacobian going from
$\mathrm{d}^{8}\\!\lambda_{0}\,\mathrm{d}^{2\mathcal{N}}\\!\eta_{0}$ to
$\prod_{l}\mathrm{d}^{4}\epsilon_{l}\,\mathrm{d}^{\mathcal{N}}\\!q_{l}$.
$\mathcal{N}=8$ is crucial here.

It remains to check $F_{\mathcal{N}}$. Inserting

$\eta_{0}(\sigma)=u_{n\\!+\\!1}q_{n\\!+\\!1}/(\sigma-\sigma_{n\\!+\\!1})+u_{n\\!+\\!2}q_{n\\!+\\!2}/(\sigma-\sigma_{n\\!+\\!2})$
(3.5)

from (3.4) into $F_{\mathcal{N}}$ in (3.2) gives $F_{\mathcal{N}}$ of the tree
amplitude (2.7) up to a missing piece:

$\displaystyle F_{\mathcal{N}}=$ $\displaystyle\sum_{1\leq i<j\leq
n\\!+\\!2}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}q_{i\mathcal{I}}q_{j}^{\mathcal{I}}-\frac{1}{2}\sum_{i=1}^{n\\!+\\!2}\braket{\xi_{i}v_{i}}q_{i\mathcal{I}}q_{i}^{\mathcal{I}}-C$
$\displaystyle
C=\frac{\braket{u_{n\\!+\\!1}u_{n\\!+\\!2}}}{\sigma_{n\\!+\\!1n\\!+\\!2}}q_{n\\!+\\!1\mathcal{I}}q_{n\\!+\\!2}^{\mathcal{I}}-\frac{1}{2}\sum_{i=n\\!+\\!1}^{n\\!+\\!2}\braket{\xi_{i}v_{i}}q_{i\mathcal{I}}q_{i}^{\mathcal{I}}.$

The second term in C vanishes because of $\braket{\xi
v}\\!\epsilon^{a}\\!=\\!v^{a}\\!-\\!\xi^{a}$ and the special relationship
between values for $\zeta_{n\\!+\\!1}$ and $\zeta_{n\\!+\\!2}$ in (3.4). The
first term can be recognized as arising from
$\frac{1}{2}\\!\braket{\eta_{0\mathcal{I}}\eta_{0}^{\mathcal{I}}}$ when going
from the torus to the Riemann sphere:

$\frac{1}{2}\\!\braket{\eta_{0\mathcal{I}}\eta_{0}^{\mathcal{I}}}\xrightarrow{\eqref{eta0RiemSph}}\frac{1}{2}(q_{n\\!+\\!1}^{\mathcal{I}}\\!\braket{u_{n\\!+\\!1}\eta_{0\mathcal{I}}(\sigma_{n\\!+\\!1})}+q_{n\\!+\\!2}^{\mathcal{I}}\\!\braket{u_{n\\!+\\!2}\eta_{0\mathcal{I}}(\sigma_{n\\!+\\!2})})=\frac{\braket{u_{n\\!+\\!1}u_{n\\!+\\!2}}}{\sigma_{n\\!+\\!1n\\!+\\!2}}q_{n\\!+\\!1\mathcal{I}}q_{n\\!+\\!2}^{\mathcal{I}}.$

This proves that the one-loop amplitude (3.2) factorizes correctly into a tree
amplitude (2.7) where two additional punctures are connected through a
complete set of intermediate physical states:

$\displaystyle\mathcal{A}_{n}^{\text{1-loop}}=$
$\displaystyle\int\prod_{l=n\\!+\\!1}^{n\\!+\\!2}\mathrm{d}^{4}\\!\epsilon_{l}^{A}\,\mathrm{d}^{\mathcal{N}}\\!q_{l}^{\mathcal{I}}\;\mathcal{A}_{n+2}^{\text{tree}},\qquad\quad\braket{\epsilon_{n+1}\epsilon_{n+2}}=1,$
$\displaystyle\epsilon^{A}_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}\\!=\\!\braket{\epsilon_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}\kappa^{A}_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}},\kappa^{1}_{n\\!+\\!1A}\\!=\\!\kappa^{1}_{n\\!+\\!2A},\kappa^{2}_{n\\!+\\!1A}\\!=\\!-\kappa^{2}_{n\\!+\\!2A},$
$\displaystyle
q^{\mathcal{I}}_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}\\!=\\!\braket{\epsilon_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}\zeta^{\mathcal{I}}_{n\\!+\\!\genfrac{}{}{0.0pt}{}{1}{2}}},\,\zeta^{1}_{n\\!+\\!1\mathcal{I}}=\\!\zeta^{1}_{n\\!+\\!2\mathcal{I}},\;\zeta^{2}_{n\\!+\\!1\mathcal{I}}=\\!-\zeta^{2}_{n\\!+\\!2\mathcal{I}}.$

###  3.2 Separating Degeneration

Pinching a separating cycle involves the degeneration of the Riemann surface
$\Sigma$ of genus $g\\!=\\!g_{L}\\!+\\!g_{R}$ into a surface $\Sigma_{L}$ of
genus $g_{L}$ and a surface $\Sigma_{R}$ of genus $g_{R}$ and can again be
modeled in terms of local coordinates by

$(\sigma-\sigma_{L})(\sigma-\sigma_{R})=q\,,$

where $\sigma_{L}$ is an extra puncture on $\Sigma_{L}$ and $\sigma_{R}$ one
on $\Sigma_{R}$ with their neighborhoods sewed together and both points
identified in the pinching limit $q\rightarrow 0$.

Proper factorization requires to show that during such a separating
degeneration an $n$-particle amplitude on $\Sigma$ can be represented as an
$n_{L}\\!+\\!1$-particle amplitude on $\Sigma_{L}$ and an
$n_{R}\\!+\\!1$-particle amplitude on $\Sigma_{R}$ with
$n\\!=\\!n_{L}\\!+\\!n_{R}$ and additional punctures $\sigma_{L}$ and
$\sigma_{R}$ which are connected through insertion of a complete set of
physical states.

First it needs to be recognized that when $\Sigma$ has an odd spin structure,
then the spin structure of $\Sigma_{L}$ can only be odd if $\Sigma_{R}$ has
even spin structure or genus $g_{R}\\!=\\!0$ and vice versa. Therefore,
because the beginning of this section has shown that one-loop amplitudes
vanish on even spin structures and the next subsection will argue that the
same is generally true for all loop amplitudes, a separating degeneration can
only occur when one side is a tree amplitude.

