# Photon Reconstruction in the Belle II Calorimeter
Using Graph Neural Networks

F. Wemmer  0000-0002-6475-0834 I. Haide  0000-0003-0962-6344 J. Eppelt
0000-0001-8368-3721 T. Ferber  0000-0002-6849-0427 A. Beaubien
0000-0001-9438-089X P. Branchini  0000-0002-2270-9673 M. Campajola
0000-0003-2518-7134 C. Cecchi  0000-0002-2192-8233 P. Cheema
0000-0001-8472-5727 G. De Nardo  0000-0002-2047-9675 C. Hearty
0000-0001-6568-0252 A. Kuzmin  0000-0002-7011-5044 S. Longo
0000-0002-8124-8969 E. Manoni  0000-0002-9826-7947 F. Meier
0000-0002-6088-0412 M. Merola  0000-0002-7082-8108 K. Miyabayashi
0000-0003-4352-734X S. Moneta  0000-0003-2184-7510 M. Remnev
0000-0001-6975-1724 J. M. Roney  0000-0001-7802-4617 J.-G. Shiu
0000-0002-8478-5639 B. Shwartz  0000-0002-1456-1496 Y. Unno
0000-0003-3355-765X R. van Tonder  0000-0002-7448-4816 R. Volpe
0000-0003-1782-2978

###### Abstract

We present the study of a fuzzy clustering algorithm for the Belle II
electromagnetic calorimeter using Graph Neural Networks. We use a realistic
detector simulation including simulated beam backgrounds and focus on the
reconstruction of both isolated and overlapping photons. We find significant
improvements of the energy resolution compared to the currently used
reconstruction algorithm for both isolated and overlapping photons of more
than 30 % for photons with energies
$E_{\gamma}<0.5\,\mathrm{\,Ge\kern-1.00006ptV}$ and high levels of beam
backgrounds. Overall, the GNN reconstruction improves the resolution and
reduces the tails of the reconstructed energy distribution and therefore is a
promising option for the upcoming high luminosity running of Belle II.

###### keywords:

calorimeter, photon reconstruction, overlapping clusters, high background,
fuzzy clustering, machine learning, deep learning, graph neural networks, end-
to-end representation spaces

##  1 Introduction

The Belle II experiment is located at the high-intensity, asymmetric electron-
positron-collider SuperKEKB in Tsukuba, Japan. SuperKEKB is colliding 4
$\mathrm{\,Ge\kern-1.00006ptV}$ positron and 7 $\mathrm{\,Ge\kern-1.00006ptV}$
electron beams at a center-of-mass energy of around 10.58
$\mathrm{\,Ge\kern-1.00006ptV}$ to search for rare meson decays and new
physics phenomena. Many of these decays include photons in the final state
that are reconstructed exclusively in the electromagnetic calorimeter. The
experimental program of Belle II targets a significantly increased
instantaneous luminosity that ultimately exceeds the predecessor experiment by
a factor of 30. This increase in luminosity also leads to a significant
increase in beam-induced backgrounds [1]. These background processes produce
both high-energy particle interactions that could be misidentified as physics
signals, but also energy depositions of low-energy particles that degrade the
energy resolution of the electromagnetic crystal calorimeter.

In this paper, we describe a fuzzy clustering algorithm based on Graph Neural
Networks (GNNs). We then study the reconstruction of photon energies in
simulated events using the GNN clustering algorithm. The term fuzzy clustering
[2] refers to the partial assignment of individual calorimeter crystals to
several clustering classes. In our case, these are potentially overlapping,
different signal photons, but also a beam background class.

The paper is organized as follows: Section 2 gives an overview of related work
on Machine Learning for calorimeter reconstruction. Section 3 describes the
Belle II electromagnetic calorimeter. The event simulation and details of the
beam background simulation are discussed in Section 4. The conventional Belle
II reconstruction algorithm and the new GNN algorithm are described in Section
5. We introduce the metrics used to measure the performance of the GNN
algorithm in Section 6. The main performance studies and results are discussed
in Section 7. We summarize our results in Section 8.

##  2 Related work

Machine Learning is widely used in high energy physics for the reconstruction
of calorimeter signals both for clustering [3, 4], energy regression [5, 6],
but also particle identification [7, 8] and fast simulation [9, 10]. Most of
the recent work has been performed in the context of the high-granularity
calorimeter (HGCAL) at CMS [11, 12]. For Belle II, the use of machine learning
utilizing the electromagnetic calorimeter is so far limited to image-based
particle identification in the barrel [13, 8].

GNNs are now widely recognized as one possible solution for irregular
geometries in high energy physics [14]. GNN architectures that are able to
learn a latent space representation of the detector geometry itself [15, 16]
are the basis of the work presented in this paper.

Previous work has focused on simplified and idealized detector geometries,
often approximated as a regular grid of readout cells expressed as 2D or 3D
images. Additionally, the presence of geometry changes and overlaps between
barrel and endcap regions, large variations of cell sizes, and the presence of
very high spatially non-uniform noise levels induced by beam background energy
depositions are neglected.

##  3 The Belle II Electromagnetic Calorimeter

The Belle II detector consists of several subdetectors arranged around the
beam pipe in a cylindrical structure that is described in detail in Ref. [17,
18]. We define the $z$-axis of the laboratory frame as the central axis of the
solenoid. The positive direction is pointing in the direction of the electron
beam. The longitudinal direction, the transverse plane with azimuthal angle
$\phi$, and the polar angle $\theta$ are defined with respect to the
detector’s solenoidal axis.

The Belle II electromagnetic calorimeter (ECL) consists of 8736 Thallium-doped
CsI (CsI(Tl)) crystals that are grouped in a forward endcap, covering a polar
angle $12.4^{\circ}<\theta<31.4^{\circ}$, a barrel, covering a polar angle
$32.2^{\circ}<\theta<128.7^{\circ}$, and a backward endcap, covering a polar
angle $130.7^{\circ}<\theta<155.1^{\circ}$. The crystals have a trapezoidal
geometry with a nominal cross-sectional area of approximately $6\times 6$ cm2
and a length of 30 cm, providing 16.1 radiation lengths of material. While
crystals in the barrel are similar in cross-section and shape, the crystals in
the endcaps vary with masses between 4.03 kg and 5.94 kg [19]; crystals in the
endcaps also have significantly more passive material in front of the
crystals. Each crystal is aligned in the direction of the collision point with
a small tilt in polar angle $\theta$. Crystals in the barrel additionally have
a small tilt in azimuthal angle $\phi$. The scintillation light produced in
the CsI(Tl) crystals is read out by two photodiodes glued to the back of each
crystal. After shaping electronics, the waveform is digitized and the crystal
energy $E^{\mathrm{crystal}}_{\mathrm{rec}}$ over baseline and time
$t^{\mathrm{crystal}}_{\mathrm{rec}}$ since trigger time of the energy
deposition are reconstructed online using FPGAs [20]. Waveforms of crystals
with energy depositions above 50 MeV are stored for offline processing to
allow for electromagnetic vs. hadronic shower identification through pulse
shape discrimination (PSD) [21]. Available information from PSD is

  * •

the fit type ID of a multi-template fit indicating which of the possible
templates provides the best goodness-of-fit,

  * •

the respective $\chi^{2}$ value as an indicator of the goodness-of-fit,

  * •

and the ratio of reconstructed hadronic and photon template energies, referred
to as PSD hadronic energy ratio in the following.

##  4 Data Set

In this work, we use simulated events to train and evaluate the reconstruction
algorithms. The detector geometry and interactions of final-state particles
with detector materials are simulated using Geant4 [22] combined with a
dedicated detector response simulation. Simulated events are reconstructed and
analyzed using the Belle II Analysis Software Framework (basf2) [23, 24]. We
simulate isolated photons, with energy $0.1<E_{\mathrm{gen}}<1.5\,\text{GeV}$,
and direction $17^{\circ}<\theta_{\mathrm{gen}}<150^{\circ}$ and
$0^{\circ}<\phi_{\mathrm{gen}}<360^{\circ}$ drawn randomly from independent
uniform distributions in $E$, $\theta$, and $\phi$. The generation vertex of
the photons is $x=0$, $y=0$, and $z=0$. For events with two overlapping
photons, we first draw randomly one photon with independent uniform
distributions as outline above. We then simulate a second photon with an
angular separation $2.9<\Delta\alpha<9.7\,^{\circ}$ drawn randomly from
uniform distributions in $\Delta\alpha$ and in $E$. This angular separation
covers approximately the distance needed to create two overlapping clusters.

As part of the simulation, we overlay simulated beam background events
corresponding to different collision conditions to our signal particles [25,
1]. The simulated beam backgrounds correspond to an instantaneous luminosity
of $\mathcal{L}_{\text{beam}}=1.06\times 10^{34}$ cm-2s-1 (called low beam
background), and $\mathcal{L}_{\text{beam}}=8\times 10^{35}$ cm-2s-1 (called
high beam background). Those two values approximately correspond to the
conditions in 2021, and the expected conditions slightly above the design
luminosity, respectively. The spatial distribution of beam backgrounds is
asymmetric: They are much higher in the backward endcap than in the forward
endcap, and they are slightly higher in the barrel than in the forward endcap.
Additional electronics noise per crystal of about 0.35 MeV is included in our
simulation as well.

The supervised training and the performance evaluation both use labeled
information that relies on matching reconstructed information with the
simulated truth information. For each of the four configurations, isolated and
overlapping photons with low and high beam backgrounds, we use 1.8 million
events for training and 200 000 events for validation. The performance
evaluation is carried out on a large number of statistically independent
samples simulated with various energies and in different detector regions.

