# Fastcc: fast colour corrections for broadband radio telescope data

Mike W. Peel Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife,
Spain  Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna,
Tenerife, Spain  Ricardo Genova-Santos Instituto de Astrofísica de Canarias,
38200 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de
La Laguna, 38206 La Laguna, Tenerife, Spain  C. Dickinson Jodrell Bank Centre
for Astrophysics, Department of Physics and Astronomy, The University of
Manchester, Oxford Road, Manchester M13 9PL, UK  J. P. Leahy Jodrell Bank
Centre for Astrophysics, Department of Physics and Astronomy, The University
of Manchester, Oxford Road, Manchester M13 9PL, UK  Carlos López-Caraballo
Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain
Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna,
Tenerife, Spain  M. Fernández-Torreiro Instituto de Astrofísica de Canarias,
38200 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de
La Laguna, 38206 La Laguna, Tenerife, Spain  J. A. Rubiño-Martín Instituto de
Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain  Departamento de
Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain  Locke
D. Spencer Institute for Space Imaging Science, University of Lethbridge,
Lethbridge, Alberta, T1K 3M4, Canada

###### Abstract

Broadband receiver data need colour corrections applying to correct for the
different source spectra across their wide bandwidths. The full integration
over a receiver bandpass may be computationally expensive and redundant when
repeated many times. Colour corrections can be applied, however, using a
simple quadratic fit based on the full integration instead. Here we describe
fastcc and interpcc, quick Python and IDL codes that return, respectively,
colour correction coefficients for different power-law spectral indices and
modified black bodies for various Cosmic Microwave Background related
experiments. The codes are publicly available, and can be easily extended to
support additional telescopes.

Unified Astronomy Thesaurus concepts: CMBR detectors (259) Radio receivers
(1355) Astronomy data analysis (1858) Photometry (1234)

##  1 Background

Cosmic Microwave Background (CMB) observations require telescopes using high
sensitivity receivers with wide bandwidths that are a large fraction of the
observing frequency. For example, the 28.4 GHz channel of the Planck satellite
was sensitive to photons at frequencies between $\sim 24-34$ GHz (see Fig. 1),
a fractional bandwidth (i.e., $\Delta\nu/\nu$) of $\sim$35 %. With such high
fractional bandwidths, the measured flux density of an astronomical source
depends on the convolution of the observing spectral bandpass with the source
spectrum. For example, a bandpass that is most sensitive at low frequencies
will result in an increase in the measured flux density of a steep spectrum
radio source compared to a flatter spectrum source. Colour corrections avoid
these effects by correcting the flux density to that which a monochromatic
receiver operating at a reference frequency $\nu_{0}$ would have measured.

The colour correction thus depends on the spectral index of the source, and if
the value for this changes (e.g., while fitting a model with multiple spectral
components, or when adding new observations), then the colour correction also
changes. For this reason, raw flux densities are published without colour
corrections, but colour corrections are applied for any scientific analysis
and for the inclusion of data in data visualization and subsequent analysis.

The fastcc and interpcc codes provide fast colour corrections for CMB
datasets, particularly in MCMC fitting techniques.

##  2 Formalism

Figure 1: Left: example bandpasses from Planck LFI, showing how these can be
complex structures that extend significantly beyond the reference frequency
for broadband detectors (dashed lines). Right: corresponding colour correction
fits for various spectral indexes.

We follow and expand on the formalism from Planck Collaboration (2016). When
the source spectrum is described by a power-law, $S\propto\nu^{\alpha}$, the
colour correction is the ratio of integrals over frequency $\nu$, with the
numerator integrating over the bandpass $g(\nu)$ (if measured in temperature
units) and the spectrum the data is calibrated to, $\delta$, and the
denominator integrating over the combination of the bandpass and the source
spectrum with spectral index $\alpha$, as

$C(\nu_{0},\alpha)=\frac{\int g(\nu)(\nu/\nu_{0})^{\delta-2}~{}d\nu}{\int
g(\nu)(\nu/\nu_{0})^{\alpha-2}~{}d\nu},$ (1)

where $\nu_{0}$ is the reference frequency. If the instrument bandpass has
been measured in intensity units, $\tau(\nu)$, this becomes:

$C(\nu_{0},\alpha)=\frac{\int\tau(\nu)(\nu/\nu_{0})^{\delta}~{}d\nu}{\int\tau(\nu)(\nu/\nu_{0})^{\alpha}~{}d\nu}.$
(2)

For instruments calibrated in CMB thermodynamic units, this becomes:

$C(\nu_{0},\alpha)=\frac{\int g(\nu)\eta_{\Delta T}(\nu)/\eta_{\Delta
T}(\nu_{0})~{}d\nu}{\int g(\nu)\left(\nu/\nu_{0}\right)^{\alpha-2}~{}d\nu},$
(3)

where $\eta_{\Delta T}(\nu)=x^{2}e^{x}/(e^{x}-1)^{2}$, with $x=h\nu/(k_{\rm
B}T_{\rm CMB})$ to adjust for the CMB spectrum in Rayleigh–Jeans temperature
units, with Planck constant $h$, Boltzmann constant $k_{\rm B}$, and CMB
temperature $T_{\rm CMB}=2.7255$ K. However, the ratio of $\eta_{\Delta T}$ is
usually small, only becoming important when $\nu_{0}$ is far from the centre
of the bandpass.

The resulting colour corrections are very smooth (see Fig. 1, right), and can
be fitted with a quadratic function:

$C(\nu_{0},\alpha)=c_{0}+c_{1}\alpha+c_{2}\alpha^{2}.$ (4)

The source flux density is then corrected as:

$S_{\mathrm{corr}}(\nu_{0})=S_{\mathrm{orig}}(\nu_{0})C(\nu_{0},\alpha)=S_{\mathrm{orig}}(\nu_{0})\times\left(c_{0}+c_{1}\alpha+c_{2}\alpha^{2}\right).$
(5)

The colour corrections for thermal dust spectra can be calculated using the
same equations as for power laws, but changing the source spectrum for a grey
body, with the dust temperature $T_{\mathrm{dust}}$ and spectral index
$\beta_{\mathrm{dust}}$ as input parameters. For equations, see the fastcc
explanatory supplement. A quadratic fit becomes increasingly different from
the colour corrections at extreme values. As such, we also include interpcc,
which uses interpolation over a set of cached colour correction values.

There are two equivalent approaches to colour correction: either change the
flux density of the source at a fixed reference frequency, or change the
reference frequency for the observations. When characterising multiple sources
within a survey, the analysis is normally simpler if all sources are at the
same frequency with corrected flux density. Alternatively, effective
frequencies $\nu_{\mathrm{eff}}$, for which $C(\nu_{\mathrm{eff}},\alpha)=1$,
can be calculated from the colour corrections (in thermodynamic units: convert
to intensity at $\nu_{\mathrm{eff}}$) via:

$\nu_{\mathrm{eff}}=\nu_{0}C(\nu_{0},\alpha)^{1/(\alpha-2)}.$ (6)

##  3 Code

The code takes input $\alpha$ for fastcc, or $T_{\mathrm{dust}}$, and
$\beta_{\mathrm{dust}}$ for interpcc, and returns the factor that the measured
flux density must be multiplied to obtain the corrected flux density. We
include both Python and IDL versions of fastcc, and interpcc in Python.
Test/example scripts are also included for both fastcc and interpcc. The code
is available at https://github.com/mpeel/fastcc and
https://doi.org/10.5281/zenodo.7376510, along with a more detailed explanatory
supplement. An IDL version for Planck LFI only was released in the Planck
Legacy Archive.111http://pla.esac.esa.int/pla/ While no dependencies are
required for fastcc, interpcc and the colour correction calculations use
Astropy (Astropy Collaboration et al., 2018), Numpy (Harris et al., 2020), and
scipy (Virtanen et al., 2020).