Now proper factorization for separate degeneration of a one-loop amplitude
into a tree and a one-loop amplitude can be simply demonstrated by first
factorizing the one-loop amplitude into a tree amplitude using nonseparating
factorization, applying factorization of tree amplitudes from subsection 2.2
with the two additional punctures from the loop amplitude necessarily ending
up on the same side, and then reversing the nonseparating degeneration for
that side, possibly with a reduced number of punctures.

###  3.3 Higher Loops and Cosmological Constant

Looking first at partition functions at two or more loops, there is a strictly
positive number of zero modes of anti-ghosts and, therefore, at the minimum
one picture-changing operator for every supersymmetric gauge symmetry
resulting either in zero because of no contractions possible or in a matrix
with a determinant that vanishes on even spin structures like at the one-loop
level. Because the partition function on odd spin structures is always zero
due to the presence of a Dirac zero mode, the cosmological constant vanishes
at every order of perturbation theory (this assumes that at the one-loop level
the Jacobi abstruse identity can be applied on even spin structures). Of
course, non-perturbative effects like IR renormalization for massless
particles cannot be excluded and might give the cosmological constant a non-
vanishing value.

Multi-loop amplitudes with a non-zero number of vertex operators will again
generally vanish on even spin structures and develop a single zero mode for
the $\lambda^{a}_{A}$ and $\eta^{a}_{\mathcal{N}}$ fields on odd spin
structures. But assuming the validity of unitary factorization it can be seen
that not even all odd spin structures contribute. The total number of odd spin
structures can be calculated by gluing Riemann surfaces of genus 1 to an
existing surface of genus $g$:

$\\#_{\text{odd}}=\sum_{m\,\text{odd}}\binom{g}{m}1^{m}3^{g-m}=2^{g-1}(2^{g}-1)$
(3.6)

When $m$ is larger than 1 in a term of (3.6) there is no way to use
generalized unitarity and factorize down to a tree amplitude using
nonseparating degeneration without going through an even spin structure as
intermediate step, i.e. such a spin structure cannot contribute to multi-loop
amplitudes. It follows that only $g\,3^{g-1}\\!$ odd spin structures have non-
vanishing multi-loop amplitudes.

It is interesting to note that many of the results in this article also apply
to the original massive ambitwistor model of [7, 9, 10]. This includes zero
cosmological constant, non-vanishing amplitudes only on the same selected
subset of odd spin structures, and proper factorization of one-loop
amplitudes.

##  4 Compton Scattering Using Massive Twistor Strings

In this section it is important to note that in the following the usual
convention is used that in amplitudes all legs are taken to be outgoing, i.e.
incoming particles will have their momenta and helicities reversed.

The 4-point amplitude for the original massive ambitwistor model of [7, 9, 10]
is calculated to be [7]

$\begin{split}&M_{4}=\frac{\langle
1234\rangle^{4}}{(2p_{1}\\!\cdot\\!p_{2})(2p_{2}\\!\cdot\\!p_{3})(2p_{2}\\!\cdot\\!p_{4})}e^{F_{\mathcal{N}}},\\\
&\langle 1234\rangle=\varepsilon^{ABCD}e_{1A}e_{2B}e_{3C}e_{4D},\\\
&F_{\mathcal{N}}=\frac{1}{2}\sum_{i,j=1}^{4}\frac{\braket{u_{i}u_{j}}}{\sigma_{ij}}q_{i\,\mathcal{I}}q_{j}^{\mathcal{I}}-\frac{1}{2}\sum_{i=1}^{4}\braket{\xi_{i}v_{i}}q_{i\,\mathcal{I}}q_{i}^{\mathcal{I}}.\end{split}$
(4.1)

More details can be found in [7]333In [7] the amplitude is given for 6
dimensions, with $\langle 1234\rangle^{4}$ replaced by $\langle
1234\rangle^{2}[1234]^{2}$, and $F_{\mathcal{N}}$ with $F_{N}+\tilde{F}_{N}$,
but the contributions become equal via dimensional reduction..

The amplitude (4.1) is valid for 4 massive particles up to spin 2. It is now
specialized to the case of Compton scattering of a massless particle with
momenta $p_{2}\\!\rightarrow\\!k_{2},p_{3}\\!\rightarrow\\!k_{3}$ against a
heavy massive particle with mass $M$ much larger than the energy $\hbar\omega$
of the massless particle. Switching from the polarized formalism to a standard
little-group based amplitude, it can readily be seen that the amplitude
becomes either

$M_{4}^{+-}=\frac{(\langle\mathbf{4}3\rangle\,[\mathbf{1}2]-\langle\mathbf{1}3\rangle\,[\mathbf{4}2])^{4}}{(2p_{1}\\!\cdot\\!k_{2})(2k_{2}\\!\cdot\\!k_{3})(2k_{2}\\!\cdot\\!p_{4})}e^{F_{\mathcal{N}}}$
(4.2)

or

$M_{4}^{++}=\frac{\langle\mathbf{1}\mathbf{4}\rangle^{4}\,[23]^{4}}{(2p_{1}\\!\cdot\\!k_{2})(2k_{2}\\!\cdot\\!k_{3})(2k_{2}\\!\cdot\\!p_{4})}e^{F_{\mathcal{N}}},$
(4.3)

depending on whether the incoming massless particle with negative helicity
preserves or flips the helicity, respectively444Remember from the note at the
beginning of the section that because of the convention to regard all
particles outgoing, conserving/flipping the helicity means that the incoming
and outgoing massless particles have the opposite/same helicity.. In order to
simplify formulas, the same notation as in [2] is used by bolding the massive
spinor helicity variables and suppressing the little group indices. (4.2)
matches the amplitude (5.35) in [2], up to the supersymmetric exponential
factor.

The supersymmetric factor is useful when trying to switch from spin 2 to lower
spin particles, by taking derivatives with respect to the supermomenta
$q_{\mathcal{I}}$ and setting them to zero. It was shown in [7] that in order
to lower the spin of the massless particle it is required to multiply the
amplitude with a factor of

$\frac{\braket{u_{2}u_{3}}}{\sigma_{23}}=-\frac{\langle
k_{2}14\rangle}{\langle 1234\rangle}$

for every spin $\frac{1}{2}$ downward change and for the massive particle with
a factor of

$\frac{\braket{u_{1}u_{4}}}{\sigma_{14}}=-\frac{\langle
p_{1}23\rangle}{\langle 1234\rangle}.$

Because $\langle k_{2}14\rangle=0$ when the massless particle flips its
helicity if follows that the amplitude (4.3) vanishes for massless particles
with spin$<\\!2$. Therefore, in this model only a graviton can scatter with
reversing its helicity [21].

