#  Few-shot bioacoustic event detection
at the DCASE 2022 challenge

###### Abstract

Few-shot sound event detection is the task of detecting sound events, despite
having only a few labelled examples of the class of interest. This framework
is particularly useful in bioacoustics, where often there is a need to
annotate very long recordings but the expert annotator time is limited. This
paper presents an overview of the second edition of the few-shot bioacoustic
sound event detection task included in the DCASE 2022 challenge. A detailed
description of the task objectives, dataset, and baselines is presented,
together with the main results obtained and characteristics of the submitted
systems. This task received submissions from 15 different teams from which 13
scored higher than the baselines. The highest F-score was of 60% on the
evaluation set, which leads to a huge improvement over last year’s edition.
Highly-performing methods made use of prototypical networks, transductive
learning, and addressed the variable length of events from all target classes.
Furthermore, by analysing results on each of the subsets we can identify the
main difficulties that the systems face, and conclude that few-show
bioacoustic sound event detection remains an open challenge.

Index Terms—  Few-shot learning, bioacoustics, sound event detection, DCASE
challenge

##  1 Introduction

The task of bioacoustic sound event detection refers to the retrieval of
animal vocalisations from audio recordings in terms of onset and offset times.
It shares a common methodology with other sound event detection (SED)
contexts, yet, the application domain of bioacoustics is particularly
challenging for SED. Deep learning contributed to overcome some of these
difficulties in bioacoustic SED, however it also established strong
requirements regarding the amount of annotated data needed [1]. Collecting and
annotating a large dataset of animal vocalisations is often not feasible given
that species are unequally abundant [2] and may be rarely observed; and audio
annotation is costly and time-consuming [3].In contrast to traditional deep
learning approaches that use a large amount of data to train models, few-shot
learning tries to build accurate models with very few training data [4]. Few-
shot learning is usually studied using $N$-way-$k$-shot classification, where
$N$ denotes the number of classes and $k$ the number of known examples for
each class.

This problem was first evaluated as a task on the DCASE 2021 challenge. This
year, the setup and goal remain the same: Given the first 5 events of a target
class, can systems detect the subsequent events of the same class in the
remaining of the audio recording? Diverse approaches have been used to address
the few-shot learning problem for classification. Some use prior knowledge
about similarity between sounds by computing embeddings (learnt representation
spaces) designed to help discriminate between unseen classes [4], while others
exploit prior knowledge about the structure of the data by using augmentation
to synthesise new data [5]. Finally, some approaches can learn models with
parameters that can be fine-tuned to smaller datasets [6]. More recent works
use meta-learning and/or prototypical networks for acoustic few-shot learning
[7], [8]. All of the above approaches deal with classification tasks rather
than detection. Indeed, SED in a few shot setup is commonly approximated as an
audio tagging task and few works have addressed the actual detection of onsets
and offsets of events [9]. At last year’s task edition, the best ranked system
improved over the baseline prototypical approach by applying a transductive
inference method and a mutual learning framework designed to make the feature
extraction network more task dependent [10]. The overall best results were
just bellow 40% f-score which indicates the difficulty of this task. This
year, we added more and diverse datasets, and increased the task difficulty
(dataset diversity); yet the task doubled the amount of participants and the
best overall f-score in the evaluation set reached the 60% level. This paper
is structured as follows. Section 2 presents the bioacoustic datasets used for
developing and evaluating submitted systems. Section 3 presents the two
baseline methods proposed for the task, followed by the evaluation procedure.
Finally, section 4 presents the results of the submitted systems and a
discussion about the overall task and future steps in the field of few-shot
bioacoustic event detection.

| Dataset | mic type | # audio files | total duration | # labels (excl. UNK) | # events
---|---|---|---|---|---|---
Development Set: Training | BV | fixed | 5 | 10 hours | 11 | 9026
HT | various | 5 | 5 hours | 5 | 611
MT | mobile | 2 | 70 mins | 4 | 1294
JD | mobile | 1 | 10 mins | 1 | 357
| WMW | various | 161 | 5 hours | 26 | 2941
Development Set: Validation | HB | handheld | 10 | 2.38 hours | 1 | 712
PB | fixed | 6 | 3 hours | 2 | 292
| ME | handheld | 2 | 20 mins | 2 | 73
Evaluation Set | CHE | fixed | 18 | 3 hours | 3 | 2550
DC | fixed | 10 | 95 mins | 3 | 967
CT | handheld | 3 | 48 mins | 3 | 365
| MS | fixed | 4 | 40 mins | 1 | 1087
| QU | handheld | 8 | 74 mins | 1 | 3441
| MGE | fixed | 3 | 32 mins | 2 | 1195
Table 1: Information on each dataset.

##  2 Datasets

A development dataset consists of predefined training and validation sets to
be used for system development. 111Dev set:
https://doi.org/10.5281/zenodo.6012309. The training set contains multi-class
temporal annotations, provided for each recording as: positive (POS), negative
(NEG) and unknown (UNK). For the validation set only single-class temporal
annotations (POS/UNK) were provided for each recording. A separate evaluation
set was kept for evaluating the performance of the systems.222Eval set:
https://doi.org/10.5281/zenodo.6517413 During the task, only the first five
POS events of the class of interest were provided for each of the recordings.
Table 1 presents an overview of all the datasets in the development and
evaluation sets.

BirdVox-DCASE-10h (BV): The BirdVox-DCASE-10h (BV) contains five audio files
from four different autonomous recording units, each lasting two hours. These
autonomous recording units are all located in Tompkins County, NY, US. They
follow the same hardware specification: the Recording and Observing Bird
Identification Node (ROBIN) developed by the Cornell Lab of Ornithology [11].
All recordings were acquired in 2015, during the fall migration season. An
expert ornithologist, Andrew Farnsworth, has annotated flight calls from four
families of passerines, namely: American sparrows, cardinals, thrushes, and
New World warblers. These flight calls have a duration in the range 50–150
milliseconds and a fundamental frequency in the range 2–10 kHz.