We then study the performance of the GNN clustering algorithm in all four
scenarios and compare it to the baseline basf2 reconstruction. Both
reconstruction algorithms are described in detail in Sec. 5.

###  4.1 Isolated Photon

To study isolated photons, we use the simulated events with a generated
isolated photon only. For each event, we select a region of interest (ROI),
defined as the $9\times 9$ array of crystals centered around a local maximum
(LM). With the irregular crystal arrangement in the endcaps, this array is not
necessarily regular but rather defined by the 80 nearest neighbors of the LM.
The LM is a crystal with at least 10 MeV of reconstructed crystal energy, and
energy higher than all its direct eight neighbors. The LM must be the only LM
in the ROI, and the matched truth particle must be a simulated photon
responsible for at least 20% of the reconstructed crystal energy. Precisely,
for the LM we require the ratio

$r^{\gamma_{1}}_{\mathrm{LM}}=\frac{E^{\gamma_{1}\mathrm{,crystal}_{\mathrm{LM}}}_{\mathrm{dep}}}{E^{\mathrm{crystal}_{\mathrm{LM}}}_{\mathrm{rec}}}\geq
0.2.$ (1)

Here, $E^{\gamma_{1}\mathrm{,crystal}_{\mathrm{LM}}}_{\mathrm{dep}}$ denotes
the truth energy deposition of photon 1 in the LM, and
$E^{\mathrm{crystal}_{\mathrm{LM}}}_{\mathrm{rec}}$ the reconstructed crystal
energy in the LM. The crystals contained in the ROI are considered for the
clustering by the GravNet algorithm and significantly extend the $5\times 5$
area considered by the baseline algorithm (Sec. 5). Furthermore, the ROI
represents the area of the local coordinate system later used as an input
feature, with the LM as the origin. Figure 2 (top) shows a typical isolated
photon event with high beam background.

###  4.2 Overlapping Photons

Two different photons that deposit some of their energy in identical crystals
are referred to as overlapping photons. To study overlapping photons, we use
the simulated events with two overlapping photons only. We select events that
have exactly two LMs that must fulfill the following selection criteria:

  1. a)

each LM must have reconstructed crystal energies greater than 10 MeV,

  2. b)

$r^{\gamma_{1}}_{\mathrm{LM_{1}}}\geq 0.2$ and
$r^{\gamma_{1}}_{\mathrm{LM_{1}}}>r^{\gamma_{2}}_{\mathrm{LM_{1}}}$,

  3. c)

$r^{\gamma_{2}}_{\mathrm{LM_{2}}}\geq 0.2$ and
$r^{\gamma_{2}}_{\mathrm{LM_{2}}}>r^{\gamma_{1}}_{\mathrm{LM_{2}}}$.

We refer to criteria a)-c) as LM separation criteria since they ensure that
the particles form two separate LMs. Additionally, events must meet the
overlap criterion:

  1. d)

each of the two photons must deposit at least 10 MeV energy in shared crystals
within a $5\times 5$ area around its respective LM.

Figure 1: Fraction of selected overlapping photon events in the barrel as a
function of generated opening angle. The orange markers correspond to events
fulfilling LM separation criteria a)-c); the blue markers correspond to events
that additionally pass the overlap criterion d) (see text for details).

Figure 1 shows the fraction of events accepted by these selections as a
function of the simulated opening angle. In the scope of this paper, we
additionally require LMs to exclusively originate from simulated particles
without additional background clusters in the ROI, that is:

  1. e)

the two LMs must be the only ones in the ROI and they must be truth-matched to
the simulated photons.

Finally, we remove rare cases of small truth energy depositions and large
backgrounds, by requiring:

  1. f)

the crystal with the largest truth energy deposition of a photon must be
within a $5\times 5$ area around its corresponding LM.

We then create a ROI centered at the midpoint between the two LMs. The
midpoint between the two LMs used to center the ROI is calculated using the
shortest distance between two LMs projected onto the surface of a sphere. The
crystal closest to the midpoint is defined as the ROI center. The LM positions
for this are determined by interpreting the global LM coordinates of their
associated crystals as latitude and longitude. Figure 2 (bottom) shows an
overlapping photon event with high beam background.

The truth energy deposition per photon and the reconstructed crystal energy
$E^{\mathrm{crystal}}_{\mathrm{rec}}$, crystal time
$t^{\mathrm{crystal}}_{\mathrm{rec}}$, crystal PSD information (see Sec. 3),
and the LM positions within the ROI are recorded for each event.

(a) Truth assignment, colors indicate the fraction belonging to each of the
photons and beam background.

(b) Reconstructed time $t$ since trigger time.

(c) Reconstructed PSD hadronic energy ratio. Gray markers indicate that no PSD
information is available.

Figure 2: Typical event displays showing (left) simulated truth assignments,
(center) input variables time, and (right) PSD hadronic energy ratio for (top)
isolated and (bottom) overlapping photons for two example events with high
beam background. The marker centers indicate the crystal centers, the marker
area is proportional to the truth energy deposition for the left plots; it is
proportional to the reconstructed crystal energy for the other plots.

##  5 Reconstruction Algorithms

Interactions of energetic photons in the Belle II ECL typically deposit energy
in up to $5\times 5$ crystals. The task of the clustering reconstruction
algorithms is to select a set of crystals that contains all the energy of the
incoming photon, but no energy from other particles or from beam background.
Low beam background results in approximately $17\,\%$ of all crystals in the
ECL having significant reconstructed energy
$E^{\mathrm{crystal}}_{\mathrm{rec}}\geq 1\,$MeV; for high beam backgrounds
this number is expected to increase to about $40\,\%$. This increase in the
number of crystals to consider in the clustering, adds to the complexity of
the reconstruction.

###  5.1 Baseline

The baseline algorithm is designed to provide maximum efficiency for cluster
finding, contain all crystals from the incoming particle for particle
identification, and select an optimal subset of the cluster crystals that
provides the best energy resolution [17]. The clustering is performed in three
steps. In the first step, all crystals are grouped into a connected set of
crystals, so-called connected regions starting with LMs, as defined
previously, that have a reconstructed crystal energy of more than 10 MeV. In
an iterative procedure all direct neighbors with energies above 0.5 MeV are
added to this LM, and the process is continued if any neighbor itself has
energy above 10 MeV. Overlapping connected regions are merged into one.

In the second step, each connected region is split into clusters, one per LM.
If there is only one LM in the connected region, up to 21 crystals in a
$5\times 5$ area excluding corners centered at the local maximum are grouped
into a cluster. If there is more than one LM in a connected region, the energy
in each crystal of the connected region is assigned a distance-dependent
weight and can be shared between different clusters. The distance is
calculated from the cluster centroid to each crystal center, where the cluster
centroid is updated iteratively using logarithmic energy weights. This process
is repeated until all cluster centroids in a connected region are stable.

In a post-processing step, an optimal subset, including the $n$ highest
energetic crystals of all non-zero weighted crystals that minimize the energy
resolution, is used to predict the cluster energy
$E_{\mathrm{rec}}^{\mathrm{basf2}}$. $n$ depends on the measured noise in the
event, and on the energy of the LM itself. The noise level is estimated by
counting the number of crystals in the event containing more than 5 MeV that
have times $t$ more than 125 ns from the trigger time.
$E_{\mathrm{rec}}^{\mathrm{basf2}}$ is also corrected already within basf2 for
possible bias using simulated events. This bias includes leakage (energy not
deposited in the crystals included in the energy sum) and beam backgrounds
(energy included in the sum that is not from the signal photon).
$E_{\mathrm{rec}}^{\mathrm{basf2}}$ is the estimator for the generated energy
of a particle.

The basf2 clustering algorithm also returns a cluster energy
$E_{\mathrm{rec,\,raw}}^{\mathrm{basf2}}$ that is not corrected for energy
bias. $E_{\mathrm{rec,\,raw}}^{\mathrm{basf2}}$ is the estimator for the
deposited energy of a particle.

###  5.2 Graph Neural Network Architecture

GNN architectures have shown that they are powerful network types to deal with
both irregular geometries and varying input sizes. In this work, all crystals
of an ROI with an energy deposition above 1 MeV are interpreted as nodes in a
graph, which leads to variable input sizes and is thus a good use case for
GNNs. The implementation of this GNN is done in PyTorch Geometric [26].

The input features consist of crystal properties and crystal measurements: The
global coordinates $\theta$ and $\phi$ of each crystal, the local coordinates
$\theta^{\prime}$ and $\phi^{\prime}$ with respect to the ROI center, the
crystal mass, and the LM(s) (in one-hot encoding) represent crystal
properties. The crystal energy $E^{\mathrm{crystal}}_{\mathrm{rec}}$ in GeV,
the time $t^{\mathrm{crystal}}_{\mathrm{rec}}$ in $\mu$s, and the PSD fit
type, PSD $\chi^{2}$, and PSD hadronic energy ratio are crystal measurements
used as input features. Pre-processing scales the input uniformly before
further processing with the GNN: All features are min-max normalized to an
interval of $[0,1]$ with the exception of
$t^{\mathrm{crystal}}_{\mathrm{rec}}$ and the PSD hadronic energy ratio which
are both normalized to the interval $[-1,1]$. The global coordinates and the
crystal masses are normalized based on the range of coordinates and masses of
all crystals in the detector instead of only the ones in the ROI.
Additionally, we average each input feature over all nodes in the ROI and
concatenate the averaged input features as additional inputs, thus enabling a
global exchange of information.