Concerning the amplitude that conserves helicity, when disregarding the
difference between $p_{1}\\!\cdot\\!k_{2}$ and $p_{1}\\!\cdot\\!k_{3}$ (valid
in the classical limit $\hbar\omega/M\rightarrow 0$) in equation (5.26) of [7]
when applied to 4 dimensions:

$\langle k_{2}14\rangle\langle
p_{1}23\rangle=-\\!2p_{1}\\!\cdot\\!k_{3}\;e_{1A}e_{2}^{A}e_{4B}e_{3}^{B}-\\!2p_{1}\\!\cdot\\!k_{2}\;e_{1A}e_{3}^{A}e_{4B}e_{2}^{B}\rightarrow-\\!2p_{1}\\!\cdot\\!k_{3}\langle
1234\rangle,$ (4.4)

and switching to the little-group helicity-based formalism the amplitude (4.2)
becomes

$M_{4}^{+-}=\frac{\langle
3|p_{1}|2]^{2h}(2p_{1}\\!\cdot\\!k_{3})^{2(2-h)}}{(2p_{1}\\!\cdot\\!k_{2})(2k_{2}\\!\cdot\\!k_{3})(2k_{2}\\!\cdot\\!p_{4})}\left(\frac{\langle\mathbf{4}3\rangle\,[\mathbf{1}2]-\langle\mathbf{1}3\rangle\,[\mathbf{4}2]}{\langle
3|p_{1}|2]}\right)^{2s},$ (4.5)

where $s$ is the spin of the massive particle and $h$ the spin of the massless
particle.

What is different with the model of this article? Assume first that the
fermionic contribution in det${}^{\prime}G$ in (2.6) is for a particle of spin
3/2 (’gravitino’) or spin 1 (’gluon’) still preserving helicity. This
represents a massive EYM amplitude as mentioned in subsection 2.1 with a
2-particle PT factor for the ’gluon’ and an open trace for the ’gravitino’:

$\text{det}^{\prime}G\rightarrow\begin{cases}\left(\\!\frac{q_{2\mathcal{I}}q_{3}^{\mathcal{I}}}{\sigma_{23}}\right)^{\\!2}\text{det}H^{[23]}_{[32]}=\text{det}^{\prime}H\\!\left(\\!\frac{\braket{u_{2}u_{3}}q_{2\mathcal{I}}q_{3}^{\mathcal{I}}}{\sigma_{23}}\right)^{\\!2},\qquad\text{'gluon'},\\\
\;\frac{q_{2\mathcal{I}}q_{3}^{\mathcal{I}}}{\sigma_{23}}\frac{\text{det}H^{[23]}_{[32]}}{\braket{u_{3}u_{2}}}=\text{det}^{\prime}H\frac{\braket{u_{2}u_{3}}q_{2\mathcal{I}}q_{3}^{\mathcal{I}}}{\sigma_{23}}\;\;,\qquad\qquad\quad\text{'gravitino'}.\end{cases}$
(4.6)

It turns out that the amplitude is the same as the helicity-conserving case
(4.5), which is good for consistency. Because a PT factor for massless
particles needs to have at least one particle of each helicity, the ’gluon’
case cannot flip the helicity, but it is possible for a ’gravitino’:

$\text{det}G^{[34]}_{[12]}=G_{13}G_{24}-G_{14}G_{23}=\frac{\epsilon_{1}^{A}\epsilon_{4A}q_{2\mathcal{I}}q_{3}^{\mathcal{I}}}{\sigma_{14}\sigma_{32}}=\frac{q_{2\mathcal{I}}q_{3}^{\mathcal{I}}}{\epsilon_{2}^{A}\epsilon_{3A}}\text{det}H^{[34]}_{[12]}.$

Therefore, going from spin 2 to 3/2 just lowers the power of $[23]$ in (4.3),
but it also lowers the power in the gravitational coupling, in contrast to the
helicity-preserving case. The same is true for the massive particle:

$M_{4}^{++}=\frac{\langle\mathbf{1}\mathbf{4}\rangle^{2s}\,[23]^{2h}}{(2p_{1}\\!\cdot\\!k_{2})(2k_{2}\\!\cdot\\!k_{3})(2k_{2}\\!\cdot\\!p_{4})},\quad
3/2\leq h,s\leq 2\;.$

Finally one can consider both, the massless and the massive, particles as
’gluons’ leading to replace det${}^{\prime}G$ with a PT factor for two
incoming and two outgoing particles, i.e. the situation is now described by
the Coulomb branch of SYM for the scattering between ’gluons’ and ’W bosons’
(see 2.1):

$<Wg^{+}g^{+}\bar{W}\\!>=\frac{\braket{\mathbf{1}\mathbf{4}}^{2}[23]}{(2k_{2}\\!\cdot\\!p_{4})\braket{23}},\quad<Wg^{+}g^{-}\bar{W}\\!>=\frac{(\langle\mathbf{4}3\rangle\,[\mathbf{1}2]-\langle\mathbf{1}3\rangle\,[\mathbf{4}2])^{2}}{(2k_{2}\\!\cdot\\!p_{4})(2k_{2}\\!\cdot\\!k_{3})}$

which is consistent with corresponding results in [1, 3, 10].

For gravitational Compton scattering on a Kerr black hole, to make contact
with the results in [22, 23, 21, 24], the kinematic variables are now taken to
be the same as in [21, 24]. The black hole is treated as a massive spin
particle with mass $M$ and a spin vector
$a^{\mu}=s^{\mu}/M=(0,a_{x},a_{y},a_{z})$. The incoming/outgoing momenta of
the scattered particle are still labeled as $k_{2}/k_{3}$ and of the black
hole as $p_{1}/p_{4}$. In the chosen reference frame the black hole is taken
to be initially at rest and the scattering process to happen in the $x-z$
plane. This leads to the following explicit representation of the kinematic
variables:

$\begin{split}p_{1}^{\mu}&=(M,0,0,0),\\\ k_{2}^{\mu}&=\hbar\omega(1,0,0,1),\\\
k_{3}^{\mu}&=\gamma\,\hbar\omega(1,\sin\theta,0,\cos\theta),\gamma=1/{(1+2\frac{\hbar\omega}{M}\sin^{2}\\!\frac{\theta}{2})}\\\
p_{4}^{\mu}&=p_{1}^{\mu}+k_{2}^{\mu}-k_{3}^{\mu}\,.\end{split}$

Here $\theta$ is the scattering angle and the value of $\gamma$ is determined
by the on-shell condition $p_{4}^{2}=M^{2}$ for the outgoing massive momentum.
It should be kept in mind that this setup violates the standard notation
mentioned at the beginning of the section, by reversing the incoming momenta
$p_{1}$ and $k_{2}$.