Hyenas (HT): Spotted hyenas are a highly social species that live in “fission-
fusion” groups and use a variety of types of vocalisations to coordinate with
one another. Spotted hyenas were recorded on custom-developed audio tags
designed by Mark Johnson and integrated into combined GPS/acoustic collars
(Followit Sweden AB) by Frants Jensen and Mark Johnson. Collars were deployed
on female hyenas of the Talek West hyena clan at the MSU-Mara Hyena Project
(directed by Kay Holekamp) in the Masai Mara, Kenya as part of a multi-species
study on communication and collective behavior. Recordings used as part of
this task contain a variety of different vocalisations which were identified
and classified into types based on the established hyena vocal repertoire
[12]. Field work was carried out by Kay Holekamp, Andrew Gersick, Frants
Jensen, Ariana Strandburg-Peshkin, and Benson Pion; labeling was done by Kenna
Lehmann and colleagues.

Meerkats (MT, ME): Meerkats are a highly social mongoose species that live in
stable social groups and use a variety of distinct vocalisations to
communicate and coordinate with one another. The meerkat vocal repertoire has
been well characterized based on previous research, allowing calls to be
reliably classified by human labellers [13, 14]. Recordings used in this task
were acquired at the Kalahari Meerkat Project (Kuruman River Reserve, South
Africa; directed by Marta Manser and Tim Clutton-Brock), as part of a multi-
species study on communication and collective behavior. Recordings of the
development set (MT) were recorded on small audio devices (TS Market, Edic
Mini Tiny+ A77, 8 kHz) integrated into combined GPS/audio collars which were
deployed on multiple members of meerkat groups. Recordings of the validation
set (ME) were recorded by an observer following a focal meerkat with a
Sennheiser ME66 directional microphone (44.1 kHz) from a distance of less than
1 m. Recordings were carried out during daytime hours while meerkats were
primarily foraging and include several different call types. Field work was
carried out by Ariana Strandburg-Peshkin, Baptiste Averly, Vlad Demartsev,
Gabriella Gall, Rebecca Schaefer and Marta Manser. Audio recordings were
labeled by Baptiste Averly, Vlad Demartsev, Ariana Strandburg-Peshkin, and
colleagues.

Jackdaws (JD): Jackdaws are corvid songbirds that usually breed, forage and
sleep in large groups. They produce thousands of vocalisations per day, but
many aspects of their vocal behaviour remain unexplored. In a multi-year field
study (Max-Planck-Institute for Ornithology, Seewiesen, Germany), wild
jackdaws were equipped with small backpacks containing miniature voice
recorders (Edic Mini Tiny A31, TS-Market Ltd., Russia) to investigate the
vocal behaviour of individuals interacting with their group and behaving
freely in their natural environment. Field work was conducted by Lisa Gill,
Magdalena Pelayo van Buuren and Magdalena Maier. Sound files were annotated by
Lisa Gill.

Western Mediterranean Wetlands Bird Dataset (WMW): This dataset contains bird
sounds from 20 endemic species that are typically inhabitants of the
“Aiguamolls de l’Empordà” natural park in Girona, Spain. Depending on the
species, audios contain vocalisations such as bird calls or songs; or sounds
such as bill clapping (Ciconia ciconia species) or drumming (Dendrocopos minor
species). The audio files that compose this dataset were originally retrieved
from the Xeno-Canto portaland were manually cleaned and labelled by Juan
Gómez-Gómez, Ester Vidaña-Vila and Xavier Sevillano using the Audacity
software [15]. In Xeno-Canto, audios are labelled by quality from A to E,
where A means that the audio quality is excellent and E means that the audio
quality is poor. This dataset contains only A and B type files which ensured
the exclusion of audios with high background noise.

HumBug (HB): Mosquitoes produce sound both as a by-product of their flight and
as a means for communication and mating. Fundamental frequencies vary in the
range of 150 to 750 Hz [16]. As part of the HumBug project, acoustic data was
recorded with a high specification field microphone (Telinga EM-23) coupled
with an Olympus LS-14. The recordings used in this challenge are a subset of
the datasets marked as _‘OxZoology’_ and _‘Thailand’_ from HumBugDB [17]333
https://github.com/HumBug-Mosquito/HumBugDB/. The recordings contain the sound
of lab-cultured _Culex quinquefasciatus_ mosquitoes from Oxford, UK, and
various species captured in the wild in Thailand, placed into plastic cups.

Polish Baltic Sea bird flight calls (PB): The PB dataset consists of bird
flight calls recordThree autonomous recording units were used with the same
hardware settings (Song Meters SM2, Wildlife Acoustics, Inc). They were
deployed close to each other ($<$100m) - near the lake, on the dune, and on
the forest clearing - to provide diverse acoustic background. The passerines
night flight calls were annotated by Hanna Pamuła. Target classes are belong
to: song thrush, _Turdus philomelos_ and blackbird, _Turdus merula_. Event
lengths vary between 8 to 400 milliseconds and the usual fundamental frequency
range for calls is 5 to 9 kHz.