As displayed in Fig. 3, our model is built out of four so-called GravNet [16]
blocks of which the concatenated outputs are passed through three dense output
layers with a final softmax activation function. Each GravNet block features
three dense layers at the beginning of the block, the initial two of which
with ELU [27] activation functions and the last one with a $\tanh$ activation
function. The dense layers feed into a GravNet layer and the overall GravNet
block is concluded by a batch normalization layer [28]. The GravNet layer is
responsible for the graph building and subsequent message passing between the
nodes of the graph. It first translates the input features into two learned
representation spaces: one representing spatial information $S$ while the
other, denoted $F_{\mathrm{LR}}$, contains the transformed features used for
message passing. In the second step, each node is connected to its $k$ nearest
neighbors defined by the Euclidean distances in $S$, thus creating an
undirected, connected graph. For each node, the input features of connected
nodes are then weighted by a Gaussian potential depending on the distance in
$S$ and aggregated by summation. The resulting features are concatenated with
the initial features and, after batch normalization, passed to the next
GravNet block and to the dense output layers.

The implementation in the present work follows the concept of fuzzy clustering
which refers to the partial assignment of individual crystals to several
clustering classes. Consequently, the GNN predicts weights
$w_{i}^{\mathrm{X}}$ that indicate the proportion of the reconstructed energy
$E^{\mathrm{crystal_{i}}}_{\mathrm{rec}}$ in a crystal $i$ that belongs to a
clustering class X. For models used with isolated photons,
$\mathrm{X}\in\\{\gamma_{1},\mathrm{background}\\}$, for models with
overlapping photons
$\mathrm{X}\in\\{\gamma_{1},\gamma_{2},\mathrm{background}\\}$. As a loss
function, we then use the Mean Squared Error (MSE) between the true and
predicted weights summed over all classes and crystals. The training is
stopped when there has been no improvement for 15 epochs in the optimization
objective. For low beam background models that objective is the MSE loss on
the validation data set, whereas the high beam background models employ the
more high-level FWHMdep (Sec. 6) on the validation data set.

Figure 3: An illustration of the GNN architecture. Each pair of gray, square brackets represents one GravNet block consisting of dense layers, a GravNet layer and a batch norm layer. Table 1: Optimized hyperparameters of the isolated photon, and overlapping photon GravNet models. The hyperparameters are the result of an optimization of the FWHMdep on the respective high background validation data set.  Hyperparameter | Isolated Photon Models | Overlapping Photon Models
---|---|---
Width of the Dense Layers, F${}_{\mathrm{IN}},\,$FOUT | 22 | 24
Feature Space Dimension FLR | 16 | 16
Spatial Information Space Dimension S | 6 | 6
Connected Nearest Neighbors $k$ | 14 | 16
Batch Norm Momentum | 0.01 | 0.4
Stacked GravNet Blocks | 4 | 4
Batch Size | 1024 | 512

(a) Truth

(b) GNN

(c) basf2

Figure 4: Comparison of (4(a)) truth energy fractions , (4(b)) reconstructed
energy fraction by the GNN , and (4(c)) reconstructed energy fraction by basf2
for an example event with high beam background. Colors indicate the fractions
belonging to each photon or background. The marker centers indicate the
crystal centers, the marker area is proportional to the truth or reconstructed
(GNN, basf2) energy deposition respectively.

Hyperparameters have been chosen through a hyperparameter optimization using
Optuna [29]. The optimization is done with respect to the FWHMdep (Sec. 6)
instead of the loss function. We optimize the two models trained for high beam
backgrounds and use the respective hyperparameters also for the corresponding
low beam background models. The final hyperparameters for both the isolated
photon models and the overlapping photon models are shown in Table 1.

The learning rate, the number of dense layers in each GravNet block, and all
dimensions of the output layers have been manually optimized by testing a
reasonable range of values. The learning rate is set to 5 $\times\,10^{-3}$
and is subject to a decay factor of 0.25 after every five epochs of stagnating
validation loss. We did not observe significant over-training and as a
consequence, we do not use dropout layers or other regularization methods but
rely on the large data set.

The GNN algorithm yields the weights $w_{i}^{\mathrm{X}}$ per crystal for all
crystals in the ROI with an energy deposition above 1 MeV. In order to
reconstruct the total cluster energy $E_{\mathrm{rec}}^{\mathrm{GNN}}$
associated with a certain particle, we then sum over all specific weights
multiplied by the reconstructed energies per crystal,
$E_{\mathrm{rec}}^{\mathrm{GNN}}=\sum
w_{i}E^{\mathrm{crystal_{i}}}_{\mathrm{rec}}$.

Figure 4 shows how the GNN and the basf2 algorithms behave in clustering a
typical case of overlapping photons.

##  6 Metrics

For performance evaluation, the reconstructed energy of a particle is compared
with two different truth targets: the total deposited truth energy
$E_{\mathrm{dep}}$ per photon in the ROI, and the generated truth energy
$E_{\mathrm{gen}}$ per photon. This results in two variants of relative
reconstruction errors. The reconstruction error on the deposited energy

$\displaystyle\eta_{\text{dep}}^{\text{basf2}}$
$\displaystyle=\frac{E_{\mathrm{rec,\,raw}}^{\mathrm{basf2}}-E_{\mathrm{dep}}}{E_{\mathrm{dep}}}\quad\text{and}$
$\displaystyle\eta_{\text{dep}}^{\text{GNN}}$
$\displaystyle=\frac{E_{\mathrm{rec}}^{\mathrm{GNN}}-E_{\mathrm{dep}}}{E_{\mathrm{dep}}}$
(2)

gives access to the energy resolution ignoring leakage and other detector
effects. It is a direct evaluation of the clustering performance of an
algorithm.

On the other hand, the reconstruction error on the generated energy

$\displaystyle\eta_{\text{gen}}^{\text{basf2}}$
$\displaystyle=\frac{E_{\mathrm{rec}}^{\mathrm{basf2}}-E_{\mathrm{gen}}}{E_{\mathrm{gen}}}\quad\text{and}$
$\displaystyle\eta_{\text{gen}}^{\text{GNN}}$
$\displaystyle=\frac{E_{\mathrm{rec}}^{\mathrm{GNN}}-E_{\mathrm{gen}}}{E_{\mathrm{gen}}}$
(3)

factors in all detector and physics effects and quantifies how much of the
improvements to the underlying clustering carry over to downstream physics
object reconstruction.

Figure 5: Example distribution of the relative reconstruction error
$\eta_{\text{gen}}$ of the generated energy and illustration of the bias
correction, the FWHM, and the tail ranges.

Evaluating both algorithms on a large number of simulated photons yields
peaking distributions in both reconstruction errors $\eta_{\text{dep}}$ and
$\eta_{\text{gen}}$. Both distributions are potentially biased because of
energy leakage and the presence of beam backgrounds (compare Sec. 5.1). We
perform a binned fit using a double-sided crystal ball [30, 31] function as
probability density function (pdf) with the kafe2 [32] framework. We shift all
reconstruction error distributions independently by a multiplicative factor to
correct the difference between the fitted peak position and zero (Fig. 5).
Since $\eta_{\text{dep}}$ and $\eta_{\text{gen}}$ are asymmetric
distributions, we repeat this procedure until the difference between the
fitted peak position and zero is less than 0.002. This procedure usually
converges within two or three iterations.

We then determine the full width half maximum (FWHM) of the final shifted
distributions in $\eta_{\text{dep}}$ and $\eta_{\text{gen}}$, yielding
$\text{FWHM}_{\text{dep}}$ and $\text{FWHM}_{\text{gen}}$ respectively. The
uncertainty on the FWHM is calculated from the uncertainties of the fit
parameters. We define the resolution as
$\sigma_{\text{gen}}=\text{FWHM}_{\text{gen}}/2.355$. In addition to the FWHM,
we determine the tails of the reconstruction error distribution. The left and
right tails $T_{\text{L,R}}$ are calculated as the 95th percentile when
ranking the unbinned events on the respective side of the peak position, as
given by the fit parameters, in ascending order ($T_{\text{R}}$) and
descending order ($T_{\text{L}}$) respectively. Propagating the uncertainty on
the peak position as given by the fit yields the uncertainty on
$T_{\text{L,R}}$.

##  7 Results

The first sections of the results focus on detailed studies of isolated
clusters. Section 7.4 then introduces overlapping clusters and their effects
on the performance. Figure 6 shows examples for the distributions of both
reconstruction errors $\eta_{\text{dep}}$ and $\eta_{\text{gen}}$, as well as
the fit results for events with low beam background. Figure 7 shows the
equivalent distributions for events with high beam background.

The $\eta_{\text{gen}}$ distributions are wider because the reconstruction
error includes the effects of leakage which result in missing energy with
respect to the generated photon energy. This only affects the left-side tails.

(a) Relative reconstruction error $\eta_{\text{dep}}$ of the deposited energy.

(b) Relative reconstruction error $\eta_{\text{gen}}$ of the generated energy.

Figure 6: Distribution of relative reconstruction errors (6(a))
$\eta_{\text{dep}}$ and (6(b)) $\eta_{\text{gen}}$ for isolated clusters for
low beam backgrounds. The first bin contains all underflow entries; the last
bin contains all overflow entries.

(a) Relative reconstruction error $\eta_{\text{dep}}$ of the generated energy.

(b) Relative reconstruction error $\eta_{\text{gen}}$ of the generated energy.

Figure 7: Distributions of relative reconstruction errors (7(a))
$\eta_{\text{dep}}$ and (7(b)) $\eta_{\text{gen}}$ for isolated clusters for
high beam backgrounds. The first bin contains all underflow entries; the last
bin contains all overflow entries.