It will be convenient to introduce a basis of spinor helicity variables in
terms of $\theta$ to represent the momentum and the polarization matrices of
the outgoing massless particle:

$\displaystyle[3|=\sqrt{2\gamma\hbar\omega}\left(\cos\frac{\theta}{2},\sin\frac{\theta}{2}\right),$
$\displaystyle[\mu|=\left(-\\!\sin\frac{\theta}{2},\cos\frac{\theta}{2}\right)\,,$
$\displaystyle|3\rangle=\sqrt{2\gamma\hbar\omega}\left(\cos\frac{\theta}{2},\sin\frac{\theta}{2}\right),$
$\displaystyle|\mu\rangle=\left(-\\!\sin\frac{\theta}{2},\cos\frac{\theta}{2}\right)\,,$
$\displaystyle
k_{3\,\alpha\dot{\alpha}}=|3\rangle_{\alpha}[3|_{\dot{\alpha}}\,,$
$\displaystyle\epsilon^{+}_{3\,\alpha\dot{\alpha}}=\sqrt{2}\frac{|\mu\rangle_{\alpha}[3|_{\dot{\alpha}}}{\langle\mu
3\rangle}\,,$
$\displaystyle\epsilon^{-}_{3\,\alpha\dot{\alpha}}=\sqrt{2}\frac{|3\rangle_{\alpha}[\mu|_{\dot{\alpha}}}{[3\mu]}\,,$

where $|\mu\rangle$ and $[\mu|$ serve as reference spinors. The incoming
massless states associated to $k_{2}$ are recovered by setting $\theta=0$.
Other choices for the reference spinors are allowed. Actually, the reference
spinor $|\mu\rangle$ in $\epsilon^{+}_{3}$ and $\epsilon^{+}_{2}$ could be
replaced by $|2\rangle$ and $|3\rangle$, respectively, and $[\mu|$ in
$\epsilon^{-}_{3}$ and $\epsilon^{-}_{2}$ by $[2|$ and $[3|$, respectively, a
choice which will be made presently.

In this kinematics the classical limit ($s\rightarrow\infty,\gamma\rightarrow
1$) of $M_{4}^{+-}$ in (4.5) can be written as [22, 23, 24]

$\begin{split}M_{4}^{+-}&=\frac{M^{2}(\cos\frac{\theta}{2})^{2h}}{\sin^{2}\frac{\theta}{2}}\exp[(k_{3}-k_{2}+2w)\cdot
a]\\\
&=\frac{M^{2}(\cos\frac{\theta}{2})^{2h}}{\sin^{2}\frac{\theta}{2}}\exp\left[-\frac{\hbar\omega}{M}\left(2a_{z}\sin^{2}\frac{\theta}{2}-a_{x}\sin\theta+2(a_{x}-ia_{y})\tan\frac{\theta}{2}\right)\right]\;,\end{split}$
(4.7)

where the vector $w$ is defined as
$w=-p_{1}\\!\cdot\\!k_{3}\frac{|3\rangle[2|}{\langle
3|p_{1}|2]}=-\frac{p_{1}\\!\cdot
k_{3}}{p_{1}\\!\cdot\epsilon_{3}^{-}}\epsilon_{3}^{-}$.

Amplitudes (4.7) and (4.5) for $s>h$ exhibit a spurious pole for $\langle
3|p_{1}|2\rangle=2M\hbar\omega\cos\frac{\theta}{2}$ that stems from the vector
$w$ and needs to be remedied with help of contact terms [24]. However, the
issue of contact terms will not be considered here. But it is intriguing to
note that the spurious pole is missing in the reverse-helicity scattering
amplitude $M_{4}^{++}$ and that indeed according to (4.4) $w$ can be rewritten
as $w=\frac{\langle k_{2}14\rangle}{2\langle
1234\rangle}|3\rangle[2|=-\\!\braket{u_{2}u_{3}}\\!/(2\sigma_{23})|3\rangle[2|$
which vanishes for massless particles when flipping the helicity, that is,
from the point of view of the ambitwistor string model the appearance or
absence of the spurious pole is directly related to solutions of the polarized
scattering equations.

In summary, although this section has not provided any new results (besides
maybe the gravitino amplitudes), it was interesting to show generally and in
the context of a black hole how, starting from a Lagrangian, the ambitwistor
string models with a single amplitude (4.1) can describe multiple aspects of
the Compton scattering of a massless particle of various spin hitting a
massive target particle with higher spin.

##  5 Summary and Outlook

This work presented an anomaly-free extension to the massive ambitwistor
string found in [8, 9, 10] as a unified model for supergravity and the Coulomb
branch of SYM. The all-multiplicity tree amplitudes were shown to have the
expected massless limit, to include EYM amplitudes for multiple gluon traces,
and to have proper unitary factorization. At every order of perturbation
theory the cosmological constant is zero and even spin structures of genus
$g\\!\geq\\!1$ Riemann surfaces do not contribute to loop amplitudes. Even odd
spin structures are restricted to contain a single odd cycle. The all-
multiplicity one-loop amplitude on an odd spin structure was calculated and
established to have unitary factorization for separating and nonseparating
degeneration. Then Compton scattering of a massless particle against a massive
target was used to obtain scattering amplitudes for both cases of conserved
and reversed helicity. They were compatible with results in the literature.
One surprise score might be counted for the non-vanishing reverse-helicity
amplitude of gravitino scattering.

Future work would include the calculation of higher loop amplitudes together
with the demonstration of proper factorization. Because the model will be IR-
divergent when massless particles like the graviton and photon are included,
IR-renormalization will be required. In order to do that correctly, a string
field theory might need to be developed. It would be interesting to see how
model features would be affected by renormalization and whether for instance
the cosmological constant could pick up a non-vanishing value. Further, the
$\mathcal{N}=8$ supergravity spectrum with SU(4) $\\!\times\\!$ SU(4)
R-symmetry should be studied for various symmetry breakdown scenarios to get
more realistic spectra555Note that the $\mathcal{N}=8$ supergravity spectrum
contains exactly the spin $\frac{1}{2}$ content for all the fermions of the
standard model and also for 8 massive gravitinos, a fact that is well known
[25, 26].. Regarding the spectrum, one issue that has come up in [14] is that
in the NS sector the spectrum is truncated because of the absence of
oscillator zero modes for the twistors, and it was argued that the issue might
be solved through spectral flow. The current work showed that because of the
vanishing of amplitudes on even spin structures the spectrum in the R sector
is sufficient up to at least the one-loop level, but is not clear whether the
NS sector obstruction would not present itself again at higher loop level.
This is another item to look into in more detail.