Transfer-Exposure-Effects project (CHE): This dataset comes from the Transfer
Exposure-Effects (TREE) research project444https://tree.ceh.ac.uk/, which was
funded by the Natural Environment Research Council (NERC), Environment Agency
and Radioactive Waste Management Ltd. Data were collected using unattended
acoustic recorders (Songmeter 3) in the Chornobyl Exclusion Zone (CEZ) to
capture the Chornobyl soundscape and investigate the longterm effects of the
Chornobyl accident on the local ecology. To date, the study has captured
approximately 10,000 hours of audio from the CEZ. The fieldwork was designed
and undertaken by Mike Wood (University of Salford), Nick Beresford (UK Centre
for Ecology & Hydrology) and Sergey Gashchak (Chornobyl Center). Common
Chiffchaff (Phylloscopus collybita) and Common Cuckoo (Cuculus canorus)
vocalisations were manually annotated and labelled from these recordings by
Helen Whitehead (University of Salford)

BIOTOPIA Dawn Chorus (DC): The Dawn Chorus project is a worldwide citizen
science and arts project bringing together amateurs and experts to experience
and record the dawn chorus at their doorstep.The DC dataset stems from dawn
chorus recordings, made using Zoom H2 recorders at 44100 Hz, at three
different locations in Southern Germany (Haspelmoor, Munich’s Nymphenburg
Schlosspark, and Nantesbuch), by Moritz Hertel and Rudi Schleich. The
vocalisations of three target species were annotated by Lisa Gill (Common
cuckoo, _Cuculus canorus_ ; European robin, _Erithacus rubecula_ ; Eurasian
wren, _Troglodytes troglodytes_). A challenging aspect of this data is related
to recordings being very busy with various other birds vocalising at the same
time.

Coati (CT): Coatis are small carnivorous mammals where females and young males
live in groups of up to 25 individuals. They communicate using a series of
different calls however their functions are still relatively unknown. The
target calls used in this dataset are growls, chitters and chirp-grunts.
Several other call types that might be confused with the targets were captured
in the recordings which configures the main challenging aspect of this data.

Manx Shearwater (MS): Manx shearwaters are procellariiform seabirds that breed
in dense island colonies in the North Atlantic and winter in the South
Atlantic off the South American coast. Adult Manx shearwaters make loud,
distinctive vocalisations while present at their breeding colony in various
contexts: to their partner in their nesting burrow, broadcasting from their
burrow, and in flight. The target class is Chick begging vocalisations which
typically comprise bouts of short, high-pitched ‘peeps’. In a multi-year
study, Audiomoth recorders were placed in burrows to record the vocalisations
during the breeding season. Fieldwork on Skomer Island was undertaken by
various members of the Oxford Navigation Group (OxNav) and annotation was
carried out by Joe Morford.

Dolphin Quacks (QU): This data consists in underwater recordings of dolphins
in their natural habitat. All files were annotated for the Quack target class
which represent calls predominantly used in a social context. Quack events can
be very short (from 30 milliseconds) and vary considerably in the course of a
single recording. This is a potential problem when the 5 initial POS events of
the target class are not good representations of the class overall.

Chick calls (MGE): Chicks of domestic chicken were recorded in the Prepared
Minds Lab from Queen Mary university of
London555https://www.preparedmindslab.org/home. The 2 target classes are
pleasure and contact calls which the chicks will produce as a response to
different stimuli. The data gathered here presents uneven time distribution,
and calls are typically short. Additionally, calls present low stereotypy
which contributes to the challenge associated with this dataset.

##  3 Baseline Methods and Evaluation

The baseline systems proposed did not change considerably from last year’s
edition [18]. Template Matching is based on spectrogram cross-correlation and
still commonly used in bioacoustics. This approach scored surprisingly well on
last edition evaluation set and thus it remains relevant as a baseline for
this task. The second system proposed is based on prototypical networks which
remain the state of the art for few-shot learning [4]. The changes from last
year’s system are the use of a ResNet, and adapting segment size depending on
the target class in the query set. These changes mainly address the problem of
high variation of event lengths and create a more adaptive system.

Team name |  | Evaluation set:
---
F-score % (95% CI)
| Validation set
---
F-score %
| Main characteristics
---
Du_NERCSLIP_2 [19] | 60.22 (59.66-60.70) | 74.4 | CNN+ProtoNet; Frame-level embeddings; PCEN;
Liu_Surrey_2[20] | 48.52 (48.18-48.85) | 50.03 |  CNN+ProtoNet; extra data; PCEN+$\triangle{MFCC}$; several post-process.
Martinsson_RISE_1 [21] | 47.97 (47.48-48.40) | 60 | ResNet+ProtoNet; Ensemble (15) based input size; logMel+PCEN
Hertkorn_ZF_2 [22] | 44.98 (44.44-45.42) | 61.76 | CNN; Frequency resolution preserving pooling; various post-process
Liu_BIT-SRCB_4 [23] | 44.26 (43.85-44.62) | 64.77 | CNN+ProtoNet; Transductive inference
Wu_SHNU_1 [24] | 40.93 (40.48-41.30) | 53.88 | ResNet+ProtoNet; Continual-learning; spectrogram
Zgorzynski_SRPOL_4 [25] | 33.24 (32.69-33.69) | 57.2 | CNN+Siamese Networks; Emsemble (3) average event-length;
Mariajohn_DSPC_1 [26] | 25.66 (25.40-25.91) | 43.89 | CNN+ProtoNet; logMel; augmentation with time-shifting and mirroring
Wilbo_RISE_4 [27] | 21.67 (21.32-21.97) | 47.94 | ResNet+ProtoNEt; Semi-supervised; Melspect+PCEN; several post-process
Zou_PKU_1 [28] | 19.20 (18.88-19.51) | 51.99 | CNN+protoNet; mutual information loss; time frequency masking + mixup
Huang_SCUT_1 [29] | 18.29 (18.01-18.56) | 54.63 | transductive inference + Adapted central difference convolution
Tan_WHU_4 [30] | 17.22 (16.82-17.55) | 54.53 | CNN+ProtoNet pretrained; transductive inference; task adaptive features
Li_QMUL_1 [31] | 15.49 (15.16-15.77) | 47.88 | CNN+protoNet; PCEN; time, frequency masking + time warping
baseline-TempMatch [18] | 12.35 (11.52-12.75) | 3.37 | Spectrogram Cross correlation
baseline-ProtoNet [18] | 5.3 (5.1-5.2) | 28.45 | ResNet+ProtoNet
Zhang_CQU_4 [32] | 4.34 (3.74-4.56) | 44.17 | CNN+protoNet; Fine tunning with MIMI; PCEN
Kang_ET_2 [33] | 2.82 (2.76-2.87) | - | CNN+ProtoNEt; pretrained ECAPA-TDNN; Fine-tuning; Specaugment
Table 2: F-score results per team (best scoring system) on evaluation and
validation sets, and summary of system characteristics. Systems are ordered by
higher scoring rank on the evaluation set.