The code now includes colour corrections for Planck LFI (Planck Collaboration,
2014a, b, 2016) and HFI (Planck Collaboration, 2014b), WMAP (Bennett et al.,
2013), IRAS (IRAS, 1988),
DIRBE,222https://lambda.gsfc.nasa.gov/product/cobe/dirbe_ancil_sr_get.cfm
QUIJOTE MFI (Rubiño-Martín et al., MNRAS accepted, Genova-Santos et al., in
prep.), and C-BASS (King et al., 2014, Taylor et al., in prep.). Additional
experiments can be added on request and provision of bandpasses.

## References

  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., SipHocz, B. M., et al. 2018, aj, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Bennett et al. (2013) Bennett, C. L., Larson, D., Weiland, J. L., et al. 2013, ApJS, 208, 20, doi: 10.1088/0067-0049/208/2/20
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * IRAS (1988) IRAS. 1988, Infrared Astronomical Satellite (IRAS) Catalogs and Atlases.Volume 1: Explanatory Supplement., Vol. 1
  * King et al. (2014) King, O. G., Jones, M. E., Blackhurst, E. J., et al. 2014, MNRAS, 438, 2426, doi: 10.1093/mnras/stt2359
  * Planck Collaboration (2014a) Planck Collaboration. 2014a, A&A, 571, A5, doi: 10.1051/0004-6361/201321527
  * Planck Collaboration (2014b) —. 2014b, A&A, 571, A9, doi: 10.1051/0004-6361/201321531
  * Planck Collaboration (2016) —. 2016, A&A, 594, A2, doi: 10.1051/0004-6361/201525818
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2


# Aggregated Multi-GANs for Controlled 3D Human Motion Prediction

Zhenguang Liu1, Kedi Lyu2, Shuang Wu3*, Haipeng Chen2*, Yanbin Hao4, Shouling
Ji5
Corresponding Authors

###### Abstract

Human motion prediction from historical pose sequence is at the core of many
applications in machine intelligence. However, in current state-of-the-art
methods, the predicted future motion is confined within the same activity. One
can neither generate predictions that differ from the current activity, nor
manipulate the body parts to explore various future possibilities.
Undoubtedly, this greatly limits the usefulness and applicability of motion
prediction. In this paper, we propose a generalization of the human motion
prediction task in which control parameters can be readily incorporated to
adjust the forecasted motion. Our method is compelling in that it enables
manipulable motion prediction across activity types and allows customization
of the human movement in a variety of fine-grained ways. To this aim, a simple
yet effective composite GAN structure, consisting of local GANs for different
body parts and aggregated via a global GAN is presented. The local GANs game
in lower dimensions, while the global GAN adjusts in high dimensional space to
avoid mode collapse. Extensive experiments show that our method outperforms
state-of-the-art. The codes are available at https://github.com/herolvkd/AM-
GAN.

## Introduction

Figure 1: Problem illustration. Mathematically, an observed motion sequence
traces out a trajectory on the skeletal pose manifold. The problem of human
motion prediction amounts to extrapolating this trajectory in a smooth and
coherent manner. We are interested in tasks: 1) predicting the most likely
motion (e.g., continuation of the walking action); 2) predicting the
transition to another specified action (e.g. transition from current action to
eating); 3) fine-grained control over body parts, e.g., requiring the upper
body performing eating action while the lower body engaging in walking.

An important component of our capacity to interact with the external world
resides in the ability to predict the future (Martinez, Black, and Romero
2017), based on existing cues and past experience. Having a mental
representation of how other humans are likely to act is crucial for us to
avoid dangers and make decisions. Likewise, the ability for machines to
anticipate and model human motion dynamics is very much coveted. Modeling and
predicting human motion is at the heart of many important applications in the
computer vision domain such as video making, human posture tracking (Taylor et
al. 2010), as well as the robotics domain such as regulating a robot’s
response and behaviour in human-machine interactions (Koppula and Saxena 2013;
Koppula, Jain, and Saxena 2016).

Human motion prediction(HMP) is challenging since humans do not follow fixed
laws of motion as is the case for inanimate objects. The inherent stochastic
nature of human behaviour invokes complexity in the form of non-linearity and
high dimensionality. Earlier approaches have typically utilized shallow
latent-variable models such as hidden Markov models (Lehrmann, Gehler, and
Nowozin 2014) or Gaussian processes (Wang, Fleet, and Hertzmann 2005) to
represent the motion dynamics with hidden states. In recent years, many deep
learning approaches were proposed and have yielded improved results. A first
class of work utilised recurrent neural networks (RNNs) such as Long Short
Term Memory (LSTM) (Fragkiadaki et al. 2015), Gated Recurrent Unit (GRU)
(Martinez, Black, and Romero 2017; Pavllo et al. 2019) or refined RNN
architectures tailored for modeling motion dynamics. Another line of work
built upon the success of deep generative models including Variational
Autoencoders (VAEs) (Habibie et al. 2017; Yan et al. 2018) and Generative
Adversarial Networks (GANs) (Barsoum, Kender, and Liu 2018; Kundu, Gor, and
Babu 2019).

A major shortcoming of prior works is that they can only predict a single
output sequence for any given observed motion sequence. However, given our
incomplete knowledge of the present, the future should not be represented as a
single deterministic state but rather a spectrum of plausible outcomes.
Considering only a single future outcome may be inadequate and lead to
misinformed decision making. For example, in safety-crucial applications such
as autonomous driving, unexpected maneuvers such as sudden overtaking must be
accounted for in guiding the course of actions.

Another severe limitation of existing works is that they can only predict one
single activity belonging to the same activity as the observed sequence, e.g.
predicting ‘walking’ motion from an observed ‘walking’ sequence. For real
applications such as video synthesis, animation, and game character movement
generation, we usually want to (1) customize a future motion with fine-grained
control on the body parts, e.g., requiring the whole body act in ‘walking’
activity but with the right arm act in ‘eating’, and (2) command the
prediction to smoothly transfer from current activity to another activity.
However, existing methods clearly fail to fulfill the above requirements. This
motivates us to propose the controlled motion prediction problem. The ability
to control and fine-tune the generated future motion would allow for a broader
research space in human-machine interaction and would certainly be desirable
for many applications.

Inspired by a divide and conquer strategy, we propose a novel Aggregated
Multi-GAN (AM-GAN) framework for the task of controlled human motion
prediction. The generator module in our architecture allows for conditional
motion generation, which may be used to realize action transfer, _i.e.,_ the
predicted motion can be controlled to transfer to a specified action instead
of continuing the original action. AM-GAN models the motion of central spine
and four limbs with separate GANs and the final complete motion is aggregated
over them. The aggregation process ensures that the overall dynamics and
interaction between kinematic chains are balanced and attuned. Dividing the
generative modeling of the entire human skeletal motion into sub kinematic
chains reduces the complexity of high dimensional motion generation into a
less complicated low dimensional one. Furthermore, such an approach also
enables us to acquire fine-grained control over the generated motion in body
chains.

To summarize, our main contributions are: 1) We propose a generalized 3D human
motion prediction problem that allows for adequate and fine-grained control
over the predictive spectrum. 2) We design a novel Aggregated Multi-GAN (AM-
GAN) framework to tackle this problem via a divide and conquer strategy. 3)
AM-GAN sets the new state-of-the-art performance for quantitative short-term
prediction and also generates natural and consistent long-term prediction.
More importantly, AM-GAN also allows control over the prediction such as
generating transitions to specified actions and fine-grained control of body
part movements. The AM-GAN is, to our knowledge, the first work that can
handle these tasks simultaneously (short-term prediction, long-term
prediction, specified action transfer, and fine-grained motion prediction
control).