And, of course, the origin of all the auxiliary spinors in the action (2) is
still a mystery. They are excluded from the physical spectrum and the question
arises whether they materialize as a ghost system used for mathematical
convenience or whether they emanate from compactifying extra twistor space
dimensions (in which case there still might not be a mandatory necessity to
relate them to extra spacetime dimensions). More knowledge in this regard
might also reveal the exact nature of the unknown six bosonic degrees of
freedom that make the model anomaly-free.

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# The closures of wreath products in product action

I. Ponomarenko  Steklov Institute of Mathematics at St. Petersburg, Russia;
and Sobolev Institute of Mathematics, Novosibirsk, Russia<EMAIL_ADDRESS>and
A. V. Vasil’ev  Sobolev Institute of Mathematics, Novosibirsk, Russia;
and Novosibirsk State University, Novosibirsk, Russia<EMAIL_ADDRESS>
###### Abstract.

Let $m$ be a positive integer and let $\Omega$ be a finite set. The
$m$-closure of $G\leq\operatorname{Sym}(\Omega)$ is the largest permutation
group on $\Omega$ having the same orbits as $G$ in its induced action on the
Cartesian product $\Omega^{m}$. The exact formula for the $m$-closure of the
wreath product in product action is given. As a corollary, a sufficient
condition is obtained for this $m$-closure to be included in the wreath
product of the $m$-closures of the factors.

The research was supported by the Mathematical Center in Akademgorodok under
agreement No. 075-15-2019-1613 with the Ministry of Science and Higher
Education of the Russian Federation.

##  1\. Introduction

Let $m$ be a positive integer and let $\Omega$ be a finite set. The
$m$-closure $G^{(m)}$ of $G\leq\operatorname{Sym}(\Omega)$ is the largest
permutation group on $\Omega$ having the same orbits as $G$ in its induced
action on the Cartesian product $\Omega^{m}$. Wielandt [13, Theorems 5.8 and
5.12] showed that

(1) $G^{(1)}\geq G^{(2)}\geq\cdots\geq G^{(m)}=G^{(m+1)}=\cdots=G$

for some $m<|\Omega|$. In this sense, the $m$-closure can be considered as a
natural approximation of $G$. One can also consider $G^{(m)}$ as the full
automorphism group of the family of all $m$-ary relations invariant with
respect to $G$.

In general, studying the $m$-closure for $m\geq 2$ is a nontrivial problem
both from theoretical and computational point of view, see, e.g., [10, 12, 8,
4, 9, 7]. Usual approach here is a reduction via the direct or wreath products
to smaller permutation groups. Let us consider these operations in more
detail.

Let $K\leq\operatorname{Sym}(\Gamma)$ and $L\leq\operatorname{Sym}(\Delta)$.
The direct product $K\times L$ has two natural actions: on the disjoint union
$\Gamma\cup\Delta$ and on the Cartesian product $\Gamma\times\Delta$. It is
well known that in both cases the $m$-closure of $K\times L$ is equal to
$K^{(m)}\times L^{(m)}$ (see, e.g., [2]). A similar formula holds for the
wreath product $K\wr L$ acting on $\Gamma\times\Delta$ [6]. However, passing
to the permutation group $K\uparrow L$ induced by the product action of $K\wr
L$ (i.e., on the Cartesian product of $|\Delta|$ copies of $\Gamma$) causes a
problem even in a small case:

(2)
$\begin{split}(\operatorname{Sym}(2)\uparrow\operatorname{Alt}(3))^{(2)}&=\operatorname{Sym}(2)\uparrow\operatorname{Sym}(3),\\\
\operatorname{Sym}(2)^{(2)}\uparrow\operatorname{Alt}(3)^{(2)}&=\operatorname{Sym}(2)\uparrow\operatorname{Alt}(3),\end{split}$

which shows that, in general, $(K\uparrow L)^{(m)}\not\leq K^{(m)}\uparrow
L^{(m)}$. The main goal of the present paper is to establish the exact formula
for the $m$-closure of $K\uparrow L$.

Apparently, the first results related to the structure of $(K\uparrow
L)^{(m)}$ were obtained in [10, Propositions 3.2, 3.3] for the case when the
group $K\uparrow L$ is primitive, but even in this case no explicit formula
was found. The inclusion $(K\uparrow L)^{(m)}\leq K^{(m)}\uparrow L^{(m)}$ was
first proved in [4, Proposition 3.1] for $m=2$ and non $2$-transitive $K$ (cf.
(2)), and then recently in [8, Theorem 3.3] for $m\geq 3$, primitive
$K\uparrow L$, and technical assumptions on the degrees of $K$ and $L$.

To state the main result, we need to define one more closure operator. Namely,
for a group $G\leq\operatorname{Sym}(\Omega)$ we denote by $G^{[m]}$ the
largest permutation group on $\Omega$ having the same orbits as $G$ in its
induced action on the ordered partitions of $\Omega$ in at most $m$ classes.
This type of closure behaves similarly to the $m$-closure (cf. formulas (1)
and (9)) and will be considered in more detail in Section 3.

######  Theorem 1.1.

Let $K$ and $L$ be permutation groups and $m\geq 2$ an integer. Then

(3) $(K\uparrow L)^{(m)}=K^{(m)}\uparrow L^{[k]},$

where $k=\min\\{k_{m},d\\}$ with $k_{m}=|\operatorname{Orb}_{m}(K)|$ and $d$
the degree of $L$.

The number $k_{m}$ defined in Theorem 1.1 is bounded from below by the number
$|\operatorname{Orb}_{m}(\operatorname{Sym}(n))|$, where $n$ is the degree of
$K$, which is equal to the number of ordered partitions of a set of
cardinality $m$ (see, e.g., [8, Example 2.1]). In particular, $k_{m}\geq m+1$
if $m\geq 3$ and $n\geq 2$.

Theorem 1.1 enables us to establish a sufficient condition for the $m$-closure
of $K\uparrow L$ to be included in the wreath product of the $m$-closures of
$K$ and $L$.

######  Theorem 1.2.

Let $K$ and $L$ be permutation groups and $m\geq 2$. Then

(4) $(K\uparrow L)^{(m)}\leq K^{(m)}\uparrow L^{(m)}$

unless $m=2$ and $K$ is $2$-transitive.