The evaluation of this task is based on an event-level F-measure with macro-
averaged metric across all classes[18]. A positive match between predicted
events and reference is found by applying the Intersection over Union (IoU)
with 30% minimum overlap, followed by Hopcroft-Karp-Karzanov algortithm for
bipartite graph matching. True positives(TP), False positives(FP) and False
negatives(FN) can be computed after the matching step. These are defined as:
TP - predicted events that match ground truth events; FP - predicted events
that do not match any ground truth events; FN - ground truth events that are
not predicted. Matches to UNK events are ignored from these counts as to not
negatively impact the systems that predict these events. Finally, the F-score
metric is computed per dataset in the evaluation set and the harmonic mean
over all is reported.

Figure 1: F-Score results by dataset. Systems are ordered by highest scoring
rank on the evaluation set.

##  4 Results

For the 2022 edition, 15 teams participated submitting a total of 46 systems.
The results for the highest scoring submission for each team are presented in
Table 2, together with the reported F-scores on the validation set and summary
of the system characteristics. Fig. 1 presents the F-scores obtained by each
team on each subset of the evaluation data. The majority of systems adopted a
prototypical network approach. Similar to last year’s results, simple
improvements over the baselines were achieved by applying data augmentation
techniques and intelligent post processing. Better ways to construct the
negative prototype were also explored by some teams who report improved
results [20, 21, 24, 27]. Transductive inference, the method used by the past
edition’s winning team, was also applied here by several participants [20, 31,
30, 28]. The highest scoring system implements a frame-level embedding
learning approach which confers to the system a high time resolution
capability [19]. Observing Fig.1, the system was particularly effective on the
QU and MGE datasets. This confirms that good time precision is fundamental,
particularly for classes with events of very short duration as the ones in
these datasets. The system ranked in second place implements a novel approach
designed to optimise the contrast between positive events and negative
prototypes [20]. This, together with an adaptive segment length dependent on
each target class, works well across all the evaluation sets. The problem of
very different length of events across target classes was also directly
addressed by other submissions. Both [21] and [25] implemented an ensemble
approach where each individual model focus on a different input size range. In
[23] this is explored through a multi-scale ResNet, and in [27] with a wide
ResNet containing many channels. Finally, it is worth mentioning the system in
[22]. Their few-shot adaptation was based on fine-tuning alone. The innovation
here is related with simple modifications to a CNN based architecture in order
to optimise the use of information, particularly in the frequency axis.
Furthermore, by allowing the network to overfitt (up to a degree) to the 5
shots, the system achieves surprisingly good performance across all the
datasets of the evaluation set. Overall, this edition saw some novel ideas
being implemented that tried to address previously identified challenges
related to this task: how to deal with very different event lengths; how to
construct a negative class when not explicit labels are given for this; and
how to bridge the gap between classification and detection for few shot sound
event detection. We believe these remain relevant questions for our goal and
for SED in general, and that the collective work developed here helped pushing
few-shot bioacoustic sound event detection into DCASE central stage.

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#  Exclusive production of $\rho$ meson in gamma-proton collisions:
$d\sigma/dt$ and the role of helicity flip processes

Anna Cisek<EMAIL_ADDRESS>College of Natural Sciences, Institute of
Physics, University of Rzeszów, ul. Pigonia 1, PL-35-310 Rzeszów, Poland
Wolfgang Schäfer<EMAIL_ADDRESS>Institute of Nuclear Physics,
Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland
Antoni Szczurek<EMAIL_ADDRESS>Institute of Nuclear Physics,
Polish Academy of Sciences, ul. Radzikowskiego 152, PL-31-342 Kraków, Poland
College of Natural Sciences, Institute of Physics, University of Rzeszów, ul.
Pigonia 1, PL-35-310 Rzeszów, Poland

###### Abstract

We calculate the differential cross section $d\sigma/dt$ for the diffractive
photoproduction process $\gamma p\to\rho p$ and compare to recent data
extracted by the CMS collaboration from ultraperipheral proton-lead
collisions. Our model is based on two-gluon exchange in the nonperturbative
domain. We take into account both helicity conserving and often neglected
helicity-flip amplitudes in the $\gamma\to V$ transition. The letter can
contribute at finite $t$. The shape of the differential cross section as well
as the role of helicity flip processes is strongly related to the dependence
of the unintegrated gluon distribution on transverse momenta in the
nonperturbative region.