In the following subsections, we are comparing the performance of the GNN and
the basf2 reconstruction algorithms for different detector regions for low and
high beam backgrounds by evaluating the energy resolution
$\text{FWHM}_{\text{gen}}/2.355$ and the tail parameters. We then analyze the
GNN in more detail by testing the input variable dependencies and the
robustness against differences in beam background levels between training and
evaluation.

###  7.1 Energy resolution and energy tails

The three detector regions barrel, forward endcap, and backward endcap
described in Sec. 3 differ in crystal geometry, levels of background, and
amount of passive material before and in between crystals. The following
section studies the variations in the energy reconstruction performance that
arise as a direct result of these differences.

(a) Low beam background.

(b) High beam background.

Figure 8: Resolution $\text{FWHM}_{\text{gen}}/2.355$ of the GNN and basf2 as
function of the simulated photon energy $E_{\mathrm{gen}}$ for both endcaps
and the barrel for (8(a)) low and (8(b)) high beam background. Each color is
associated with one detector region; the light color indicates basf2, the dark
color the GNN. The bands indicate the uncertainty of the fits, see text for
details. The fit parameters are summarized in Tab. 2.

In order to access the energy dependence of the resolution and tail parameters
we simulate test data sets of photons at various fixed energies. The FWHM for
each simulated data set is then determined according to Sec. 6. Plotting the
resolutions $\text{FWHM}_{\text{gen}}/2.355$ over the generated photon
energies $E_{\mathrm{gen}}$ reveals a characteristic relationship that is
parameterized by the function $a/E_{\mathrm{gen}}\oplus
b/\sqrt{E_{\mathrm{gen}}}\oplus c$, where $\oplus$ indicates addition in
quadrature.

Both the GNN as well as the baseline algorithm perform differently in regards
to the energy resolution in all three detector parts, as can be seen in Fig.
8(a) for low beam background and as Fig. 8(b) for high beam background. Table
2 reports the parameters of the fitted parameterization of the resolution. We
attribute these difference to the large spread of both shape and size of
crystals in the endcaps, the asymmetric distribution of beam backgrounds, and
the different amount of passive material in front of the different detector
regions.

Overall, the energy resolution of the GNN algorithm is significantly better
than the baseline algorithm for all photon energies. The GNN energy resolution
is better by more than 30 % for photon energies below
$500\,\mathrm{\,Me\kern-1.00006ptV}$ which is the energy range of more than 90
% of all photons in $B$-meson decay chains. The higher the beam background,
the larger the difference between the GNN and the baseline algorithm. The
difference between the two algorithms decreases with energy because the
relative contribution of beam backgrounds to the photon energy resolution
decreases.

The shape of the left-side tails is dominated by passive material and is hence
expected to be different in the different detector regions. The left-side
tails are almost independent of beam backgrounds as can be seen by comparing
Fig. 9(a) for low beam background and Fig. 9(c) for high beam background. The
GNN and the baseline algorithm both show the smallest tail length for the
barrel region with decreasing tail lengths for increasing energy. The left-
side tails are largest in the backward endcap due to the highest ratio of
passive to active material as expected. The right-side tails are mostly
originating from beam background being wrongly added to photon clusters. The
GNN produces shorter tails than the baseline algorithm for all energies and
for both low and high beam backgrounds, with the performance difference
increasing for lower energies and higher beam backgrounds.

Table 2: Fit results ($a/E_{\mathrm{gen}}\oplus b/\sqrt{E_{\mathrm{gen}}}\oplus c$) of the fits shown in Fig. 8. Region | Algorithm | Low Beam Background | High Beam Background
---|---|---|---
a ($\times 10^{-2}$) | b ($\times 10^{-2}$) | c ($\times 10^{-2}$) | a ($\times 10^{-2}$) | b ($\times 10^{-2}$) | c ($\times 10^{-2}$)
Barrel | GNN | 0.23$\pm$0.02 | 1.32$\pm$0.02 | 1.00$\pm$0.01 | 1.25$\pm$0.02 | 2.39$\pm$0.02 | 0.75$\pm$0.03
basf2 | 0.35$\pm$0.02 | 1.54$\pm$0.02 | 0.91$\pm$0.02 | 1.88$\pm$0.02 | 3.11$\pm$0.03 | 0.31$\pm$0.10
Forward | GNN | 0.00$+$0.14 | 1.11$\pm$0.01 | 1.49$\pm$0.00 | 0.61$\pm$0.03 | 2.23$\pm$0.02 | 1.20$\pm$0.02
basf2 | 0.00$+$0.37 | 1.51$\pm$0.01 | 1.38$\pm$0.01 | 1.11$\pm$0.03 | 2.92$\pm$0.03 | 0.84$\pm$0.03
Backward | GNN | 0.50$\pm$0.02 | 1.69$\pm$0.03 | 1.59$\pm$0.02 | 2.18$\pm$0.03 | 2.51$\pm$0.05 | 2.28$\pm$0.02
basf2 | 0.78$\pm$0.03 | 2.12$\pm$0.04 | 1.50$\pm$0.03 | 2.72$\pm$0.05 | 4.64$\pm$0.05 | 0.91$\pm$0.08

(a) Left tail length $T_{L}$.

(b) Right tail length $T_{R}$.

(c) Left tail length $T_{L}$.

(d) Right tail length $T_{R}$.

Figure 9: 95 % left- and right tail lengths $T_{L}$ and $T_{R}$ of
$\eta_{\text{gen}}$ for the GNN and basf2 as function of the simulated photon
energy $E_{\mathrm{gen}}$ for both endcaps and the barrel for (9(a) and 9(b))
low and (9(c) and 9(d)) high beam background. Each color is associated with
one detector region.

###  7.2 Beam Background Robustness

The beam background levels are changing continuously during detector
operations. Ideally, reconstruction algorithms at Belle II are insensitive to
such changes. The basf2 baseline algorithm achieves robustness against
increasing beam backgrounds by adaptively including fewer crystals in the
energy sum calculation. Since our GNN is trained with a large number of events
with event-by-event fluctuations of beam backgrounds, we expect robustness
against varying beam backgrounds if the GNN generalizes well enough. We test
the robustness of our GNN by comparing GNNs trained and tested on the same
backgrounds, against GNNs trained and tested on the two different beam
backgrounds (Fig. 10, parameterization in Tab. 3). While the GNNs trained on
the same beam backgrounds achieve a better resolution than the ones trained on
different beam backgrounds, the GNN still outperforms the baseline algorithm
even for networks trained on the different beam backgrounds. This demonstrates
an excellent generalization with respect to different levels of beam
backgrounds.

Figure 10: Resolution $\text{FWHM}_{\text{gen}}/2.355$ as a function of the simulated photon energy $E_{\mathrm{gen}}$ for the GNNs trained with low beam background (LBB GNN) and high beam background (HBB GNN) in the barrel. The color is associated with the evaluation on either beam background; the dark color indicates the model trained with the beam background identical to the evaluation, and the light color indicates the model trained with the respective other beam background. The bands indicate the uncertainty of the fits, see text for details. The fit parameters are summarized in Tab. 3. Table 3: Fit results ($a/E_{\mathrm{gen}}\oplus b/\sqrt{E_{\mathrm{gen}}}\oplus c$) of the fits shown in Fig. 10 for the GNN trained with low beam background (LBB GNN) and high beam background (HBB GNN). The values for the LBB GNN inferred on low beam background test samples, and for the HBB GNN inferred on high beam background are identical to the ones reported in Tab. 2. Region | Algorithm | Low Beam Background | High Beam Background
---|---|---|---
a ($\times 10^{-2}$) | b ($\times 10^{-2}$) | c ($\times 10^{-2}$) | a ($\times 10^{-2}$) | b ($\times 10^{-2}$) | c ($\times 10^{-2}$)
Barrel | LBB GNN | 0.23$\pm$0.02 | 1.32$\pm$0.02 | 1.00$\pm$0.01 | 1.59$\pm$0.02 | 2.27$\pm$0.03 | 1.32$\pm$0.02
HBB GNN | 0.28$\pm$0.02 | 1.58$\pm$0.01 | 0.85$\pm$0.02 | 1.25$\pm$0.02 | 2.39$\pm$0.02 | 0.75$\pm$0.03

###  7.3 Input Parameter Dependency

As discussed in Sec. 3, multiple input features are available for the GNN,
while the basf2 algorithm uses crystal position and energy only. This section
presents a study of the influence of the input features on the FWHM. For that,
the architecture described in Sec. 5.2 is trained on isolated photon events
with low or high beam backgrounds using different combinations of input
features. The 200 000 events from the respective validation data set, as
described in Sec. 4, are used for inference. The data set covers an energy
range of $0.1<E_{\mathrm{gen}}<1.5\,\text{GeV}$ and the full detector range
$17^{\circ}<\theta_{\mathrm{gen}}<150^{\circ}$ and
$0^{\circ}<\phi_{\mathrm{gen}}<360^{\circ}$, each of which in uniform
distribution. The FWHM of $E_{\mathrm{gen}}$ and $E_{\mathrm{dep}}$ is
calculated as described in Sec. 6. All GNNs use the global crystal
coordinates, the LM position, and the crystal mass as input features. A
comparison of the FWHM for the different additional input features is shown in
Tab. 4. The results show, that even for the minimal set of input variables,
the GNN’s FWHM is smaller than basf2’s for both the deposited and the
generated energy in both beam background scenarios. Adding local coordinates
leads to small improvements and using time information brings significant
improvement in the GNN performance. PSD information has almost no effect on
the FWHM. Since the main purpose of the PSD information is to differentiate
electromagnetic and hadronic interactions per crystal, this is expected. In
anticipation of future extensions of the GNN to hadronic interactions as well,
the PSD information is kept throughout this work.