## Related Work

RNN based motion prediction. Deep learning has achieved great success in
various fields (Kingma et al. 2014; Butepage et al. 2017; Liu et al. 2017; Zhu
et al. 2020; Xu et al. 2020). RNN is a natural candidate for modeling human
motion sequence due to their suitability in handling time series. Each time
step of a motion sequence is encoded as a hidden state that learns the
dynamics of the motion up to the current. The final hidden state corresponding
to the last observed frame is then recurrently updated to generate a future
sequence. This approach has been adopted in several works (Fragkiadaki et al.
2015; Martinez, Black, and Romero 2017). A similar work (Li et al. 2018)
employs convolutional layers instead of RNN units. In general, the progress
within this line of work seeks to better model long-term temporal dependency
and overcome the issue that the current hidden state tends to be overwhelmed
by the last observed frame. (Martinez, Black, and Romero 2017) proposes to
incorporate a residual connection between the RNN units while (Liu et al.
2019) proposes a hierarchical RNN architecture. The major shortcoming of this
class of works is that the output is fixed to be a single deterministic future
motion sequence, and would be inadequate for obtaining a representation of the
spectrum of future possibilities.
GAN based human motion prediction. GAN (Goodfellow et al. 2014) provides an
alternative to the RNN based approach for HMP. In an adversarial process, a
generator network outputs possible future motion sequences conditioned on an
observed input sequence and a discriminator network assesses how likely these
generated sequences are real or fake. In contrast to RNN-based approaches,
GAN-based approaches (Barsoum, Kender, and Liu 2018; Kundu, Gor, and Babu
2019) for HMP are probabilistic in that the learned generative model can
output different sequences of possible future human poses from the same input
sequence by sampling an additional latent noise vector. A challenging problem
with GAN approaches is mode-collapsing during training, which leads to
generated outputs all lying within certain modes. (Kundu, Gor, and Babu 2019)
proposed a modification of the discriminator so as to incorporate both
adversarial loss as well as content loss on the prediction.
Controllable motion prediction. To our knowledge, few existing works
assimilate control for motion prediction tasks. (Holden, Saito, and Komura
2016; Pavllo et al. 2019) incorporate control parameters in the form of path
trajectories along which a predicted walking or running action is bound to
traverse. Another related work (Holden et al. 2017) adapts the technique of
neural style transfer for motion generation. For example, a walking motion
sequence may be manifested in a jubilant fashion or a depressed manner and
these different styles can be transferred onto a walking sequence without
altering the essential content. However, the problem of generating transitions
between different class of actions has not yet been studied in the setting of
HMP. Different from previous works, our approach does not focus on controlling
the trajectory or expression style of a walking action. Rather, we seek to
allow for predicting transitions between different actions and fine-grained
control over body parts in executing a prediction.

## Our Approach

Figure 2: The proposed AM-GAN architecture. Each kinematic chain is modeled by
its own sub-GAN (with matching colors). Each sub-GAN has a generator in the
form of a VAE network as well as a local critic. The combined output from all
sub-GANs is then aggregated and passed to a global critic. The aggregation
layer and the global critic constitute the global GAN.

In this section, we present the details of our proposed AM-GAN. The overview
of our approach is inspired by the divide and conquer strategy. We observe
that by reducing the general motion generation task entailing of the entire
human skeleton to its constituent skeletal kinematic chains, the issue of mode
collapsing can be effectively alleviated, leading to more accurate and diverse
motion sequence forecasts.

### Problem Formulation

Presented with an observed human motion sequence parameterized as
$\left(\mathbf{p}_{1},\mathbf{p}_{2},\cdots,\mathbf{p}_{t}\right)$ with
optional control parameters, we are interested in generating a future pose
sequence
$\left(\mathbf{p}_{t+1},\mathbf{p}_{t+2},\cdots,\mathbf{p}_{t+T}\right)$.

Each human skeletal pose $\mathbf{p}_{i}$ is composed of distinct key joints.
For simplicity, we adopt 3D coordinates to characterize the locations of each
joint. Mathematically, the skeletal human pose lies on a manifold, and a
motion sequence traces out a trajectory on this pose manifold, which is
depicted in Fig. 1. The problem of HMP is to find a smooth extension to the
observed trajectory, while action transfer is to find a smooth transfer from
current trajectory to the target one.

### Method Overview and Rationale

Divide and conquer with multiple GANs and aggregation. Existing methods
attempt to model the dynamics of entire human skeletal motion in its entirety.
(Jain et al. 2016; Li et al. 2018) further seek to model spatial dependencies
between joints. These attempts set the problem of motion generation on an
unnecessarily high dimensional manifold. Optimizing for both the adversarial
and content loss on a high dimensional manifold is harder and more likely to
incur issues of non-convergence, gradient diminishing and mode collapse
(Kodali et al. 2017).

Taking into account the anatomical separability of different chains and the
difficulty of motion generation in high dimensional manifolds, we propose to
divide the motion generation problem into that of the constituent chains and
conquer the complete problem through an aggregation process that takes into
account the interaction between these chains.

Corresponding to each pose $\mathbf{p}_{i}$, the constituent chains will be
denoted as $\mathbf{c}_{i}^{j}$ where $1\leq j\leq 5$ is the chain index
(torso and four limbs). Note that we also experimented with other body
divisions such as dividing each limb into two parts, but with no significant
performance gain obtained.

Controlled prediction across action types. It is not realistic to confine
human motion prediction within a single class of action, as is the case for
existing methods. We propose a method that explicitly allows for predictions
where one action may transit and evolve into another specified action.
Therefore, instead of purely RNNs, we engage generative modeling which are
better suited for this objective.

Fine-grained control of body parts. Previous motion prediction works are
lacking in their abilities to control individual body parts in the predicted
motion. We attempt to attain fine-grained control of body components and
again, modeling each kinematic chain with its own specific GAN plays a major
role in this. The capacity to control fine details of body parts in motion is
undoubtedly a very desirable influence in fostering the versatility and
usability in human motion prediction research. For example, different
stylistic peculiarities or gait patterns across different individuals maybe
directly integrated into the final prediction.