##  I Introduction

The exclusive photoproduction of vector mesons is one of the intensively
studied processes at high energies. It can be seen as a diffractive excitation
of a photon into a vector meson state. Being a diffractive process it shows a
prominent forward peak, and within the forward cone $s$-channel helicity is
expected to be conserved to very good accuracy. For the light vector mesons,
like the $\rho$ which we discuss in this work, the energy dependence displays
a “soft pomeron” behaviour and follows the one of the total $\gamma p$
photoabsorption cross section. At high $\gamma^{*}p$ cm-energies and large
virtualities $Q^{2}$ of the photon, a QCD factorization approach based on a
twist expansion of meson distribution amplitudes and unintegrated gluon
distributions (UGDs) Anikin  _et al._ (2010) can be used to constrain the
gluon UGD Bolognino  _et al._ (2018, 2021). On the other hand in the
photoproduction limit, $Q^{2}=0$ many theoretical approaches to diffractive
$\rho$-meson production are based on the vector meson dominance ansatz and
essentially describes the process as an elastic scattering of a $\rho$-meson
on the proton. Besides being problematic in the continuation to large
$Q^{2}\gg m_{\rho}^{2}$, in this approach also the helicity and transverse
momentum dependence of the amplitude have to be inserted “by hand”. Indeed,
within the diffraction cone, models of soft diffractive processes often impose
the $s$-channel helicity conservation, following Gilman  _et al._ (1970).

Our work is motivated by a recent measurement of the differential cross
section $d\sigma/dt$ [CMS] (2018). This observable has been advocated as a
probe of gluon saturation effects Armesto and Rezaeian (2014); Gonçalves  _et
al._ (2019). These calculations, which are formulated in the color dipole
approach also restrict themselves to the helicity conserving part of the
amplitude. In this work we wish to investigate the shape of the differential
cross section, some questions that come to mind are: do we expect dips in the
differential cross section, or what is the role of spin-flip transitions at
intermediate and large $t$? We will see, that the answer to these questions is
indeed strongly dependent on the modelling of the unintegrated glue. We
compare our calculations for a variety of unintegrated gluon distributions
available in the literature, and compare to the recent (still preliminary)
data of the CMS collaboration [CMS] (2018) which were obtained from the
ultraperipheral $pPb$-collisions.

##  II Sketch of the formalism

Figure 1: Sample Feynman diagram for the $\gamma p\to\rho p$ diffractive
amplitude.

The imaginary part of the amplitude for the reaction
$\gamma(\lambda_{\gamma})p\to V(\lambda_{V})p$ depicted in Fig. 1 can be
written as Ivanov  _et al._ (2006):

$\displaystyle\Im m\,{\cal
M}_{\lambda_{V},\lambda_{\gamma}}(W,\mbox{\boldmath$\Delta$})$
$\displaystyle=$ $\displaystyle
W^{2}\frac{c_{V}\sqrt{4\pi\alpha_{em}}}{4\pi^{2}}\,\int{d\kappa^{2}\over\kappa^{4}}\alpha_{S}(q^{2}){\cal{F}}(x,{\mbox{\boldmath$\Delta$}\over
2}+\mbox{\boldmath$\kappa$},{\mbox{\boldmath$\Delta$}\over
2}-\mbox{\boldmath$\kappa$})$ (1) $\displaystyle\times$
$\displaystyle\int\frac{dzd^{2}\mbox{\boldmath$k$}}{z(1-z)}I(\lambda_{V},\lambda_{\gamma};z,\mbox{\boldmath$\kappa$},\mbox{\boldmath$k$},\mbox{\boldmath$\Delta$})\psi_{V}(z,k)\;.$

Here $W$ is the $\gamma p$-cm energy, $\Delta$ is the transverse momentum
transfer, and for the $\rho$-meson $c_{V}=1/\sqrt{2}$. Furthermore,
$\psi_{V}(z,k)$ is a (radial part of the) light cone wave funtion of the
vector meson, and ${\cal
F}(x,\mbox{\boldmath$\kappa$}_{1},\mbox{\boldmath$\kappa$}_{2})$ is a
generalization of the unintegrated gluon distribution valid off the forward
direction. Its first argument has to be understood as $x=M_{V}^{2}/W^{2}$,
where $M_{V}$ is the mass of the vector meson.

The impact factors $I(\lambda_{V},\lambda_{\gamma})$ beyond $s$-channel
helicity conservation have been obtained in Ivanov and Kirschner (1998);
Kuraev  _et al._ (1998) and are conveniently summarized in Ivanov  _et al._
(2006).

We are interested in the case of real, transversely polarized photons, so that
the following amplitudes depending on the helicities $\lambda_{\gamma}$ and
$\lambda_{V}$ of photon and meson, contribute:

  1. 1.

The $s$-channel helicity conserving $T\to T$ transition, where
$\lambda_{\gamma}=\lambda_{V}$: ,

$\displaystyle I(T,T)_{(\lambda_{V}=\lambda_{\gamma})}$ $\displaystyle=$
$\displaystyle
m_{q}^{2}\Phi_{2}+\Big{[}z^{2}+(1-z)^{2}\Big{]}(\mbox{\boldmath$k$}\mbox{\boldmath$\Phi$}_{1})$
$\displaystyle+\frac{m_{q}}{M+2m_{q}}\Big{[}\mbox{\boldmath$k$}^{2}\Phi_{2}-(2z-1)^{2}(\mbox{\boldmath$k$}\mbox{\boldmath$\Phi$}_{1})\Big{]}\;.$

  2. 2.

the helicity flip by one unit, i.e. from the transverse photon
$\lambda_{\gamma}=\pm 1$ to the longitudinally polarized meson,
$\lambda_{V}=0$.