Table 4: Comparison of the performances of GNN models with different additional input features, and the performance of the basf2 baseline. Shown are the $\mathrm{FWHM_{dep}}$ and $\mathrm{FWHM_{gen}}$ (see Sec. 6), for 200 000 events in the validation data sets (see Sec. 4) with low and high beam background. The data sets cover an energy range of $0.1<E_{\mathrm{gen}}<1.5\,\text{GeV}$ and the full detector range $17^{\circ}<\theta_{\mathrm{gen}}<150^{\circ}$ and $0^{\circ}<\phi_{\mathrm{gen}}<360^{\circ}$, each of which in uniform distribution. The uncertainties of the FWHM in each column are correlated since they use the same simulated events. The input features are described in detail in Sec. 3. Input Features | Low Beam Background |  High Beam Background
---|---|---
$\mathrm{FWHM_{dep}}$ $\times 10^{-2}$ |  $\ \ \mathrm{FWHM_{gen}}$ $\ \ \times 10^{-2}$ |  $\mathrm{FWHM_{dep}}$ $\times 10^{-2}$ |  $\ \ \mathrm{FWHM_{gen}}$ $\ \ \times 10^{-2}$
Energy |  2.17$\pm$0.01 |  5.25$\pm$0.02 |  5.05$\pm$0.03 |  8.08$\pm$0.04
Energy, local coordinates |  2.11$\pm$0.02 |  5.19$\pm$0.02 |  5.04$\pm$0.04 |  8.04$\pm$0.04
Energy, local coordinates, PSD |  2.19$\pm$0.01 |  5.20$\pm$0.02 |  5.06$\pm$0.03 |  8.07$\pm$0.04
Energy, local coordinates, time |  1.72$\pm$0.01 |  4.85$\pm$0.02 |  4.52$\pm$0.03 |  7.63$\pm$0.03
Energy, local coordinates, time, PSD |  1.72$\pm$0.01 |  4.85$\pm$0.02 |  4.51$\pm$0.03 |  7.62$\pm$0.03
basf2 |  2.32$\pm$0.02 |  5.13$\pm$0.02 |  6.73$\pm$0.05 |  8.97$\pm$0.07

###  7.4 Overlapping Photons

When discussing overlapping photon events, it is important to note that the
FWHM of the photon energy distribution not only depends on its own properties
but also on the properties of the second photon present. To account for that,
the evaluation is split in energy bins of [0.1, 0.2], [0.2, 0.5], [0.5, 1.0],
and [1.0, 1.5] $\mathrm{\,Ge\kern-1.00006ptV}$ for both photons respectively.
We then show the FWHM of the first photon for different simulated energies of
the second photon for low beam backgrounds (see Tab. 5) and high beam
backgrounds (Tab. 6).

The GNN provides a better FWHM for all combinations, but the improvement is
most significant if the photon is low energetic. For low beam backgrounds, the
GNN improves the FWHM by up to 20 % for photons with simulated energies
between $0.1<E_{\mathrm{gen}}<0.2$ $\mathrm{\,Ge\kern-1.00006ptV}$. For high
beam backgrounds, the GNN improves the FWHM by more than 35 % for photons with
simulated energies between $0.1<E_{\mathrm{gen}}<0.2$
$\mathrm{\,Ge\kern-1.00006ptV}$.

The result shows that the significant performance improvement observed for
isolated photons can also be achieved for the more complicated overlapping
photon signatures.

##  8 Conclusion and Outlook

(a) Truth

(b) GravNet

(c) basf2

Figure 11: Comparison of truth energy fractions (11(a)), the reconstructed
energy fraction by the GNN (11(b)), and the reconstructed energy fraction by
basf2 (11(c)) for one example event with only one local maximum. Colors
indicate the fractions belonging to each photon or background. The marker
centers indicate the crystal centers, the marker area is proportional to the
reconstructed energy in each crystal.

In this work, we have presented a complete study of a GNN-based clustering
algorithm for the Belle II electromagnetic calorimeter. We have been using a
realistic full detector simulation and simulated beam background for low and
high luminosity conditions of Belle II. The GNN algorithm has been compared to
the currently used basf2 baseline algorithm. We find a significantly improved
resolution of more than 30 % for high beam backgrounds, but also improved
performance in reducing the right-side tails of the reconstruction errors that
are caused by beam background. Such significant improvements in photon
reconstruction performance directly improve the physics reach of Belle II for
almost all final states with photons, but also analyses that use missing
energy information [17]. We also trained different GNNs to separate energy
depositions of overlapping photon clusters. The improvement of the energy
resolution is up to 30 % for the low energy photon in asymmetric photon pairs.
Any improvement in overlapping photon reconstruction has direct implications
for the reconstruction of boosted $\pi^{0}$ mesons or axion-like particles
with couplings to photons [33].

While the basf2 algorithm strictly reconstructs one cluster for each LM, the
GNN algorithm only uses the LMs to center the ROI. The GNN algorithm can
therefore in principle also be used to reconstruct overlapping photons that
only produced one LM (Fig. 11). The extension of the GNN algorithm to such
overlapping signatures as well as to charged particles and neutral hadrons
will be the focus of follow-up work.

This is the first application of a GNN-based clustering algorithm at Belle II
for a realistic detector geometry and realistic and high beam backgrounds.
This is also the first time that an algorithm has shown to improve the
performance of the photon reconstruction by explicitly including timing
information on clustering level at Belle II.

Table 5: $\mathrm{FWHM_{gen}}\times 10^{2}$ of one photon with photon energy $E_{\gamma}^{(1)}$ in dependence of the second photon energy $E_{\gamma}^{(2)}$ for low beam background for the full detector (barrel and endcaps combined). The uncertainties of the FWHM for the two algorithms are correlated for each energy interval since they use the same simulated events. The improvement over the basf2 baseline algorithm is stated in percent for each energy interval. $E_{\gamma}^{(1)}$ (GeV) $\downarrow$ |  $E_{\gamma}^{(2)}$ (GeV) $\rightarrow$ |  [0.1, 0.2] |  [0.2, 0.5] |  [0.5, 1.0] |  [1.0, 1.5]
---|---|---|---|---|---
[0.1, 0.2] | GNN |  11.04$\pm$0.79 |  11.98$\pm$0.40 |  11.94$\pm$0.31 |  13.25$\pm$0.34
basf2 |  12.72$\pm$0.80 |  13.93$\pm$0.55 |  14.32$\pm$0.41 |  15.16$\pm$0.48
Improvement |  15.2 % |  16.3 % |  20.0 % |  14.4 %
[0.2, 0.5] | GNN |  7.38$\pm$0.18 |  7.57$\pm$0.12 |  8.23$\pm$0.09 |  8.38$\pm$0.12
basf2 |  8.48$\pm$0.22 |  8.30$\pm$0.14 |  8.84$\pm$0.12 |  8.96$\pm$0.12
Improvement |  14.9 % |  9.7 % |  7.5 % |  7.0 %
[0.5, 1.0] | GNN |  5.22$\pm$0.08 |  5.43$\pm$0.05 |  5.69$\pm$0.04 |  5.89$\pm$0.04
basf2 |  5.58$\pm$0.10 |  5.71$\pm$0.06 |  5.85$\pm$0.05 |  6.17$\pm$0.05
Improvement |  6.7 % |  5.1 % |  2.8 % |  4.9 %
[1.0, 1.5] | GNN |  4.24$\pm$0.06 |  4.43$\pm$0.04 |  4.67$\pm$0.03 |  4.77$\pm$0.03
basf2 |  4.55$\pm$0.07 |  4.58$\pm$0.04 |  4.74$\pm$0.04 |  4.85$\pm$0.04
Improvement |  7.3 % |  3.4 % |  1.4 % |  1.8 %
Table 6: $\mathrm{FWHM_{gen}}\times 10^{2}$ of one photon with photon energy $E_{\gamma}^{(1)}$ in dependence of the second photon energy $E_{\gamma}^{(2)}$ for high beam background for the full detector (barrel and endcaps combined). The uncertainties of the FWHM for the two algorithms are correlated for each energy interval since they use the same simulated events. The improvement to the basf2 baseline is stated in percent for each energy interval. $E_{\gamma}^{(1)}$ (GeV) $\downarrow$ |  $E_{\gamma}^{(2)}$ (GeV) $\rightarrow$ |  [0.1, 0.2] |  [0.2, 0.5] |  [0.5, 1.0] |  [1.0, 1.5]
---|---|---|---|---|---
[0.1, 0.2] | GNN |  24.77$\pm$0.83 |  24.10$\pm$0.76 |  24.02$\pm$0.60 |  24.72$\pm$0.63
basf2 |  33.12$\pm$1.08 |  32.82$\pm$1.38 |  31.28$\pm$0.79 |  32.42$\pm$0.88
Improvement |  33.7 % |  36.2 % |  30.3 % |  31.1 %
[0.2, 0.5] | GNN |  13.16$\pm$0.30 |  13.96$\pm$0.20 |  14.17$\pm$0.16 |  14.17$\pm$0.16
basf2 |  17.73$\pm$0.47 |  17.56$\pm$0.31 |  17.62$\pm$0.24 |  16.88$\pm$0.23
Improvement |  34.8 % |  25.8 % |  24.3 % |  19.1 %
[0.5, 1.0] | GNN |  8.07$\pm$0.12 |  8.56$\pm$0.08 |  8.71$\pm$0.06 |  8.84$\pm$0.06
basf2 |  10.53$\pm$0.19 |  10.77$\pm$0.12 |  10.75$\pm$0.09 |  10.73$\pm$0.08
Improvement |  30.6 % |  25.8 % |  23.4 % |  21.4 %
[1.0, 1.5] | GNN |  6.05$\pm$0.08 |  6.33$\pm$0.05 |  6.42$\pm$0.04 |  6.54$\pm$0.04
basf2 |  7.52$\pm$0.12 |  7.56$\pm$0.07 |  7.60$\pm$0.06 |  7.68$\pm$0.06
Improvement |  24.2 % |  19.6 % |  18.3 % |  17.4 %

Data Availability StatementThe datasets generated during and analysed during
the current study are property of the Belle II collaboration and not publicly
available. Acknowledgments The authors would like to thank the Belle II
collaboration for useful discussions and suggestions on how to improve this
work. The authors would like to thank Jan Kieseler for helpful discussions.