Methods \Time (ms) | Walking | Eating
---|---|---
80 | 160 | 320 | 400 | 1000 | 80 | 160 | 320 | 400 | 1000
ERD (Fragkiadaki et al. 2015) | 0.93 | 1.18 | 1.59 | 1.78 | 2.24 | 1.27 | 1.45 | 1.66 | 1.80 | 2.02
SRNN (Jain et al. 2016) | 0.81 | 0.94 | 1.16 | 1.30 | 1.80 | 0.97 | 1.14 | 1.35 | 1.46 | 2.11
RRNN (Martinez, Black, and Romero 2017) | 0.33 | 0.56 | 0.78 | 0.85 | 1.14 | 0.26 | 0.43 | 0.66 | 0.81 | 1.34
LSTM-AE (Tu et al. 2018) | 1.00 | 1.11 | 1.39 | —– | 1.39 | 1.31 | 1.49 | 1.86 | —– | 2.01
CEM (Li et al. 2018) | 0.33 | 0.54 | 0.68 | 0.73 | 0.92 | 0.22 | 0.36 | 0.58 | 0.71 | 1.24
HP-GAN (Barsoum, Kender, and Liu 2018) | 0.95 | 1.17 | 1.69 | 1.79 | 2.47 | 1.28 | 1.47 | 1.70 | 1.82 | 2.51
SKelNet (Guo and Choi 2019) | 0.34 | 0.52 | 0.69 | 0.70 | 0.90 | 0.23 | 0.39 | 0.50 | 0.71 | 1.23
BiHMP-GAN (Kundu, Gor, and Babu 2019) | 0.33 | 0.52 | 0.63 | 0.67 | 0.85 | 0.20 | 0.33 | 0.54 | 0.70 | 1.20
HMR (Liu et al. 2019) | 0.34 | 0.52 | 0.67 | 0.69 | 0.90 | 0.22 | 0.31 | 0.51 | 0.69 | 1.21
QuaterNet (Pavllo et al. 2019) | 0.23 | 0.37 | 0.59 | 0.66 | 0.86 | 0.20 | 0.33 | 0.54 | 0.68 | 1.17
AM-GAN (Ours) | 0.23 | 0.51 | 0.62 | 0.66 | 0.84 | 0.20 | 0.31 | 0.49 | 0.66 | 1.15
Methods \Time (ms) | Smoking | Discussion
80 | 160 | 320 | 400 | 1000 | 80 | 160 | 320 | 400 | 1000
ERD (Fragkiadaki et al. 2015) | 1.66 | 1.95 | 2.35 | 2.42 | 3.14 | 2.27 | 2.47 | 2.68 | 2.76 | 3.11
SRNN (Jain et al. 2016) | 1.45 | 1.68 | 1.94 | 2.08 | 2.57 | 1.22 | 1.49 | 1.83 | 1.93 | 2.19
RRNN (Martinez, Black, and Romero 2017) | 0.35 | 0.64 | 10.3 | 1.15 | 1.83 | 0.37 | 0.77 | 1.06 | 1.10 | 1.79
LSTM-AE (Tu et al. 2018) | 0.92 | 1.03 | 1.15 | —– | 1.77 | 1.11 | 1.20 | 1.38 | —– | 1.73
CEM (Li et al. 2018) | 0.26 | 0.49 | 0.96 | 0.92 | 1.62 | 0.32 | 0.67 | 0.94 | 1.01 | 1.86
HP-GAN (Barsoum, Kender, and Liu 2018) | 1.71 | 1.89 | 2.33 | 2.42 | 3.20 | 2.29 | 2.61 | 2.79 | 2.88 | 3.67
SKelNet (Guo and Choi 2019) | 0.26 | 0.47 | 0.91 | 0.89 | 1.56 | 0.29 | 0.62 | 0.88 | 1.00 | 1.68
BiHMP-GAN (Kundu, Gor, and Babu 2019) | 0.26 | 0.50 | 0.91 | 0.86 | 1.11 | 0.33 | 0.65 | 0.91 | 0.95 | 1.77
HMR (Liu et al. 2019) | 0.26 | 0.47 | 0.90 | 0.91 | 1.49 | 0.29 | 0.55 | 0.83 | 0.94 | 1.61
QuaterNet (Pavllo et al. 2019) | 0.26 | 0.49 | 0.92 | 0.89 | 1.67 | 0.28 | 0.62 | 0.87 | 0.95 | 1.89
AM-GAN (Ours) | 0.25 | 0.46 | 0.88 | 0.88 | 1.10 | 0.28 | 0.55 | 0.81 | 0.92 | 1.58
Table 1: Performance evaluation (in MAE) of comparison methods over 4
different action types on the H3.6m dataset

### Architecture Overview

The overall framework of our method is illustrated in Fig. 2. There are five
sub-GANs (local GANs), each taking care of a kinematic chain. The sub-GANs
will now model motion generation on a lower dimensional manifold which is
advantageous from a computational perspective and may be favorable in avoiding
mode collapse. After a local discrimination process, the outputs from all sub-
GANs are then combined via an aggregation layer. The output of the aggregation
layer is then passed to the global critic for assessing the quality of the
generated motion sequence in its entirety.

The conditional probability distributions learnt through the generative
modeling process may then be invoked to realize action transfer in motion
prediction. The sub-GANs in the AM-GAN network can further be mobilized to
control the skeletal pose at a fine-grained scale.

Generator. In order to effectively interpolate between latent representations
for generating transitions across different action types, we have adopted a
Variational Auto-encoder (VAE) architecture (Kingma et al. 2014) as the
generator module for each of our sub-GAN. Since the task entails sequential
timeseries data, the encoder and decoder engage GRU for modeling the motion
sequences.

Formally, the encoder (in sub-GAN $j$) maps the observed sequence
$\mathbf{C}^{j}=\left(\mathbf{c}_{1}^{j},\mathbf{c}_{2}^{j},\cdots,\mathbf{c}_{t}^{j}\right)$
to a latent variable $\mathbf{z}^{j}$.

$\displaystyle\mathbf{z}^{j}$ $\displaystyle=$
$\displaystyle\text{EN}(\mathbf{C}^{j})$ (1)

The decoder then recursively generates a future prediction sequence based on
this latent vector $\mathbf{z}^{j}$.

$\displaystyle\mathbf{C}^{\prime j}$ $\displaystyle=$
$\displaystyle\text{DE}(\mathbf{z}^{j})$ (2)

Figure 3: We perform motion interpolation in the latent space (i.e., $z$ in
VAE) and insert these motion frames (red) to fill up the discontinuity between
two action sequences. Different from other two tasks, here the AM-GAN network
is trained to predict an immediate future pose rather than a sequence.

Aggregation. Dynamically, there is interconnection and interdependence between
different kinematic chains, as typified in the synchronized movements of the
right arm and left leg in walking. Therefore, the individual latent motion
contexts from each kinematic chain is mapped through a fully connected
aggregation layer, to the final predicted pose sequence output as follows:

$\displaystyle\mathbf{P}^{\prime}$ $\displaystyle=$
$\displaystyle\text{Agg}\left(\sum_{j=1}^{5}\mathbf{C}^{\prime j}\right)$ (3)

where Agg denotes the mapping function of the aggregation layer.
$\mathbf{P}^{\prime}$ is shorthand for the future pose sequence.

Critic. The critic module consists of Local Critics for each sub-GAN
$\text{D}^{j}$ and a Global Critic $\text{D}^{G}$ after the aggregation layer.
A three-layers fully connected feedforward network is used in both types of
critics. The local critics judge the generation of the five branches of the
human body to ensure the accuracy of local prediction. The rationale is to
first focus on the local context and reduce interference between local
components or body parts at the base level. However, we cannot simply ignore
the correlation and synchronization between different body parts in executing
a motion. We also include a global critic to judge the feasibility of the
motion and to ensure the naturalness and accuracy of the entire pose.

### Loss Function

Now, we zoom in on the loss function for training AM-GAN. To account for
chain-level and pose-level accuracy, we consider both local loss and global
loss. We further engage a stability loss at the chain-level and a consistency
loss at the pose-level to ensure the authenticity and temporal smoothness of
the motion.

Chain-level loss. Within each sub-GAN, a generator outputs the motion
prediction of a kinematic chain $j$. A critic network $\mathbf{D}^{j}$ learns
to assign higher scores to real future sequences and lower scores to the
generated ones. This gives the Wasserstein GAN loss for kinematic chain $j$
as:

$\displaystyle\mathcal{L}_{\text{Chain}^{j}}=\mathop{\mathbb{E}}_{\mathbf{C}^{j}}\mathbf{D}^{j}[C^{j}_{t+1:t+T}]-\mathop{\mathbb{E}}_{\mathbf{C^{\prime}}^{j}}\mathbf{D}^{j}[C^{\prime
j}_{t+1:t+T}]$ (4)

where $\mathbf{C}^{j}$ and $\mathbf{C}^{\prime j}$ respectively denote the
real and generated future sequences for chain $j$.

On top of the Wasserstein GAN loss, we further introduce a stability loss
given by the variance over $\mathcal{L}_{\text{Chain}^{j}}$.

$\displaystyle\mathcal{L}_{\text{Stability}}=\frac{1}{5}\sum_{j=1}^{5}(\mathcal{L}_{\text{Chain}^{j}}-\overline{\mathcal{L}_{\text{Chain}^{j}}})^{2}.$
(5)

The local or chain level loss for chain $j$ is given by a weighted sum of the
Wasserstein GAN loss for the chain as well as the stability loss,

$\displaystyle\mathcal{L}_{\text{Local}}^{j}=\alpha\mathcal{L}_{\text{Chain}^{j}}+\beta\mathcal{L}_{\text{Stability}}+\gamma\mathcal{L}_{\text{GroundTruth}}^{j}$
(6)

where $\alpha$, $\beta$, and $\gamma$ are weight hyperparameters.
$\mathcal{L}_{\text{GroundTruth}}^{j}$ is the difference between ground truth
of chain $j$ and its prediction.