Proof. Without loss of generality, we assume that $K$ is of degree at least
$2$. Then $k_{m}\geq m+1$: this follows from above if $m\geq 3$, and from the
fact that $K$ is not $2$-transitive if $m=2$. Now if $d\geq m+1$, then
$k-1\geq m$ and Lemma 3.2 together with (1) yields $L^{[k]}\leq L^{(k-1)}\leq
L^{(m)}$. On the other hand, if $d\leq m$, then $k=d$ and formulas (9) and (1)
yield $L^{[k]}=L^{[d]}=L\leq L^{(m)}$. Thus in any case, inclusion (4) holds
by Theorem 1.1. $\square$

Theorem 1.2 gives a natural generalization of the two mentioned results from
[4] and [8]. The exceptional case $m=2$ and $K$ is $2$-transitive cannot be
avoided, see example in (2). Some other examples of primitive groups $L$ for
which $L^{(2)}=L$ and $L^{[2]}>L$ can be found among the groups listed in [11,
Theorem 2]. We believe that there are infinitely many such examples where $L$
is imprimitive.

When the group $L$ is primitive and $m\geq 3$, the right-hand side of the
equality in Theorem 1.1 can be made more precise with the help of the main
results of [11].

######  Theorem 1.3.

Let $K$ and $L$ be permutation groups and $m\geq 3$. Assume that $L$ is
primitive and is not an alternating group in standard action. Then

$(K\uparrow L)^{(m)}=K^{(m)}\uparrow L.$

Proof. Follows from Theorem 1.1 and Lemma 3.3. $\square$

The authors are grateful to S. V. Skresanov for very useful comments to the
first draft of the paper.

##  2\. Preliminaries

We start with some basic facts of Wielandt’s theory of $m$-closures. First,
note that taking the $m$-closure is a closure operator:

$G\leq G^{(m)},\quad G^{(m)}=(G^{(m)})^{(m)},\quad G\leq H\ \Rightarrow\
G^{(m)}\leq H^{(m)},$

see [13, Theorem 5.4, 5.9, 5.7], respectively. Second, there is a clear
sufficient condition for a permutation to lie in $m$-closure.

######  Lemma 2.1.

(The closure argument, see [13, Theorem 5.6]) Let
$G\leq\operatorname{Sym}(\Omega)$, $f\in\operatorname{Sym}(\Omega)$, and $m$ a
natural number. Then $f\in G^{(m)}$ if and only if for every
$\alpha\in\Omega^{m}$ there is $g\in G$ such that $\alpha^{f}=\alpha^{g}$.

The closure argument is crucial in finding $m$-closures of products of
permutation groups, see, e.g., the detailed proof of the following folklore
result in [2, Lemma 2.4].

######  Theorem 2.2.

Let $K\leq\operatorname{Sym}(\Gamma)$, $L\leq\operatorname{Sym}(\Delta)$, and
let $K\times L$ act on the Cartesian product $\Gamma\times\Delta$. For every
integer $m\geq 2$,

$(K\times L)^{(m)}=K^{(m)}\times L^{(m)}.$

We use the same argument in the proof of our main result for the wreath
products of permutation groups in product action. Let us take a closer look at
such a product.

Let $K\leq\operatorname{Sym}(\Gamma)$ and $L\leq\operatorname{Sym}(\Delta)$.
Without loss of generality, we assume that $\Delta=\\{1,\ldots,d\\}$. The
wreath product $K\wr L$ induces a permutation group $G=K\uparrow L$ on the
Cartesian product

(5) $\Omega=\underbrace{\Gamma\times\cdots\times\Gamma}_{d\ \,\text{copies}}.$

Every permutation of $G$ can be written in the form

(6) $g=(g_{1},\ldots,g_{d};\overline{g})$

for some $g_{1},\ldots,g_{d}\in K$ and $\overline{g}\in L$. The action of $g$
on the point

$\omega=(\omega_{1},\ldots,\omega_{d})\in\Omega$

is defined as follows (see, e.g., [3, Section 2.7]):

(7)
$(\omega^{g})_{i}=(\omega_{i^{\overline{g}^{-1}}})^{g_{i^{\overline{g}^{-1}}}},\quad
1\leq i\leq d.$

It is well known (see, e.g., [1, Theorem 9.2.1]) that the automorphism group
of the Hamming graph is the wreath product of two symmetric groups in product
action. When the vertex set of this graph is of the form (5), the edge set is
an orbit of $\operatorname{Sym}(\Gamma)\uparrow\operatorname{Sym}(\Delta)$. It
almost immediately follows that

(8)
$(\operatorname{Sym}(\Gamma)\uparrow\operatorname{Sym}(\Delta))^{(2)}=\operatorname{Sym}(\Gamma)\uparrow\operatorname{Sym}(\Delta).$

All undefined notation for permutation groups used in the paper are mostly
standard and can be found in [3].

##  3\. Closure with respect to partitions

Let $\Omega^{[m]}$ be the set of all ordered partitions $\Pi$ of $\Omega$ such
that $|\Pi|\leq m$. For a group $G\leq\operatorname{Sym}(\Omega)$, we denote
by $G^{[m]}$ the largest permutation group on $\Omega$ having the same orbits
as $G$ in its induced action on $\Omega^{[m]}$. Obviously,

(9) $\operatorname{Sym}(\Omega)=G^{[1]}\geq G^{[2]}\geq\cdots\geq
G^{[m]}=G^{[m+1]}=\cdots=G$

for some $m\leq|\Omega|$.

The groups of series (9), except for the first one, are orbit equivalent to
$G$ in the sense of [11], i.e., for all $m\geq 2$,

(10)
$\operatorname{Orb}(G^{[m]},2^{\Omega})=\operatorname{Orb}(G,2^{\Omega}).$

Indeed, let $S\subseteq\Omega$, and let $\Pi=(\Pi_{1},\Pi_{2})$ the partition
of $\Omega$ in two classes $\Pi_{1}=S$ and $\Pi_{2}=\Omega\setminus S$. Now if
$H=G^{[m]}$, then $\Pi^{G}=\Pi^{H}$ and hence
$S^{G}=\Pi_{1}^{G}=\Pi_{1}^{H}=S^{H}$, as required.

The following statement is an analog of the closure argument (Lemma 2.1).

######  Lemma 3.1.

Let $G\leq\operatorname{Sym}(\Omega)$, $f\in\operatorname{Sym}(\Omega)$, and
$m$ a natural number. Then $f\in G^{[m]}$ if and only if for every
$\Pi\in\Omega^{[m]}$ there is $g\in G$ such that $\Pi^{f}=\Pi^{g}$.