The training of the models was performed on the TOpAS GPU cluster at the
Steinbuch Centre for Computing (SCC) at KIT. This work is funded by Helmholtz
(HGF) Young Investigators Group VH-NG-1303 and BMBF ErUM-Pro 05H23VKKBA. I.
Haide is supported by the Landesgraduiertenförderung Baden-Württemberg.

## Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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January 2021

HU-EP-20/46-RTG

String Dualities at Order $\alpha^{\prime\,3}$

Tomas Codina†, Olaf Hohm† and Diego Marques∗

† Institute for Physics, Humboldt University Berlin,
Zum Großen Windkanal 6, D-12489 Berlin, Germany

∗ Instituto de Astronomía y Física del Espacio,
Casilla de Correo 67 - Suc. 28 (C1428ZAA), Buenos Aires, Argentina

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We compute the cosmological reduction of the fourth powers of the Riemann
> tensor claimed to arise in string theory at order $\alpha^{\prime\,3}$, with
> overall coefficient proportional to $\zeta(3)$, and show that it is
> compatible with an $O(9,9)$ symmetry. This confirms the general result in
> string theory, due to Sen, that classical string theory with $d$-dimensional
> translation invariance admits an $O(d,d)$ symmetry to all orders in
> $\alpha^{\prime}$.

##  1 Introduction

String theory continues to be a most promising framework for a consistent
theory of quantum gravity. At low energies string theories are described by
Einstein’s theory of general relativity coupled to matter fields, which
universally include an antisymmetric tensor (B-field) and a scalar (dilaton).
Intriguingly, however, even classical string theory modifies general
relativity in two important respects: it includes an infinite number of
higher-derivative corrections governed by the (inverse) string tension
$\alpha^{\prime}$ [1, 2], and it permits dualities identifying solutions that
are drastically different in standard geometry. These general features suggest
promising scenarios for cosmology [3, 4, 5, 6, 7], but an immediate obstacle
is that the explicit form of the $\alpha^{\prime}$ corrections is at best
known to the first few orders. For Type II string theories not even the first
non-trivial higher-derivative corrections, which arise at order
$\alpha^{\prime\,3}$, are known completely. Moreover, notwithstanding early
important work in [8, 9, 10], the compatibility of $\alpha^{\prime}$
corrections with string dualities such as T-duality has only in recent years
become the focus of attention. The ‘space of duality invariant cosmologies’
has been explored to all orders in $\alpha^{\prime}$ and shown to permit novel
features [11, 12], but it is not known which points in this theory space
actual string theories inhabit. It is thus a matter of some urgency to find
efficient methods to deal with $\alpha^{\prime}$ corrections.

In this letter we investigate T-duality at order $\alpha^{\prime\,3}$. Our
goal is to analyze whether the eight-derivative corrections (quartic in the
Riemann tensor) with overall coefficient proportional to the transcendental
$\zeta(3)$ are compatible with the $O(9,9)$ T-duality invariance upon
reduction to one dimension (cosmic time). An important issue is that one
should allow for the $O(d,d)$ transformations themselves to receive
$\alpha^{\prime}$ corrections [13]. It indeed follows from Meissner’s work on
the cosmological reduction to first order in $\alpha^{\prime}$ that the
T-duality transformations in terms of standard supergravity fields are
$\alpha^{\prime}$-deformed [9, 14, 15]. However, it is possible to find new
field variables for which the T-duality transformations take the standard
form. A general framework was developed in [12, 11] that systematically uses
field redefinitions to bring both the dimensionally reduced action and the
most general $O(d,d)$ invariant action to a form that involves only first
order derivatives. The claim is that this procedure eliminates all ambiguities
resulting from the freedom to perform integrations by part and to use lower-
order equations of motion to modify higher-derivative terms. Upon passing to
this canonical field basis the $O(d,d)$ invariance, if present, should take
the standard form. This yields a systematic procedure to test the reduced
actions for $O(d,d)$ invariance, which has been successfully applied to first
order in $\alpha^{\prime}$ in cosmological reductions and, more recently, for
general torus compactifications [16]. While the complete higher-derivative
corrections at order $\alpha^{\prime\,3}$ are not known, the eight-derivative
terms involving only the metric are believed to be known completely. We will
see that this is sufficient to show compatibility with $O(d,d)$, which in turn
determines the B-field and dilaton couplings that survive upon cosmological
reduction.

Apart from potential applications in cosmology, this result is of conceptual
interest in view of these eight-derivative corrections being proportional to
the transcendental number $\zeta(3)$. The transcendentality implies that these
corrections cannot be linked by conventional symmetry transformations to
corrections with rational coefficients, as present in bosonic and heterotic
string theory starting at first order in $\alpha^{\prime}$. It is sometimes
questioned whether the $\zeta(3)$ couplings are compatible with the continuous
$O(d,d,\mathbb{R})$. In a previous version of this letter we arrived at the
incorrect conclusion that these couplings are not compatible with
$O(d,d,\mathbb{R})$ invariance, but this would have been in quite serious
conflict with basic principles of string theory. As shown by Sen, classical
(tree-level) string theory truncated to states of zero momentum along $d$
directions admits an $O(d,d,\mathbb{R})$ invariance to all orders in
$\alpha^{\prime}$ [18]. While the original proof was couched in the language
of string field theory the argument only relies on the symmetries of the
S-matrix of this consistently truncated sector. At tree-level, holomorphic
factorization yields two independent manifest $O(d,\mathbb{R})$ symmetries,
and combining this $O(d,\mathbb{R})\times O(d,\mathbb{R})$ invariance with the
$GL(d,\mathbb{R})$ symmetry following from diffeomorphism invariance and
constant shifts of the B-field implies $O(d,d,\mathbb{R})$ invariance [19].
Thus, the tree-level corrections at order $\alpha^{\prime\,3}$ proportional to
$\zeta(3)$ really ought to be consistent with $O(d,d,\mathbb{R})$, and indeed
they are.

##  2 Cosmological Reduction

We now review the leading corrections in Type II string theory, and compute
the minimal form of the one-dimensional effective action obtained after a
cosmological reduction. The $\alpha^{\prime}$ corrections in Type II string
theory begin at $\alpha^{\prime}{}^{3}$. The couplings for the gravitational
sector were originally computed from four-point scattering amplitudes [1], and
later from the sigma-model $\beta$-function [20, 21, 22]. They take the
compact form

$J(c)\equiv t_{8}t_{8}R^{4}+\frac{c}{8}\epsilon_{10}\epsilon_{10}R^{4}\ ,$
(2.1)

where $c=1$ has been determined in the literature, but here we keep it more
general in order to see whether this is fixed by duality arguments. The first
term in (2.1) is

$\displaystyle t_{8}t_{8}R^{4}=\ $ $\displaystyle
t^{\mu_{1}\dots\mu_{8}}t_{\nu_{1}\dots\nu_{8}}R_{\mu_{1}\mu_{2}}{}^{\nu_{1}\nu_{2}}R_{\mu_{3}\mu_{4}}{}^{\nu_{3}\nu_{4}}R_{\mu_{5}\mu_{6}}{}^{\nu_{5}\nu_{6}}R_{\mu_{7}\mu_{8}}{}^{\nu_{7}\nu_{8}}$
$\displaystyle=\ $ $\displaystyle 3\cdot
2^{7}\left[R_{\alpha\beta\mu\nu}R^{\beta\gamma\nu\rho}R^{\sigma\mu}{}_{\gamma\delta}R^{\delta\alpha}{}_{\rho\sigma}+\frac{1}{2}R_{\alpha\beta\mu\nu}R^{\beta\gamma\nu\rho}R_{\gamma\delta\rho\sigma}R^{\delta\alpha\sigma\mu}\right.$
$\displaystyle-\frac{1}{2}R_{\alpha\beta\mu\nu}R^{\beta\gamma\mu\nu}R_{\gamma\delta\rho\sigma}R^{\delta\alpha\rho\sigma}-\frac{1}{4}R_{\alpha\beta\mu\nu}R^{\beta\gamma\rho\sigma}R^{\mu\nu}{}_{\gamma\delta}R^{\delta\alpha}{}_{\rho\sigma}$
$\displaystyle\left.+\frac{1}{16}R_{\alpha\beta\mu\nu}R^{\beta\alpha\rho\sigma}R^{\gamma\delta\mu\nu}R_{\delta\gamma\rho\sigma}+\frac{1}{32}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}R_{\gamma\delta\rho\sigma}R^{\gamma\delta\rho\sigma}\right]\
,$ (2.2)

where the $t_{8}$ tensor can be defined by its action over generic matrices
[23, 24]