Pose-level loss. The aggregation layer aggregates and balances the five
kinematic chains to give a generated pose sequence
$\mathbf{P}^{\prime}_{t+1:t+T}$. A global critic network $\mathbf{D}^{G}$
learns the authenticity of this pose, incurring a pose-level Wasserstein GAN
loss as:

$\displaystyle\mathcal{L}_{\text{Pose}}=\mathop{\mathbb{E}}_{\mathbf{P}}\mathbf{D}^{G}[P_{t+1:t+T}]-\mathop{\mathbb{E}}_{\mathbf{P^{\prime}}}\mathbf{D}^{G}[P^{\prime}_{t+1:t+T}]$
(7)

where $\mathbf{P}$ denotes the real future pose sequence.

To ensure temporal smoothness, a consistency loss is incorporated as

$\displaystyle\mathcal{L}_{\text{Consistency}}=\sum_{i=1}^{T}{|\mathbf{P}^{\prime}_{t+i}-\mathbf{P}^{\prime}_{t+i-1}|^{2}}.$
(8)

The global or pose-level loss for the global GAN is given by a weighted sum of
the pose Wasserstein GAN loss and the consistency loss:

$\displaystyle\mathcal{L}_{\text{Global}}=\lambda\mathcal{L}_{\text{Pose}}+\mu\mathcal{L}_{\text{Consistency}}+\eta\mathcal{L}_{\text{GroundTruth}}$
(9)

$\mathcal{L}_{\text{GroundTruth}}$ is the difference between the ground truth
pose sequence $\mathbf{P}$ and its prediction $\mathbf{P^{\prime}}$.

Action transfer. The training for action transfer is challenging since there
are no public MoCap datasets involving transitions between different actions.
We can only train in an unsupervised way. As illustrated in Fig. 3, we perform
interpolation in the latent space to fill in the discontinuity between two
sequences. Specifically, let
$\mathbf{P}=(\mathbf{P}_{1},\cdots,\mathbf{P}_{t})$ be a sequence of action
$A_{1}$ and
$\mathbf{P}^{\prime}=(\mathbf{P}^{\prime}_{m+1},\cdots,\mathbf{P}^{\prime}_{m+T})$
a sequence of action $A_{2}$. We are to transfer from $\mathbf{P}$ to
$\mathbf{P}^{\prime}$. We may first train the VAE generator using large-scale
ground truth sequences of all actions, making it able to predict the immediate
future pose (not future sequence). Then, feed $\mathbf{P}$ and
$(\mathbf{P}^{\prime}_{m-t+1},\cdots,\mathbf{P}^{\prime}_{m})$ respectively to
the trained VAE network, obtaining their latent vectors $\mathbf{z}_{1}$ and
$\mathbf{z}_{2}$. We perform interpolation between $\mathbf{z}_{1}$ and
$\mathbf{z}_{2}$ to fill in the discontinuity between $\mathbf{P}_{t}$ to
$\mathbf{P}^{\prime}_{m+1}$. Given that $\mathbf{z}_{2}$ contains motion
contexts towards generating prediction $\mathbf{P}^{\prime}_{m+1}$, the
decoded interpolations will gradually transfer from $\mathbf{P}_{t}$ to
$\mathbf{P}^{\prime}_{m+1}$.

Time | 80 | 160 | 400 | 1000 | 80 | 160 | 400 | 1000 | 80 | 160 | 400 | 1000 | 80 | 160 | 400 | 1000
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
| Directions | Greeting | Phoning | Posing
HMR | 0.38 | 0.58 | 0.90 | 1.43 | 0.52 | 0.85 | 1.40 | 1.73 | 0.56 | 1.09 | 1.61 | 1.80 | 0.24 | 0.53 | 1.42 | 2.50
QNet | 0.36 | 0.57 | 0.90 | 1.42 | 0.52 | 0.81 | 1.38 | 1.73 | 0.58 | 1.12 | 1.60 | 1.82 | 0.26 | 0.56 | 1.41 | 2.56
Ours | 0.36 | 0.57 | 0.89 | 1.41 | 0.51 | 0.86 | 1.36 | 1.70 | 0.54 | 1.05 | 1.58 | 1.79 | 0.22 | 0.51 | 1.41 | 2.48
| Purchases | Sitting | Sittingdown | Takingphoto
HMR | 0.58 | 0.83 | 1.25 | 1.99 | 0.38 | 0.61 | 1.15 | 1.66 | 0.39 | 0.75 | 1.28 | 1.91 | 0.21 | 0.44 | 1.01 | 1.42
QNet | 0.59 | 0.83 | 1.24 | 2.37 | 0.37 | 0.58 | 1.11 | 1.62 | 0.37 | 0.73 | 1.23 | 1.94 | 0.24 | 0.48 | 1.02 | 1.43
Ours | 0.55 | 0.85 | 1.23 | 1.95 | 0.35 | 0.60 | 1.13 | 1.60 | 0.36 | 0.72 | 1.20 | 1.85 | 0.23 | 0.41 | 0.99 | 1.40
| Waiting | Walkingdog | Walkingtogether | Average
HMR | 0.27 | 0.59 | 1.33 | 2.21 | 0.55 | 0.87 | 1.36 | 1.84 | 0.26 | 0.50 | 0.71 | 1.21 | 0.39 | 0.69 | 1.46 | 1.79
QNet | 0.29 | 0.58 | 1.03 | 2.38 | 0.53 | 0.89 | 1.34 | 1.97 | 0.24 | 0.45 | 0.68 | 1.31 | 0.38 | 0.69 | 1.47 | 1.82
Ours | 0.23 | 0.56 | 1.29 | 2.15 | 0.53 | 0.85 | 1.33 | 1.80 | 0.22 | 0.45 | 0.73 | 1.19 | 0.37 | 0.67 | 1.43 | 1.75
Table 2: Performance evaluation (in MAE) over 11 remaining action types on the H3.6m dataset.The unit of time is ms. Time (ms) | 80 | 160 | 320 | 400 | 1000
---|---|---|---|---|---
| Walking
Remove Local Critic | 0.40 | 0.63 | 0.73 | 0.85 | 1.01
Remove Global Critic | 0.35 | 0.61 | 0.72 | 0.84 | 1.92
Remove sub-GANs | 0.90 | 1.10 | 1.61 | 1.79 | 2.01
AM-GAN (Complete) | 0.23 | 0.51 | 0.62 | 0.66 | 0.84
| Discussion
Remove Local Critic | 0.40 | 0.71 | 0.89 | 1.20 | 2.89
Remove Global Critic | 0.29 | 0.62 | 0.87 | 1.00 | 1.19
Remove sub-GANs | 2.20 | 2.51 | 2.68 | 2.71 | 3.20
AM-GAN (Complete) | 0.28 | 0.55 | 0.81 | 0.92 | 1.58
Table 3: Ablation studies on different components of our AM-GAN evaluated over
‘Walking’ and ‘Discussion’. Figure 4: Some visual results. Figure 5: Action
transfer examples. Figure 6: Fine-grained control over different chains in the
predicted motion. In the left figure, we predict a ‘walking’ action (purple)
with the right leg following the action of ‘eating’ (red). The result is shown
on the bottom row with matching colors. In the right figure, we predict a
’posing’ action with the left hand following the ‘direction’ action.

Fine-grained pose control. Breaking down the motion generation to chain level
representations allows us to naturally incorporate fine-grained manipulation
of each chain. Specifically, assuming that we are to generate the future
sequence of an observed ‘walking’ sequence P and wants to control the right
arm to act like ‘direction’. We can train the proposed network for each action
independently. Then, we simply feed P into the trained ‘walking’ network to
generate future ‘walking’ action for the five chains. Similarly, we feed P
into the trained ‘direction’ network to generate future ‘direction’ action for
the right arm chain. We replace the right arm chain generated by the ‘walking’
network with that of the ‘direction’ network, and feed the new five chains to
the aggregation layer to generate a natural final pose sequence.