Proof. The membership $f\in G^{[m]}$ is equivalent to the equality
$(\Pi^{G})^{f}=\Pi^{G}$ for all $\Pi\in\Omega^{[m]}$, which in turn is
equivalent to the existence of $g\in G$ such that $\Pi^{f}=\Pi^{g}$ for each
$\Pi\in\Omega^{[m]}$. $\square$

Lemmas 2.1 and 3.1 enable us to establish a simple relationship between two
types of closures.

######  Lemma 3.2.

$G^{[m+1]}\leq G^{(m)}$.

Proof. Let $g\in G^{[m+1]}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\Omega^{m}$. Put
$\Pi=(\Pi_{1},\ldots,\Pi_{m+1})$, where $\Pi_{i}=\\{\alpha_{i}\\}$,
$i=1,\ldots,m$, and
$\Pi_{m+1}=\Omega\setminus\\{\alpha_{1},\ldots,\alpha_{m}\\}$. Then
$\Pi\in\Omega^{[m+1]}$. By Lemma 3.1, there exists $h\in G$ such that
$\Pi^{g}=\Pi^{h}$. It follows that

$\\{\alpha_{i}^{g}\\}=\Pi_{i}^{g}=\Pi_{i}^{h}=\\{\alpha_{i}^{h}\\},\quad
i=1,\ldots,m.$

Thus, $\alpha^{g}=\alpha^{h}$, and we are done by the closure argument.
$\square$

In general, we cannot improve Lemma 3.2 by replacing $m+1$ by $m$ in
$G^{[m+1]}$. Indeed, let $G=\operatorname{Alt}(n)$, $n\geq 3$. Then the
stabilizer $G_{1,\ldots,n-2}$ is trivial implying $G^{(n-1)}=G$, see [13,
Theorem 5.12]. On the other hand, the group $G$ is $(n-2)$-transitive. Using
this, it is not hard to verify that for every ordered partition $\Pi$ with at
most $n-1$ classes, $\Pi^{G}=\Pi^{\operatorname{Sym}(n)}$. Together with
formula (9), this implies

(11) $G^{[m]}=\begin{cases}\operatorname{Sym}(n)&\text{if $m\leq n-1$},\\\
G&\text{otherwise}.\\\ \end{cases}$

It follows that $G^{[n-1]}\not\leq G^{(n-1)}$.

######  Lemma 3.3.

Let $G\leq\operatorname{Sym}(n)$ be a primitive group and $m\geq 3$. Then
$G^{[m]}=G$, unless $G=\operatorname{Alt}(n)$ and $m\leq n-1$.

Proof. In view of (11), we may assume that $G\not\geq\operatorname{Alt}(n)$.
The group $G^{[m]}\geq G$ is primitive, and orbit equivalent to $G$, i.e.,
equality (10) holds. By [11, Corollary 3] this implies that $G=G^{[m]}$,
unless

$(G,G^{[m]})\in{\mathfrak{S}},$

where ${\mathfrak{S}}$ consists of explicitly described pairs $(H,H^{*})$ of
primitive groups (of degree at most $10$) with $H<H^{*}$.

A straightforward computation in computer package GAP [5] shows that for every
pair $(H,H^{*})\in{\mathfrak{S}}$ there is an ordered partition
$\Pi=\Pi(H,H^{*})$ with three classes such that $H^{*}$ acts on the orbit
$\Pi^{H^{*}}$ regularly. Now let $\Pi=\Pi(G,G^{[m]})$. Since $|\Pi|=3\leq m$,
we have $\Pi^{G^{[m]}}=\Pi^{G}$. Consequently,

$|G|\geq|\Pi^{G}|=|\Pi^{G^{[m]}}|=|G^{[m]}|,$

which is possible only if $G^{[m]}=G$. $\square$

##  4\. Proof of Theorem 1.1

Let $K\leq\operatorname{Sym}(\Gamma)$ and $L\leq\operatorname{Sym}(\Delta)$,
where $\Delta=\\{1,\ldots,d\\}$. Put $G=K\uparrow L$ and $\Omega=\Gamma^{d}$.
Then $G\leq\operatorname{Sym}(\Omega)$.

Every $m$-tuple $\alpha\in\Omega^{m}$ is written in the form
$\alpha=(\alpha^{(1)},\ldots,\alpha^{(m)})$ where
$\alpha^{(j)}=(\alpha^{(j)}_{1},\ldots,\alpha^{(j)}_{d})\in\Gamma^{d}$,
$j=1,\ldots,m$. It is convenient to treat $\alpha$ as a $d\times m$ matrix

$\begin{pmatrix}\alpha^{(1)}_{1}&\alpha^{(2)}_{1}&\cdots&\alpha^{(m)}_{1}\\\
\alpha^{(1)}_{2}&\alpha^{(2)}_{2}&\cdots&\alpha^{(m)}_{2}\\\
\cdots&\cdots&\cdots&\cdots\\\
\alpha^{(1)}_{d}&\alpha^{(2)}_{d}&\cdots&\alpha^{(m)}_{d}\\\ \end{pmatrix}$

where the $j$th column is $\alpha^{(j)}$. In fact, we are rather interested in
rows of this matrix which are $m$-tuples of points of $\Gamma$; denote the
$i$th row by $\alpha[i]=(\alpha^{(1)}_{i},\ldots,\alpha^{(m)}_{i})$.

Now let $\operatorname{Orb}_{m}(K)=\\{s_{1},\ldots,s_{a}\\}$, where $a=k_{m}$.
For the $m$-tuple $\alpha$, there are uniquely determined numbers $1\leq
a_{1}<\cdots<a_{r}\leq a$ such that $\alpha[i]\in s_{a_{\ell}}$ for every
$i\in\Delta$ and some $1\leq\ell\leq r$. Denote by $\Pi(\alpha)$ the ordered
partition of $\Delta$ with classes

$\Pi_{\ell}(\alpha)=\\{i\in\Delta:\ \alpha[i]\in
s_{a_{\ell}}\\},\qquad\ell=1,\ldots,r.$

Note that the integer $k$ from the statement of the theorem is equal to
$\min\\{a,d\\}$.

######  Lemma 4.1.

The mapping $\Omega^{m}\to\Delta^{[k]}$, $\alpha\mapsto\Pi(\alpha)$, is well
defined and surjective.