$\displaystyle I(L,T)$ $\displaystyle=$
$\displaystyle-2Mz(1-z)(2z-1)(\mbox{\boldmath$e$}\mbox{\boldmath$\Phi$}_{1})\Big{[}1+\frac{(1-2z)^{2}}{4z(1-z)}\frac{2m_{q}}{M+2m_{q}}\Big{]}$
$\displaystyle+\frac{Mm_{q}}{M+2m_{q}}(2z-1)(\mbox{\boldmath$e$}\mbox{\boldmath$k$})\Phi_{2}\;.$

  3. 3.

the helicity flip by two units, from the transverse photon
$\lambda_{\gamma}=\pm 1$ to the transversely polarized meson with
$\lambda_{V}=\mp 1$:

$\displaystyle I(T,T)_{(\lambda_{V}=-\lambda_{\gamma})}$ $\displaystyle=$
$\displaystyle 2z(1-z)(\Phi_{1x}k_{x}-\Phi_{1y}k_{y})$ (4)
$\displaystyle-\frac{m_{q}}{M+2m_{q}}\Big{[}(k_{x}^{2}-k_{y}^{2})\Phi_{2}-(2z-1)^{2}(k_{x}\Phi_{1x}-k_{y}\Phi_{1y})\Big{]}\;.$

Here

$\displaystyle M=\sqrt{\mbox{\boldmath$k$}^{2}+m_{q}^{2}\over z(1-z)}\,$ (5)

is the invariant mass of the $q\bar{q}$ pair. In the expressions above
$\mbox{\boldmath$\Phi$}_{1}$, $\Phi_{2}$ are given by (see e.g. Ref. Ivanov
_et al._ (2006)):

$\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle-{1\over(\mbox{\boldmath$r$}+\mbox{\boldmath$\kappa$})^{2}+\varepsilon^{2}}-{1\over(\mbox{\boldmath$r$}-\mbox{\boldmath$\kappa$})^{2}+\varepsilon^{2}}+{1\over(\mbox{\boldmath$r$}+\mbox{\boldmath$\Delta$}/2)^{2}+\varepsilon^{2}}+{1\over(\mbox{\boldmath$r$}-\mbox{\boldmath$\Delta$}/2)^{2}+\varepsilon^{2}}\,,$
$\displaystyle\mbox{\boldmath$\Phi$}_{1}$ $\displaystyle=$
$\displaystyle-{\mbox{\boldmath$r$}+\mbox{\boldmath$\kappa$}\over(\mbox{\boldmath$r$}+\mbox{\boldmath$\kappa$})^{2}+\varepsilon^{2}}-{\mbox{\boldmath$r$}-\mbox{\boldmath$\kappa$}\over(\mbox{\boldmath$r$}-\mbox{\boldmath$\kappa$})^{2}+\varepsilon^{2}}+{\mbox{\boldmath$r$}+\mbox{\boldmath$\Delta$}/2\over(\mbox{\boldmath$r$}+\mbox{\boldmath$\Delta$}/2)^{2}+\varepsilon^{2}}+{\mbox{\boldmath$r$}-\mbox{\boldmath$\Delta$}/2\over(\mbox{\boldmath$r$}-\mbox{\boldmath$\Delta$}/2)^{2}+\varepsilon^{2}}\,.$
(6)

Quark and antiquark carry a fraction $z$ and $1-z$ of the vector meson
lightcone plus-momentum as well as the transverse momenta
$\mbox{\boldmath$k$}_{q}=\mbox{\boldmath$k$}+z\mbox{\boldmath$\Delta$}$ and
$\mbox{\boldmath$k$}_{\bar{q}}=-\mbox{\boldmath$k$}+(1-z)\mbox{\boldmath$\Delta$}$.
Furthermore, we introduced
$\mbox{\boldmath$r$}=(\mbox{\boldmath$k$}_{q}-\mbox{\boldmath$k$}_{\bar{q}})/2=\mbox{\boldmath$k$}+(z-\frac{1}{2})\mbox{\boldmath$\Delta$}$,
and $\varepsilon^{2}=m_{q}^{2}+z(1-z)Q^{2}$. The $\rho$–meson is treated as
the pure $s$-wave bound state of light quarks with the constituent quark mass
taken as $m_{q}=0.22\,\rm{GeV}$. As to the vector meson radial light-front
wave function (LFWF), we use the parametrization

$\displaystyle\psi_{V}(z,k)=C\exp\Big{[}-{1\over 2}a\,\vec{k}^{2}\Big{]},$ (7)

with

$\displaystyle\vec{k}^{2}={1\over 4}(M^{2}-4m_{q}^{2})\,.$ (8)

The wave function parameter $a$ is adjusted such that the leptonic decay width
of the $\rho$ meson is reproduced.

We now wish to specify the function ${\cal
F}(x,\mbox{\boldmath$\kappa$}_{1},\mbox{\boldmath$\kappa$}_{2})$. Here the
transverse momenta sum up to the total transverse momentum transfer
$\mbox{\boldmath$\kappa$}_{1}+\mbox{\boldmath$\kappa$}_{2}=\mbox{\boldmath$\Delta$}$.
We parametrize them as

$\displaystyle\mbox{\boldmath$\kappa$}_{1}={\mbox{\boldmath$\Delta$}\over
2}+\mbox{\boldmath$\kappa$}\,,\,\mbox{\boldmath$\kappa$}_{2}={\mbox{\boldmath$\Delta$}\over
2}-\mbox{\boldmath$\kappa$}\,,$ (9)

in terms of the integration variable $\kappa$. Then, we write

$\displaystyle{\cal F}\Big{(}x,{\mbox{\boldmath$\Delta$}\over
2}+\mbox{\boldmath$\kappa$},{\mbox{\boldmath$\Delta$}\over
2}-\mbox{\boldmath$\kappa$}\Big{)}=f(x,\mbox{\boldmath$\kappa$})\,G(\mbox{\boldmath$\Delta$}^{2})\,,$
(10)

where $f(x,\mbox{\boldmath$\kappa$})$ is an unintegrated gluon distribution,
which in the perturbative domain at large $\mbox{\boldmath$\kappa$}^{2}$ is
related to the standard gluon distribution as