## Experiments

### Dataset and Preprocessing

In order to verify the ability of our model, we selected the largest and
widely used data set: H3.6m (Ionescu et al. 2014) that contains 3.6 million
human images with 3D poses (comprising 25 distinct skeletal joints) obtained
via the Vicon mocap system. There are 15 different action classes performed by
7 subjects. Following the protocol of previous methods (Fragkiadaki et al.
2015; Jain et al. 2016), subject 5 is used for testing while the data for the
other 6 subjects are used for training. The observed sequence has 25 frames
and the predicted output sequence is also set to 25 frames.

### Implementation Details

We use Tensorflow to implement our method. For each of our sub-GAN, we have
adopted a Variational Auto-encoder (VAE) architecture (Kingma et al. 2014) as
the generator module. The encoder and decoder of VAE consist of a GRU with
1,024 hidden units. The training of the GANs follows that of Wasserstein GANs
(Gulrajani et al. 2017) but with our customized loss function outlined in the
Loss Function section. The aggregation module is a full connected layer with
1,024 units. The local critic is a three-layer fully connected feedforward
network with 512 units, while the global critic possesses 1,024 units. The
Adam Optimizer (Kingma and Ba 2014) is employed with initial learning 5e-5 and
the batch size is set to 16.

### Quantitative Evaluation on H3.6M Dataset

We benchmark our method against previous methods, including ERD (Fragkiadaki
et al. 2015), SRNN (Jain et al. 2016), RRNN (Martinez, Black, and Romero
2017), LSTM-AE (Tu et al. 2018), HP-GAN (Barsoum, Kender, and Liu 2018), CEM
(Li et al. 2018), BiHMP-GAN (Kundu, Gor, and Babu 2019), SKel-Net (Guo and
Choi 2019), HMR (Liu et al. 2019) and QuaterNet (Pavllo et al. 2019).
Following previous work, the mean angle error (MAE) is adopted as the
evaluation metric. We report the results in Table 1 over four actions
‘Walking’, ‘Eating’, ‘Smoking’, ‘Discussion’.

As shown in Table 1, our method generally achieves superior performance in
quantitative accuracy over the state-of-the-arts. One case in point is that
the actions have different levels of complexity. For example, walking is a
more regular action and demonstrates a high degree of periodicity whereas
discussion is more complex and basically aperiodic. As would be expected,
prediction accuracy falls with complexity of the action, which is observed
throughout the empirical results. One noteworthy remark is that our method
demonstrates significant improvements for the more complex action types such
as discussion. This can be attributed to the fact that by modeling body parts
and skeletal kinematic chains separately, our AM-GAN approach is less affected
by lack of regularity and periodicity in the overall motion pattern.
Furthermore, comparing against the other two GAN frameworks HP-GAN and BiHMP-
GAN, the superior performance of our AM-GAN for both short-term and long-term
prediction is also an indicator to the effectiveness of the divide and conquer
approach in modeling local body parts and merging them at the final stage.

For the sake of completeness, we benchmark our method over the remaining 11
H3.6m actions against the two most recent and state-of-the-art works HMR and
QuaterNet(QNet). The comparison results are shown in Table 2. Obviously,our
method achieves new state-of-the-art performance quite consistently for short
term and long term predictions. As shown in Fig. 4, we also visualize the
results of our method against current excellent HMR, observing that our method
delivers more natural and smooth sequences.

### Action Transfer

We now consider scenarios where we control the specific action performed in
the future. Some generated sequences depicting transitioning back and forth
between different actions are showcased in Fig. 5, where the middle six frames
are generated transition frames. Empirical results show that our method is
able to smoothly and authentically transfer from one action to another.

### Human Pose Control

We can assimilate high level controls over specific body parts with AM-GAN.
The local and global level generation can be seamlessly combined to create
natural and consistent motion sequences. As an example, we illustrate in the
left figure of Fig. 6 on how we may control the movements of the right leg in
the predicted sequence. Specifically, given a walking sequence, we generate
its future ‘walking’ sequence but enforce the right leg act like ‘eating’
action prediction. Similarly, in the right figure of Fig.6, we show another
example where we predict the future ’posing’ action sequence but command the
left hand to follow a ‘direction’ action.

### Ablation Studies

We quantitatively analyze the contribution of various design components,
including removing the 5 sub-GANs, removing the local critic and global
critic. The empirical results are reported in Table 3. One crucial finding is
that incorporating sub-GANs and a divide and conquer strategy indeed improves
over the baseline with a single GAN for modeling the motion generation.
Modeling the constituent chains instead of the entire pose directly reduces
complexity, thus improving the short-term and long-term prediction
capabilities. Furthermore, it may be observed that the improvement over the
baseline is more significant in the case of aperiodic actions such as
‘Discussion’. This is likely due to the fact that zooming in on individual
body parts is more effective for modeling complex and irregular motion, where
different body parts usually engage differently in the motion. Another
observation is that the local critic enhances the short-term prediction
whereas the global critic elevates long-term prediction performance.

## Conclusion

In this paper, we propose a novel human motion prediction problem to
incorporate control. In order to reduce the complexity of optimizing over the
entire human pose manifold and to achieve fine-grained control, we employed a
divide and conquer strategy and modeled each kinematic chain separately with
our AM-GAN. On top of improving the state-of-the-art for both short and long-
term predictions, AM-GAN also allows for controlling of the prediction
spectrum. This includes generating transitions to a desired action or fine-
grained control over specific body parts. We are positive that controllable
motion generation would open many doors in adjacent fields.

## Acknowledgements

This work was partly supported by the National Key Research and Development
Program of China under No. 2020AAA0140004. The Natural Science Foundation of
Zhejiang Province, China No. LQ19F020001. The National Natural Science
Foundation of China No. 61902348 and No. 61772466. The Zhejiang Provincial
Natural Science Foundation for Distinguished Young Scholars No. LR19F020003,
and the Fundamental Research Funds for the Central Universities (Zhejiang
University NGICS Platform).

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# An array-oriented Python interface for FastJet

Aryan Roy1 Jim Pivarski2 Chad Wells Freer3 1 Manipal Institute of Technology,
Manipal, India  2 Princeton University, Princeton NJ, USA  3 Massachusetts
Institute of Technology, Boston MA, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Analysis on HEP data is an iterative process in which the results of one step
often inform the next. In an exploratory analysis, it is common to perform one
computation on a collection of events, then view the results (often with
histograms) to decide what to try next. Awkward Array is a Scikit-HEP Python
package that enables data analysis with array-at-a-time operations to
implement cuts as slices, combinatorics as composable functions, etc. However,
most C++ HEP libraries, such as FastJet, have an imperative, one-particle-at-
a-time interface, which would be inefficient in Python and goes against the
grain of the array-at-a-time logic of scientific Python. Therefore, we
developed fastjet, a pip-installable Python package that provides FastJet C++
binaries, the classic (particle-at-a-time) Python interface, and the new
array-oriented interface for use with Awkward Array. The new interface
streamlines interoperability with scientific Python software beyond HEP, such
as machine learning. In one case, adopting this library along with other
array-oriented tools accelerated HEP analysis code by a factor of 20. It was
designed to be easily integrated with libraries in the Scikit-HEP ecosystem,
including Uproot (file I/O), hist (histogramming), Vector (Lorentz vectors),
and Coffea (high-level glue). We discuss the design of the fastjet Python
library, integrating the classic interface with the array oriented interface
and with the Vector library for Lorentz vector operations. The new interface
was developed as open source.

##  1 Introduction

Jets, an important signal in High Energy Physics (HEP) collisions, are
collimated bunches of particles produced by the hadronization of a quark or
gluon. They are measured in particle detectors and studied in order to
determine the properties of the original quarks. Many jet-finding algorithms
were developed independently since 1977, until the FastJet package [1] was
introduced in 2006, containing fast implementations of the most commonly used
algorithms. All jet-finding algorithms take a collection of particle 4-momenta
(“pseudojets”) and return a collection of jet 4-momenta for each collision
event.