Proof. Obviously, $|\Pi(\alpha)|\leq d$ and $|\Pi(\alpha)|\leq a$, and hence
$|\Pi(\alpha)|\leq k$ for all $\alpha$. So $\Pi(\alpha)\in\Delta^{[k]}$ and
the mapping is well defined. Let
$\Pi=(\Pi_{1},\ldots,\Pi_{k})\in\Delta^{[k]}$. Then every $i\in\Delta$ belongs
to some $\Pi_{\ell}$; choose arbitrary $\beta_{i}\in s_{\ell}$, which is
possible because $k\leq a$. Now let $\alpha$ be the unique $m$-tuple of
$\Omega$ for which

$\alpha[i]=\beta_{i},\quad i\in\Delta.$

Then $\Pi(\alpha)=\Pi$, as required. $\square$

In accordance with formula (7), the permutation
$g\in\operatorname{Sym}(\Gamma)\uparrow\operatorname{Sym}(\Delta)$ acts on an
$m$-tuple $\alpha$ as follows:

(12)
$\alpha^{g}[i]=(\alpha[i^{\overline{g}^{-1}}])^{g_{i^{\overline{g}^{-1}}}},\quad
1\leq i\leq d.$

Thus, $\overline{g}$ permutes the rows of the $d\times m$ matrix $\alpha$,
while $g_{i}$ permutes elements of the $i$th row coordinatewise.

From (1), monotonicity of $m$-closure operator, and (8) it follows that

$H=G^{(m)}\leq
G^{(2)}\leq(\operatorname{Sym}(\Gamma)\uparrow\operatorname{Sym}(\Delta))^{(2)}=\operatorname{Sym}(\Gamma)\uparrow\operatorname{Sym}(\Delta).$

As in (6), every permutation of $H$ is written in the form

$h=(h_{1},\ldots,h_{d};\overline{h})$

for some $h_{1},\ldots,h_{d}\in\operatorname{Sym}(\Gamma)$ and
$\overline{h}\in\operatorname{Sym}(\Delta)$.

######  Lemma 4.2.

$\Pi(\alpha)^{\overline{h}}=\Pi(\alpha^{h})$ for all $h\in
K^{(m)}\uparrow\operatorname{Sym}(\Delta)$ and $\alpha\in\Omega^{m}$.

Proof. Let $i\in\Delta$. There is $\ell\in\\{1,\ldots,r\\}$ such that
$i\in\Pi_{\ell}(\alpha)$, i.e., $\alpha[i]\in s_{a_{\ell}}$. By formula (12),

(13)
$\alpha^{h}[i^{\overline{h}}]=(\alpha[i])^{h_{i}}\in(s_{a_{\ell}})^{h_{i}}=s_{a_{\ell}}.$

Let us emphasize that although the indices $a_{\ell}$, $\ell=1,\ldots,r,$ have
been defined for the tuple $\alpha$, formula (13) shows that they remain the
same for $\alpha^{h}$. Consequently,
$i^{\overline{h}}\in\Pi_{\ell}(\alpha^{h})$ implying
$\Pi_{\ell}(\alpha)^{\overline{h}}=\Pi_{\ell}(\alpha^{h})$ for all $\ell$.
$\square$

Let us prove that $K^{(m)}\uparrow L^{[k]}\leq H$ . First, we note that

(14) $1\uparrow L^{[k]}\leq H.$

Indeed, let $h\in 1\uparrow L^{[k]}$ and $\alpha\in\Omega^{m}$. By Lemma 3.1,
there exists $\overline{f}\in L$ such that
$\Pi(\alpha)^{\overline{h}}=\Pi(\alpha)^{\overline{f}}$. Clearly,
$f=(1,\ldots,1;\overline{f})\in 1\uparrow L$. By Lemma 4.2,

$\Pi(\alpha^{hf^{-1}})=\Pi(\alpha)^{\overline{h}\,\overline{f}^{-1}}=\Pi(\alpha).$

It follows that for every $i\in\Delta,$ the tuples $\alpha^{hf^{-1}}[i]$ and
$\alpha[i]$ belong to the same $m$-orbit of $K$. Therefore,

(15) $\alpha^{hf^{-1}}[i]=\alpha[i]^{k_{i}}$

for some $k_{i}\in K$. Then $k=(k_{1},\ldots,k_{d})\in K^{d}$ and hence $kf\in
K\uparrow L$. Furthermore, formula (15) yields $\alpha^{hf^{-1}}=\alpha^{k}$
implying $\alpha^{h}=\alpha^{kf}$. Thus, $h\in H$ by Lemma 2.1, which proves
(14).

By Theorem 2.2, we have $K^{(m)}\uparrow 1=(K\uparrow 1)^{(m)}\leq H$.
Together with (14), this yields

$K^{(m)}\uparrow L^{[k]}=\langle{K^{(m)}\uparrow 1,1\uparrow
L^{[k]}}\rangle\leq H=(K\uparrow L)^{(m)}.$

To prove the reverse inclusion, let $h\in H$. For an arbitrary $m$-orbit $s$
of $K$, the set

$X_{s}=\\{\alpha\in\Omega^{m}:\ \alpha[i]\in s,\ i=1,\ldots,d\\}$

is invariant with respect to the group $G$. Consequently, $(X_{s})^{h}=X_{s}$.
By formula (12) this yields

$\alpha[i]^{h_{i}}=\alpha^{h}[i^{\overline{h}}]\in X_{s}$

for all $\alpha\in X_{s}$ and $1\leq i\leq d$. Since $\alpha[i]\in s$, this
implies $s^{h_{i}}=s$. Thus, $h_{i}$ preserves each $m$-orbit of $K$ and so
$h_{i}\in K^{(m)}$ for all $i$.

It remains to prove that $\overline{h}\in L^{[k]}$ for all $h\in H$. To this
end, let $\Pi\in\Delta^{[k]}$. By Lemma 4.1, there is $\alpha\in\Omega^{m}$
such that $\Pi=\Pi(\alpha)$. By the closure argument, there is $g\in K\uparrow
L$ such that $\alpha^{h}=\alpha^{g}$. Since $h_{i}\in K^{(m)}$ for all $i$,
Lemma 4.2 yields

$\Pi^{\overline{h}}=\Pi(\alpha)^{\overline{h}}=\Pi(\alpha^{h})=\Pi(\alpha^{g})=\Pi(\alpha)^{\overline{g}}=\Pi^{\overline{g}}.$

Thus, $\overline{h}\in L^{[k]}$ by Lemma 3.1. It follows that $h\in
K^{(m)}\uparrow L^{[k]}$. $\square$

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