$\displaystyle f(x,\mbox{\boldmath$\kappa$})\to{\partial
xg(x,\mbox{\boldmath$\kappa$}^{2})\over\partial\log\mbox{\boldmath$\kappa$}^{2}}\,.$
(11)

For the function $G(\mbox{\boldmath$\Delta$}^{2})$, which satisfies $G(0)=1$
and takes into account the momentum transfer dependence of the proton-Pomeron
coupling, we adopt two options:

  1. 1.

an exponential parametrization:

$\displaystyle G(\mbox{\boldmath$\Delta$}^{2})=\exp\Big{[}-{1\over
2}B\mbox{\boldmath$\Delta$}^{2}\Big{]}\,,$ (12)

with a diffraction slope of $B=4\,\rm{GeV}^{-2}$.

  2. 2.

a dipole form factor parametrization often used in nonperturbative Pomeron
models Donnachie  _et al._ (2004):

$\displaystyle
G(\mbox{\boldmath$\Delta$}^{2})={4m_{p}^{2}+2.79\mbox{\boldmath$\Delta$}^{2}\over
4m_{p}^{2}+\mbox{\boldmath$\Delta$}^{2}}\,{1\over\Big{(}1+{\mbox{\boldmath$\Delta$}^{2}\over\Lambda^{2}}\Big{)}^{2}}\,,$
(13)

with $\Lambda^{2}=0.71\,\rm{GeV}^{2}$.

Besides the model of Eq.( 10), we also tried a parametrization of the off-
diagonal UGD of the type

$\displaystyle{\cal
F}(x,\mbox{\boldmath$\kappa$}_{1},\mbox{\boldmath$\kappa$}_{2})={2\mbox{\boldmath$\kappa$}_{1}^{2}\mbox{\boldmath$\kappa$}_{2}^{2}\over\mbox{\boldmath$\kappa$}_{1}^{4}+\mbox{\boldmath$\kappa$}_{2}^{4}}\,f\Big{(}x,{\mbox{\boldmath$\kappa$}_{1}^{2}+\mbox{\boldmath$\kappa$}_{2}^{2}\over
2}\Big{)}\,G(\mbox{\boldmath$\Delta$}^{2})\,,$ (14)

which had been proposed in Ref. Cudell  _et al._ (2009). As we found only
small differences, we stick to the simpler form of Eq.( 10).

Before we come to a discussion of our results for the differential cross
section, let us briefly discuss the different models for the unintegrated
gluon distribution which we applied. Once again we stress that our process is
dominated by the nonperturbative domain. Here our assumption is simply that
this effective two-gluon exchange couples to the $q\bar{q}$ pair conserving
helicities of quarks. This rather economical choice avoids the introduction of
additional parameters. Let us remind the reader that also a class of
nonperturbative models can be cast into the form based on a nonperturbative
gluon distribution Shoshi  _et al._ (2002). Other models including an
anomalous chromomagnetic moment of quarks or other couplings motivated by an
instanton model of the QCD vacuum can be found for example in Korchagin  _et
al._ (2013); Kochelev  _et al._ (2016).

Figure 2: Different models of UGDs in $\log\kappa^{2}$ distributions. The
results for different x (or energy) are shown.

We will perform our numerical calculations for a number of UGDs. Those will
be:

  1. 1.

a UGD obtained from the Fourier transform of the Golec-Biernat–Wüsthoff dipole
cross section Golec-Biernat and Wüsthoff (1998). We slightly change the
normalization and use fixed value of $\alpha_{s}=0.3$.

  2. 2.

a UGD by Ivanov and Nikolaev Ivanov and Nikolaev (2002), which contains
explicit modeling of the nonperturbative low $\kappa^{2}$ part. Here, as for
all the UGDs below, a freezing of $\alpha_{s}$ in the soft region is employed.

  3. 3.

a UGD of the Durham group constructed by the so-called Kimber-Martin-Ryskin
(KMR) Kimber  _et al._ (2001) procedure starting from collinear partons of
Ref. Harland-Lang  _et al._ (2015). The UGD is extended to the soft region in
a simple way, such that the distribution in $\log\kappa^{2}$ levels off to an
$x$-dependent plateau at small $\kappa^{2}$.

  4. 4.

The Kutak-Staśto (KS) nonlinear UGD from Ref. Kutak and Stasto (2005).

In Fig. 2 we show the four different versions of the UGDs which we use for the
$x$-values corresponding to the $\gamma p$-energies relevant to the CMS data
[CMS] (2018).

##  III Results

Most phenomenological studies of vector meson photoproduction concentrate on
the modelling of the forward amplitude or on the region of small momentum
transfers within the diffraction cone. The CMS data for $d\sigma/dt$ up to
$-t\sim 1\,\rm{GeV}^{2}$ present us with welcome opportunity to test popular
models of vector meson production in a broader kinematic region.

It should be noted, that we are still far away from the regime of the “hard
exclusive” process for which $s\sim|t|\sim|u|\gg\Lambda_{QCD}^{2}$, and where
a perturbative approach based on a different type of factorization becomes
applicable Lepage and Brodsky (1980). Rather, although we go beyond the
diffraction cone, we are still in the domain of Regge asymptotics, $-t/s\ll
1$.