The mostly-C++ FastJet package includes a Python interface, which addresses
each particle of each event as a separate Python object. Loops over Python
objects are considerably slower than loops over C++ objects, so many
scientific libraries in Python opt for an “array-oriented” approach, in which
large datasets of numerical arrays are referenced by a single Python object
and operations on whole arrays are initiated by individual Python function
calls. The pyjet package [2] in Scikit-HEP [3] has a partially array-oriented
interface to FastJet: it refers to each event as a NumPy array of particles
(with 4 momentum components each). NumPy cannot describe collections of
events, since each event has a different number of input particles and returns
different numbers of output jets. Awkward Array [4], an extension of NumPy,
can represent arrays of varying numbers of particles, and hence it can provide
an interface to FastJet that operates on collections of events in a single
function call: see Figure 1.

In this work, we present a new Python package for FastJet called fastjet
(lowercase) with a fully array-oriented interface through Awkward Array. It
will be replacing the pyjet package in Scikit-HEP since pyjet diverged from
the original FastJet interface and implements only a subset of its algorithms.
This new fastjet package contains the original FastJet bindings, making them
available on PyPI (the Python Package Index) so that they can be installed
with pip for the first time. From a user’s perspective, fastjet is FastJet in
Python, with the original particle-oriented interface, which we call the
“classic” interface, and the new array-oriented interface in the same
namespace. The array-oriented interface uses the same idioms as the classic
interface wherever possible, but replaces lists of lists of PseudoJet objects
with efficient Awkward Arrays of Vectors for high-volume data.

Figure 1: Degree of vectorization in FastJet’s classic interface (left), in
which every particle is represented by a Python object, pyjet (middle), in
which every event is a Python object, and Awkward Array (right), in which an
entire dataset may be in a single Python object.

##  2 Features required for the new fastjet package

The fastjet package was designed with the following constraints:

  * •

It must handle deeply nested, variable length lists in an efficient way.

  * •

It must perform Lorentz vector calculations on the array elements, including
coordinate system transformations.

  * •

It must be pip-installable with as many of the FastJet plugins enabled as
possible.

##  3 The Scikit-HEP ecosystem

The requirements outlined in the previous section can be satisfied by
leveraging other packages in the Scikit-HEP ecosystem. The following are
directly or indirectly used by fastjet.

###  3.1 Awkward Array

Awkward Array represents and manipulates arrays of data with complex data
types, including the lists (and lists of lists) of 4-component momentum
records that are the inputs and outputs of jet-finding (including lists of
constituents for each jet). Awkward Array can also implement cuts and operate
on combinations of particles (“combinatorics”), and it is used for general
data analysis by a community of physicists, particularly through the Coffea
framework [5]. Building a jet-finding interface on Awkward Array gets these
benefits for free.

The new fastjet package uses Awkward Arrays in the following form (typically
with larger numbers of particles) to represent a collection of events. Non-
kinematic fields, like “charge,” are not required for jet-finding, but do not
need to be excluded from the particle records.

⬇ array = ak.Array([  [  {”px”: 1.2, ”py”: 3.2, ”pz”:5.4, ”E”: 2.5, ”charge”:
1},  {”px”: 32.2, ”py”: 64.21, ”pz”: 543.34, ”E”: 24.12, ”charge”: -1},
{”px”: 32.45, ”py”: 63.21, ”pz”: 543.14, ”E”: 24.56, ”charge”: 1},  ],  [], #
empty event [  {”px”: 2.95, ”py”: -0.35, ”pz”: 0.62, ”E”: 2.86, ”charge”: 1},
{”px”: 4.33, ”py”: 0.53, ”pz”: 1.47, ”E”: 3.95, ”charge”: -1},  {”px”: 0.32,
”py”: 0.06, ”pz”: 0.12, ”E”: 0.21, ”charge”: 1},  {”px”: 0.32, ”py”: 0.01579,
”pz”: 0.01, ”E”: 0.32, ”charge”: 1},  ],  [  {”px”: 9.74, ”py”: -0.01, ”pz”:
0.23, ”E”: 9.73, ”charge”: -1},  ],  ])  (1)

###  3.2 Vector

Vector [6] is a library for array-oriented operations on 2D, 3D and Lorentz
vectors. The new fastjet library uses Vector to perform coordinate
transformations, calculate angles between vectors, $\Delta R$, etc. Since our
arrays of particles bypass FastJet’s PseudoJet class, Vector is needed to
replace the functionality that PseudoJet’s methods would ordinarily be used
for.

##  4 The particle-oriented interface

Clustering many events in one function call reduces the amount of code users
need to write, thereby improving the ergonomics of the analysis workflow. A
typical analysis using the classic FastJet interface with the `array} variable
defined in Equation~(\ref`eqn:array) would look like this:

⬇ jetdef = fastjet.JetDefinition(fastjet.antikt_algorithm, 0.6)  for
original_event in array:  fastjet_event = []  for p in original_event:
fastjet_event.append(  fastjet.PseudoJet(p[”px”], p[”py”], p[”pz”], p[”E”])  )
cluster = fastjet.ClusterSequence(fastjet_event, jetdef)  jets =
cluster.inclusive_jets()  (2)

But the same analysis in the array-oriented interface looks like this:

⬇ jetdef = fastjet.JetDefinition(fastjet.antikt_algorithm, 0.6)  cluster =
fastjet.ClusterSequence(array, jetdef)  jets = cluster.inclusive_jets()  (3)

The classic interface requires explicit loops over particle data, while the
array-oriented interface achieves the same result with considerably less code.
Since the loops are now in the C++ code invoked by the ClusterSequence
constructor, it is also much faster.

Figure 2: The internal structure of fastjet; each box represents a distinct
part of the code and arrows represent dependencies between them.

##  5 The internal structure

The array-oriented interface is implemented with new Python-C++ bindings,
written with pybind11 [7]. The classic interface is implemented with SWIG [8],
another (much older) tool for interfacing Python and C++. Both of these
bindings are included in the fastjet package so that users can access either
one. This two-bindings architecture is depicted in Figure 2. At its core,
FastJet is a C++ library with a C++ interface. The Python bindings that are
included with it are a direct mirror of the C++ interface: every C++ class and
method has a corresponding Python class and method. This allows for easy
porting between C++ and Python and requires only one suite of documentation,
though the resulting Python code is slower and more verbose. Since the fastjet
package includes FastJet shared object libraries, it also includes these
original Python bindings and makes them visible in the public namespace. The
array-oriented bindings directly connect new Python functions to C++, without
going through the original bindings. Awkward Arrays can have a variety of
structures, and jet-finding should be possible for any structure that at least
has a collection of records with 4-momentum attributes (in any coordinate
system Vector supports). The most basic case is “array of lists (events) of
particle records.” Other cases can be implemented in terms of this one, such
as an “array of particle records (representing one event),” which can be
viewed as a basic case of length 1. Also, general data structures that
eventually contain “lists (events) of particle records,” possibly within other
records and lists, can be implemented from the basic case while preserving the
structures it is nested within. A tree containing arrays of the right form is
spliced to provide the right output. Jet-finding only needs 4 kinematic
variables, $p_{x}$, $p_{y}$, $p_{z}$, $E$ in Cartesian coordinates. Any other
fields are allowed and passed through fastjet functions that return the
original constituents, without interpretation. Including extra fields, such as
the “charge” in Equation 1, is equivalent to setting up user data in FastJet’s
PseudoJet class, though the latter relies on

    void*} pointer handling. The array-oriented view provides this necessary feature in a natural, less rigid way.
    To adhere to the original interface as much as possible, arguments that do not need to be arrays accept the same argument types as the corresponding classic functions, such as JetDefinition in Equation~\ref

eqn:array-oriented. SWIG objects can supply their C++ pointer values through
Python and pybind11 can call Python, so code written with pybind11 can
directly access the C++ objects that SWIG wraps: the two-bindings architecture
does not prevent interoperability.