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!t^{\alpha\beta\gamma\delta\mu\nu\rho\sigma}M^{1}_{\alpha\beta}M^{2}_{\gamma\delta}M^{3}_{\mu\nu}M^{4}_{\rho\sigma}=\
8\,{\rm
Tr}\\{M^{1}M^{2}M^{3}M^{4}+M^{1}M^{3}M^{2}M^{4}+M^{1}M^{3}M^{4}M^{2}\\}$ (2.3)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-2\left({\rm
Tr}\\{M^{1}M^{2}\\}{\rm Tr}\\{M^{3}M^{4}\\}+{\rm Tr}\\{M^{1}M^{3}\\}{\rm
Tr}\\{M^{2}M^{4}\\}+{\rm Tr}\\{M^{1}M^{4}\\}{\rm Tr}\\{M^{2}M^{3}\\}\right)\
.$

For the second term in (2.1) we have the following convention for the Levi-
Civita tensor

$\displaystyle\epsilon_{10}\epsilon_{10}R^{4}=$
$\displaystyle\epsilon^{\alpha\beta\mu_{1}\dots\mu_{8}}\epsilon_{\alpha\beta\nu_{1}\dots\nu_{8}}R_{\mu_{1}\mu_{2}}{}^{\nu_{1}\nu_{2}}R_{\mu_{3}\mu_{4}}{}^{\nu_{3}\nu_{4}}R_{\mu_{5}\mu_{6}}{}^{\nu_{5}\nu_{6}}R_{\mu_{7}\mu_{8}}{}^{\nu_{7}\nu_{8}}$
(2.4) $\displaystyle=$ $\displaystyle-2\cdot
8!R_{[\alpha\beta}{}^{\alpha\beta}R_{\gamma\delta}{}^{\gamma\delta}R_{\mu\nu}{}^{\mu\nu}R_{\rho\sigma]}{}^{\rho\sigma}$
$\displaystyle=$ $\displaystyle
3.2^{10}\left[R^{\alpha\beta}{}_{\gamma\delta}R^{\gamma\nu}{}_{\mu\beta}R^{\sigma\mu}{}_{\alpha\rho}R^{\delta\rho}{}_{\sigma\nu}+R^{\gamma\delta}{}_{\alpha\beta}R^{\mu\nu}{}_{\gamma\delta}R^{\alpha\rho}{}_{\sigma\mu}R^{\sigma\beta}{}_{\nu\rho}\right.$
$\displaystyle\
-\frac{1}{2}R_{\alpha\beta\mu\nu}R^{\beta\gamma\nu\rho}R_{\gamma\delta\rho\sigma}R^{\delta\alpha\sigma\mu}+\frac{1}{2}R_{\alpha\beta\mu\nu}R^{\beta\gamma\mu\nu}R_{\gamma\delta\rho\sigma}R^{\delta\alpha\rho\sigma}$
$\displaystyle\
\left.-\frac{1}{16}R_{\alpha\beta\mu\nu}R^{\beta\alpha\rho\sigma}R^{\gamma\delta\mu\nu}R_{\delta\gamma\rho\sigma}-\frac{1}{32}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}R_{\gamma\delta\rho\sigma}R^{\gamma\delta\rho\sigma}+\dots\right]\
,$

where the dots stand for terms containing Ricci tensors and scalars, which can
be eliminated by using field redefinitions, at the expense of introducing
dilaton couplings that we will ignore at the moment.

The couplings given by $t_{8}t_{8}$ have nonzero contribution at four-graviton
level [1], while the $\epsilon_{10}\epsilon_{10}$ interactions have nonzero
contributions starting only at five-graviton level [25]. The presence of this
term in the tree-level effective action was inferred by the $\beta$-function
approach in [20, 21, 22], predicting $c=1$. This prediction was confirmed in
[26] through sphere-level scattering amplitudes of five gravitons. The
literature also suggests that this value for $c$ is required by supersymmetry
[28, 27] and the emergence of T-duality symmetry in a circle compactification
[30, 29]. For the specific value $c=1$ it can be shown using Bianchi
identities that the corrections are given by only two terms [21]

$J(1)=-3\cdot
2^{6}\left[R^{\alpha\beta\mu\nu}R_{\mu\nu}{}^{\gamma\delta}R_{\alpha\gamma}{}^{\rho\sigma}R_{\rho\sigma\beta\delta}-4R_{\alpha\beta}{}^{\gamma\delta}R_{\delta\mu}{}^{\alpha\nu}R_{\nu\rho}{}^{\beta\sigma}R_{\sigma\gamma}{}^{\mu\rho}\right]\
.$ (2.5)

The consensus is that these are the unique purely gravitational terms
appearing in the leading $\alpha^{\prime}$ corrections in Type II string
theory.

The simplest way to test for $O(d,d)$ invariance is by performing a
cosmological reduction in which the $D=10$ dimensional target space splits
into a single temporal external direction and $d=9$ internal ones. The fields
only depend on time and we use the following ansatz

$\displaystyle G_{\mu\nu}=\text{diag}\left(-n^{2},g_{ij}\right)\
,\quad\quad\phi=\frac{1}{2}\Phi+\frac{1}{2}\log(\sqrt{g})\ ,\quad\quad
B_{\mu\nu}=\text{diag}\left(0,b_{ij}\right)\,,$ (2.6)

where $\mu,\nu$ are $D=10$ indices and $i,j$ are $d=9$ indices. All partial
derivatives but $\partial_{0}\Psi=\partial_{t}\Psi\equiv\dot{\Psi}$ are set to
zero. After the reduction, the effective one-dimensional action can be cast in
terms of the following quantities

$L^{i}{}_{j}\equiv g^{ik}\dot{g}_{kj}\ ,\quad\quad M^{i}{}_{j}\equiv
g^{ik}\dot{b}_{kj}\ ,$ (2.7)

plus the lapse function $n$, the lower-dimensional dilaton $\Phi$ and their
time derivatives.

A method to bring the effective action to a minimal form that makes it
systematic to assess its $O(9,9)$ invariance was introduced in [12, 11]. The
idea is that the lower dimensional equations of motion (where we have gauge
fixed $n=1$ after varying the action)

$\displaystyle\dot{L}$ $\displaystyle=M^{2}+\dot{\Phi}L\ ,$ (2.8)
$\displaystyle\dot{M}$ $\displaystyle=ML+\dot{\Phi}M\ ,$
$\displaystyle\ddot{\Phi}$
$\displaystyle=\frac{1}{2}\left(\dot{\Phi}^{2}+\frac{1}{4}{\rm
Tr}\left(L^{2}-M^{2}\right)\right)\ ,$ $\displaystyle\dot{\Phi}^{2}$
$\displaystyle=\frac{1}{4}{\rm Tr}\left(L^{2}-M^{2}\right)\ ,$

can be combined with integrations by part to remove all higher derivative
terms containing dilatons, and also allow one to remove the derivatives from
$L$ and $M$, leaving a final minimal form containing only powers of $L$ and
$M$. It was then shown which of these interactions can be cast in terms of the
generalized metric

$\mathcal{S}\equiv\mathcal{H}\eta^{-1}=\begin{pmatrix}bg^{-1}&g-bg^{-1}b\\\
g^{-1}&-g^{-1}b\end{pmatrix}\ ,$ (2.9)

so as to make the $O(d,d)$ symmetry manifest, if present. We refer to [12, 11]
for details on this procedure.

In the two-derivative case, the parent action

$S_{0}=\int
d^{D}x\sqrt{-G}e^{-2\phi}\left[R+4\left(\nabla\phi\right)^{2}-\frac{1}{12}H^{2}\right]\
,$ (2.10)

compactifies to an action where the $O(9,9)$ symmetry is manifest [9, 12, 11]

$S_{0}=\int dt\,e^{-\Phi}\left[-\dot{\Phi}^{2}-\frac{1}{8}{\rm
Tr}\big{(}\dot{\mathcal{S}}^{2}\big{)}\right]\,,$ (2.11)

where we used that ${\rm Tr}\big{(}\dot{\mathcal{S}}^{2}\big{)}=2\,{\rm
Tr}\left(M^{2}-L^{2}\right)$.

In the following we will simplify the problem by setting the B-field to zero,
which is sufficient in order to display the cosmological effective action in a
duality invariant form. In this case, the zeroth order equations of motion
(2.8) allow for the redefinitions

$\dot{L}\rightarrow\dot{\Phi}L\
,\quad\ddot{\Phi}\rightarrow\frac{1}{2}\left(\dot{\Phi}^{2}+\frac{1}{4}\left(L^{2}\right)\right)\
,\quad\dot{\Phi}^{2}\rightarrow\frac{1}{4}\left(L^{2}\right)\ ,$ (2.12)

where from now on we will denote the traces of $d\times d$ matrices by
parenthesis, i.e.,

$L^{i}{}_{i}=\text{Tr}(L)\equiv(L)\ ,\quad(L^{2})^{i}{}_{i}=(L^{2})\
,\quad\dots$ (2.13)

but we will keep the Tr notation for the duality covariant $2d\times 2d$
matrix $\mathcal{S}$. Indices are raised and lowered with $g$, namely
$L_{ij}=g_{ik}L^{k}{}_{j}=\dot{g}_{ij}$.

In the simplified case with vanishing two-form, the generalized metric is
related to $L$ by (with $M,N$ denoting doubled internal indices)

$\mathcal{S}_{M}{}^{N}=\begin{pmatrix}0&g_{ij}\\\ g^{ij}&0\end{pmatrix}\
,\quad(\dot{\mathcal{S}}^{2m})_{M}{}^{N}=\begin{pmatrix}(-1)^{m}(L^{2m})_{i}{}^{j}&0\\\
0&(-1)^{m}(L^{2m})^{i}{}_{j}\end{pmatrix}\ ,$ (2.14)

and so

${\rm Tr}\big{(}\dot{\mathcal{S}}^{2m}\big{)}=2\,(-1)^{m}\,(L^{2m})\
,\quad\quad{\rm Tr}\big{(}\dot{\mathcal{S}}^{2m-1}\big{)}=0\,,\quad
m\in\mathbb{N}\ .$ (2.15)

This shows that only traces containing even powers of $L$ can be written in
terms of the generalized metric; those involving odd powers do not admit a
duality covariant expression.