We come to the comparison with the results of the CMS collaboration [CMS]
(2018). The differential cross section $d\sigma/dt$ was extracted from
ultraperipheral proton-lead collisions for four different $\gamma p$ cm-
energies: $W=35,59,108,179\,\rm{GeV}$. At these energies we can safely
approximate $t\sim-\mbox{\boldmath$\Delta$}^{2}$.

In Fig. 3, we show our results for the IN UGD using the exponential
parametrization for $G(\mbox{\boldmath$\Delta$}^{2})$. We show the $T\to T$
helicity conserving contribution by the long dashed line. The dotted line
shows the $T\to L$ transition, where helicity is changed by one unit. Finally,
by the dash-dotted line we display the $T\to T^{\prime}$ transition where
$|\lambda_{\gamma}-\lambda_{V}|=2$. We stress, that we take the Pomeron
coupling on the proton side to be helicity conserving, in accord with the very
small spin-flip coupling found in Regge analyses Buttimore  _et al._ (1999).
In this case, in agreement with general requirements of angular momentum
conservation, helicity flip amplitudes vanish in the forward direction
$\propto|\mbox{\boldmath$\Delta$}|^{|\lambda_{\gamma}-\lambda_{V}|}$. Notice
that this vanishing of the spin flip contributions is not visible at the
resolution of the plot in Fig. 4.

We observe, that the $T\to T$ contribution has a dip at $-t\sim 0.5\div
0.7\,\rm{GeV}^{-2}$, which position is slightly dependent on energy. It
derives from a zero in the corresponding amplitude. We note that the dip is
partially filled by the $T\to L$ contribution and turns into a shoulder with
increasing energy. The $T\to L$ transition itself possesses a dip at
$-t\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$<$}}0.2\,\rm{GeV}^{-2}$ but in this region the $T\to T$ transition
vastly dominates. The double helicity flip contribution is very small
throughout the whole kinematic region.

In Fig. 4 we show the results for the same IN UGD, but now using the dipole
parametrization for $G(\mbox{\boldmath$\Delta$}^{2})$. We observe that some
qualitative features, like the position of the dip are very similar to the
previous case, however the cross section develops a much harder tail at large
$-t$. In general, the IN UGD gives a reasonable description of data,
especially within the diffraction cone, however a diffractive minimum or
shoulder predicted by this glue are not borne out by the data.

We now turn to the results for other UGDs, where for definiteness we stick to
the dipole parametrisation of $G(\mbox{\boldmath$\Delta$}^{2})$.

In Fig. 5 we show the results for the KMR UGD. Here the helicity conserving
part dominates throughout. In this case, there is no dip in the differential
cross section. The description of data is very good except for the highest
energy, where the $t$-dependence is too hard.

In Fig. 6 the results for the Kutak-Staśto UGD are shown. Also here we observe
no dip and a complete dominance of the helicity conserving process. The CMS
data are somewhat underpredicted with a slightly too hard $t$-dependence.

The results for the GBW UGD are shown in Fig. 7. They show generally the best
agreement with the data. Also here, there is no dip within the measured
region, and again helicity flip transitions are negligible.

Finally, we show for convenience the results for all UGDs summed over all
helicity combinations in Fig. 8.

Figure 3: Distribution in $t$, the four-momentum transfer squared in the
$\gamma p\to\rho p$ reaction for different energies and the Ivanov-Nikolaev
UGD. Here the exponential parametrization of the form factor
$G(\mbox{\boldmath$\Delta$}^{2})$ was used.

Figure 4: Distribution in $t$, the four-momentum transfer squared in the
$\gamma p\to\rho p$ reaction for different energies and the Ivanov-Nikolaev
UGDF. Here the dipole parametrization of the form factor
$G(\mbox{\boldmath$\Delta$}^{2})$ was used.

Figure 5: Distribution in $t$ for different energies and the KMR UGD. Here the
dipole parametrization of the form factor $G(\mbox{\boldmath$\Delta$}^{2})$
was used.

Figure 6: Distribution in $t$ for different energies and the Kutak-Staśto
nonlinear UGD. Here the dipole parametrization of the form factor
$G(\mbox{\boldmath$\Delta$}^{2})$ was used.

Figure 7: Distribution in $t$ for different energies and the GBW UGD. Here the
dipole parametrization of the form factor $G(\mbox{\boldmath$\Delta$}^{2})$
was used.

Figure 8: Distribution in $t$ for different energies for the different UGDs.

##  IV Conclusions

In this work we have investigated the differential cross section $d\sigma/dt$
for diffractive $\rho$-meson photoproduction at high energies in a model based
on two-gluon exchange in the nonperturbative domain. We have studied the role
played by the often neglected helicity-flip amplitudes, which can contribute
at finite $t$.

We have found that the large $|t|$-behaviour $d\sigma/dt$ depends on the form
factor describing the coupling of the pomeron to the $p\to p$ transition,
while the dip-bump structure depends rather on the UGD used.

We have included traditional $T\to T$ contribution as well as somewhat smaller
$T\to L$ and $T\to T^{\prime}$ (double spin-flip) contributions. The relative
amount and differential shape of the subleading contributions depends on the
UGD used.

Some of the UGDs generate dips for $T\to T$ transition. A good example is the
Ivanov-Nikolaev UGD. All UGDs generate dips for $T\to L$ transition.

Although helicity flip contributions are negligible for most of the models
that we have employed, they can be relevant, depending very much on the
transverse momentum dependence of the UGD. This should be kept in mind
especially when a specific model predicts diffractive dips and account only
for the helcity conserving pieces, as is the common practice with most current
models.

## Acknowledgements

This work was partially supported by the Polish National Science Center grant
UMO2018/31/B/ST2/03537 and by the Center for Innovation and Transfer of
Natural Sciences and Engineering Knowledge in Rzeszów.

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