In this language, the reduced Riemann tensor reads

$\displaystyle R_{ijkl}$ $\displaystyle=\frac{1}{2}L_{i[k}L_{l]j}\ ,\quad\quad
R_{i0j0}=-\frac{1}{2}\dot{L}_{ij}-\frac{1}{4}(L^{2})_{ij}\ .$ (2.16)

The reduced action takes the form

$S=\int dt\,e^{-\Phi}\,\Big{(}-\dot{\Phi}^{2}-\frac{1}{8}{\rm
Tr}\big{(}\dot{\mathcal{S}}^{2}\big{)}+\alpha^{\prime
3}\,\frac{\zeta(3)}{3\cdot 2^{14}}\,J(c)\Big{)}\,,$ (2.17)

where the normalization of [2] is recovered upon setting $\alpha^{\prime}=1$.
Here $J(c)$, which was defined in (2.1), is evaluated using (2.16). We now
briefly explain how this computation is performed following the general
procedure of [12, 11]. Inserting (2.16) into (2.1) one obtains terms involving
traces of products of $L$ and $\dot{L}$. One may then systematically eliminate
all terms that contain $\dot{L}$ and $(L^{2})$ as follows: First, one uses the
EOM (2.12) to replace $\dot{L}\to\dot{\Phi}L$, leaving terms involving traces
of products of $L$ and powers of $\dot{\Phi}$. Even powers of $\dot{\Phi}$ can
then be eliminated by use of the third equation in (2.12),
$\dot{\Phi}^{2}\to{\tiny\frac{1}{4}}(L^{2})$. Those containing odd powers of
$\dot{\Phi}$ vanish. (To see this use repeatedly the substitution
$\dot{\Phi}^{2}\to{\tiny\frac{1}{4}}(L^{2})$ to arrive at terms with a single
$\dot{\Phi}$, then integrate it by parts to get $\int
dte^{-\Phi}\dot{\Phi}X(L)=\int dte^{-\Phi}\dot{X}(L)$. Now using
$\dot{L}\to\dot{\Phi}L$ gives the first term back with a different
coefficient, thus proving that these terms vanish.) At this point we are left
with traces of powers of $L$, and we now argue that terms containing $(L^{2})$
vanish,

$\int dte^{-\Phi}(L^{2})X(L)=0\ .$ (2.18)

This is proved as follows. One uses the lapse EOM to replace $(L^{2})$ by
$4\dot{\Phi}^{2}$, after which one integrates by part one $\dot{\Phi}$ factor,
using $e^{-\Phi}\dot{\Phi}=-\frac{d}{dt}(e^{-\Phi})$. This creates terms with
$\ddot{\Phi}$ and $\dot{L}$, for which ones uses again the EOMs (2.12) to
write the result in terms of $\dot{\Phi}^{2}$ and $(L^{2})$. Finally, one
replaces $\dot{\Phi}^{2}$ by $\frac{1}{4}(L^{2})$, using the lapse equation.
This reproduces the original integral, but with a different coefficient, hence
proving that it is zero.

The upshot of the above procedure is that we may ignore from the beginning all
$\dot{L}$ terms, setting $R_{i0j0}=-\frac{1}{4}(L^{2})_{ij}$, and then
eliminate all $(L^{2})$ contributions at the end of the computation. For the
two contributions to $J(c)$ one then finds

$\begin{split}t_{8}t_{8}R^{4}&\simeq\frac{9}{4}(L^{8})+\frac{51}{16}(L^{4})^{2}-6(L^{3})(L^{5})\
,\\\
e_{10}e_{10}R^{4}&\simeq-90(L^{8})+\frac{45}{2}(L^{4})^{2}+48(L^{3})(L^{5})\
,\end{split}$ (2.19)

where the symbol $\simeq$ indicates that these equalities hold up to EOMs and
integration by parts inside the integral $\int dte^{-\Phi}$. We then find that
the reduction of (2.1) is given by

$J(c)\simeq\frac{1}{4}(9-45c)(L^{8})+\frac{1}{16}(51+45c)(L^{4})^{2}-6(1-c)(L^{3})(L^{5})\
.$ (2.20)

As explained, while the first two terms can be written in a $O(9,9)$ invariant
form using the identities (2.15), the last one involves traces of odd powers
of $L$ and is then non-invariant. These contributions come from the second
term in the second line of (2.2) and the first term in the fourth line of
(2.4), respectively.

Given that these contributions are unambiguous, meaning that they cannot be
modified through EOMs nor integrations by parts, T-duality then fixes the
coefficient to its expected value $c=1$, for which

$\displaystyle J(1)$ $\displaystyle\simeq$
$\displaystyle-9(L^{8})+6(L^{4})^{2}=-\frac{9}{2}{\rm
Tr}(\dot{\mathcal{S}}^{8})+\frac{3}{2}\big{(}{\rm
Tr}(\dot{\mathcal{S}}^{4})\big{)}^{2}\ .$ (2.21)

Using this in (2.17) we then finally arrive at the minimal form of the
cosmological effective action written in a manifestly $O(9,9)$ invariant way

$S=\int dt\,e^{-\Phi}\left\\{-\dot{\Phi}^{2}-\frac{1}{8}{\rm
Tr}\big{(}\dot{\cal
S}^{2}\big{)}+\frac{\alpha^{\prime}{}^{3}\zeta(3)}{2^{15}}\left[-3\,{\rm
Tr}\big{(}\dot{\cal S}^{8}\big{)}+{\rm Tr}\big{(}\dot{\cal S}^{4}\big{)}{\rm
Tr}\big{(}\dot{\cal S}^{4}\big{)}\right]\right\\}\ .$ (2.22)

Let us finally point out that while the above result was computed for
vanishing B-field and dilaton, by duality invariance (2.22) must be the
complete cosmological action including B-field and dilaton. One might be
worried that there could be dilaton couplings in higher dimensions that upon
cosmological reduction and field redefinitions contribute to the gravitational
terms and change the above coefficients, but one may convince oneself that
this cannot happen. Following the steps outlined above it can be seen that a
generic term of the form
$\nabla_{\mu_{1}}\nabla_{\mu_{2}}\dots\nabla_{\mu_{n}}\phi\,X^{\mu_{1}\mu_{2}\dots\mu_{n}}$,
where $X$ is a tensor depending on $G$ and $\phi$, compactifies to terms that
can be redefined away, up to terms containing traces $(L)$, which violate
duality invariance.

##  3 Discussion

In this letter we have analyzed the compatibility of the continuous
$O(d,d,\mathbb{R})$ symmetry with the eight-derivative couplings quartic in
the Riemann tensor proportional to $\zeta(3)$ that arise in string theory at
order $\alpha^{\prime\,3}$. We have shown that demanding $O(9,9,\mathbb{R})$
invariance upon cosmological reduction to one timelike dimension fixes the
relative coefficient between the couplings $t_{8}t_{8}R^{4}$ and
$\epsilon_{10}\epsilon_{10}R^{4}$ uniquely to the value determined
independently by other methods. This result illustrates the strength of
duality invariance, for it allows one to reconstruct from the
$t_{8}t_{8}R^{4}$ term, which was computed by Gross and Witten from the four-
point amplitude [2], the complete gravitational couplings in ten dimensions,
in addition to determining the B-field and dilaton couplings that survive upon
cosmological reduction. This gives strong constraints on the possible B-field
and dilaton couplings also in ten dimensions, but we do not expect these to be
determined completely, since there could be couplings that disappear upon
reduction.

The results presented here are useful, in particular, in that they determine
the first two non-trivial coefficients for Type II string theory in the
general cosmological classification to all orders in $\alpha^{\prime}$ [11].
There have already been a number of papers exploring cosmological consequences
of this $\alpha^{\prime}$-complete cosmology, see [32, 33, 34, 35, 36], and
here we have further constrained the ‘space of duality covariant string
cosmologies’. Moreover, our results provide a non-trivial test for the core
assumption underlying this classification: that there is a field basis for
which the $O(d,d,\mathbb{R})$ symmetry, expected to exist in string theory to
all orders in $\alpha^{\prime}$, takes the standard form. This is in contrast
to double field theory [31] and conventional dimensional reduction with a
generic number of external dimensions [17, 16], where a Green-Schwarz-type
mechanism needs to be invoked that can be viewed as
$\alpha^{\prime}$-deforming the $O(d,d)$ transformations.

Note added: Upon completion of this letter we became aware of the results in
[37], which exhibit obstacles for a double field theory formulation of the
$\zeta(3)$ couplings in ten dimensions. The first version of this letter also
identified an obstacle for the conventional realization of $O(d,d)$ in
dimensional reduction, but this was due to a computational mistake, and the
corrected results presented here fully confirm the presence of $O(d,d)$. We
therefore expect that there is also a double field theory formulation of the
$\zeta(3)$ couplings, perhaps involving novel structures.

### Acknowledgements

We thank Juan Maldacena, Ashoke Sen and Pierre Vanhove for discussions and
correspondence, Stanislav Hronek and Linus Wulff for giving us access to [37]
before submission, and Heliudson Bernardo, Axel Kleinschmidt, Krzysztof
Meissner, Gabriele Veneziano and Linus Wulff for comments on the first version
of this letter.

This work is supported by the ERC Consolidator Grant “Symmetries & Cosmology”.
D. M. is supported by CONICET. T. C. is supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer
417533893/GRK2575 “Rethinking Quantum Field Theory”.

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