##  6 Case Study

The new fastjet package is an essential part of a search for soft unclustered
energy patterns (SUEPs) in the CMS detector at the LHC. This search is
characterized by a large number of low-pT tracks [9]. As such, non-standard
jet-finding features prominently in this analysis. The analysis was originally
performed using pyjet in the NanoAODTools framework [10], but was recently
rewritten with Awkward Array for data manipulation, Vector for Lorentz
kinematics, Coffea for scale-out, and fastjet for jet-clustering. The new
workflow can analyze 50 thousand events in under a minute, as opposed to 17
minutes in the original workflow, while producing exactly the same results.
The throughput also increased from 40 Hz to 1000 Hz. Used in conjunction with
the other Scikit-HEP packages, fastjet simplifies the user experience by not
having to switch from an array-oriented perspective to a particle-oriented
perspective just for jet-finding, and this encourages the use of higher-
performance tools.

##  7 Conclusion

In this paper, we presented the design choices and performance indicators of a
new array-oriented interface for jet-finding with FastJet. The new package is
pip-installable and seemlessly integrates with other Scikit-HEP packages such
as Awkward Array and Vector. The array-oriented design results in improved
performance when dealing with large sets of events, such as the factor of 20
observed in the search for SUEPs at the LHC. Representing kinematic data in
Awkward record arrays, which are less rigid than C++ PseudoJet instances,
allows user data to be attached in a natural way. This package brings the new
paradigm of array-oriented analysis to jet clustering, allowing users to think
about these workflows in a new way. Making temporary arrays of jets clustered
with different settings becomes very easy in an array-based interface, and
fastjet’s connections to other libraries in the Scikit-HEP ecosystem opens the
door to a wider set of possibilities.

##  8 Acknowledgements

We would like to thank Matteo Cacciari, Gavin Salam, Gregory Soyez, Salvatore
Rappoccio, Patrick Komiske, and Eduardo Rodrigues for helpful discussions.
This work was supported by the National Science Foundation under Cooperative
Agreement OAC-1836650 (IRIS-HEP). moreauthor morelabelname moreauthor
morelabelname

## References

  * [1] Matteo Cacciari, Gavin P Salam and Gregory Soyez  “FastJet user manual”  In _The European Physical Journal C_ 72.3 Springer, 2012, pp. 1–54
  * [2] Noel Dawe et al.  “pyjet”  Zenodo, 2018  DOI: 10.5281/zenodo.1197493
  * [3] Rodrigues, Eduardo  “The Scikit-HEP Project”  In _EPJ Web Conf._ 214, 2019, pp. 06005  DOI: 10.1051/epjconf/201921406005
  * [4] Jim Pivarski et al.  “Awkward Array”, 2018  DOI: 10.5281/zenodo.4341376
  * [5] Lindsey Gray and Nicholas Smith  “Coffea”  Zenodo, 2019  DOI: 10.5281/zenodo.3266454
  * [6] Henry Schreiner and Jim Pivarski  “Vector”  Zenodo, 2021  DOI: 10.5281/zenodo.5942082
  * [7] Wenzel Jakob, Jason Rhinelander and Dean Moldovan  “pybind11 – Seamless operability between C++11 and Python” https://github.com/pybind/pybind11, 2017
  * [8] David M. Beazley  “SWIG: An Easy to Use Tool for Integrating Scripting Languages with C and C++”  In _Proceedings of the 4th Conference on USENIX Tcl/Tk Workshop, 1996 - Volume 4_ , TCLTK’96  Monterey, California: USENIX Association, 1996, pp. 15
  * [9] Simon Knapen, Simone Pagan Griso, Michele Papucci and Dean J. Robinson  “Triggering Soft Bombs at the LHC”  In _JHEP_ 08, 2017, pp. 076  DOI: 10.1007/JHEP08(2017)076
  * [10] Andrea Rizzi, Giovanni Petrucciani and Marco Peruzzi  “A further reduction in CMS event data for analysis: the NANOAOD format”  In _EPJ Web Conf._ 214, 2019, pp. 06021  DOI: 10.1051/epjconf/201921406021


Leading all-loop quantum contribution to the effective potential in the
inflationary
cosmology

D.I. Kazakov1,2, R. M. Iakhibbaev1 and D.M. Tolkachev1,3

1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Russia, 141980
2Moscow Institute of Physics and Technology, Dolgoprudny, Russia, 141701
3Stepanov Institute of Physics, Minsk, Belarus, 220072

###### Abstract

In this paper, we have constructed quantum effective potentials and used them
to study slow-roll inflationary cosmology. We derived the generalised RG
equation for the effective potential in the leading logarithmic approximation
and applied it to evaluate the potentials of the $T^{2}$ and $T^{4}$-models,
which are often used in modern models of slow-roll inflation. We found that
while the one-loop correction strongly affects the potential, breaking its
original symmetry, the contribution of higher loops smoothes the behaviour of
the potential. However, unlike the $\phi^{4}$-case, we found that the
effective potentials preserve spontaneous symmetry breaking when summing all
the leading corrections. We calculated the spectral indices $n_{s}$ and $r$
for the effective potentials of both models and found that they are consistent
with the observational data for a wide range of parameters of the models.

## Introduction

The inflationary model based on the assumption of the existence of an
accelerated expansion stage in the early Universe, became one of the
foundations of modern cosmology [1, 2]. This area is rapidly developing
because it explains the observed flatness, homogeneity and isotropy of the
observable Universe. The advantage of the inflationary theory is also related
to its successful explanation of the peculiarities of the CMB spectrum [3].

The most common way to realise the accelerated expansion is to use a scalar
field (inflaton or inflationary field) with self-interaction and to consider
the solution of the field equations in slow-roll regime. Obtaining an inflaton
potential is a rather difficult problem, since at present the observational
data give a wide scope for consideration of a large number of potentials [4].
Recently, $\alpha$-attractor type models in inflationary cosmology have
aroused considerable interest, mainly because they are justified in
supergravity and satisfy observational data. [5, 6, 7]. Therefore, their (or
their possible modifications) detailed study in the context of inflation is
quite an interesting task [6].

In quantum theory, the classical inflaton potential acquires quantum loop
corrections. Such corrections, first considered in the paper by Coleman and
Weinberg [8], can change the form of the potential and lead to the appearance
of a new vacuum, which means spontaneous symmetry breaking takes place. The
inflationary model based on the $\phi^{4}$ theory with a one-loop quantum
correction was considered in [9]. Within the framework of this model, the old
theory of inflation with tunneling [1] and then the new inflation in models
with slow rolling [9] was formulated. Note, however, that taking into account
only one-loop corrections is not always justified. As was shown in the
original work [8], summing the leading terms with the renormalization group
significantly changes the behavior of the potential. Hence, the construction
of an effective potential that takes into account all leading corrections is a
current task.

However, the all-loop effective potentials are not often used in cosmology.
Existing literature (see e.g.[10, 11]) deals with effective potentials in
renormalisable cases when the formalism of the usual renormalisation group
formalism is applicable. At the same time, the most popular cosmological
potentials from the point of view of quantum field theory, are non-
renormalizable. This leads to all the problems typical of non - renormalizable
theories such as infinite arbitrariness in normalisation of counter terms,
absence of standard renormalization group (RG) equations for summation of
leading corrections, scheme dependence of results of calculations, etc.
However, if one considers only the leading quantum corrections (the leading
logarithmic approximation), these problems can be avoided. Indeed, the leading
divergences (they are also the leading orders of $\log\phi^{2}$) are
universal, they do not depend on the arbitrariness of the subtraction
procedure (they are not scheme-dependent). Moreover, this property is
independent of the renormalizability or non-renormalizability of the theory.
Thus, one can try to calculate the main quantum corrections to the effective
potential and apply the obtained expression for the description of inflation,
ignoring the main problems of non-renormalizable theories so far.

In the present paper we take into account the leading quantum corrections to
the classical potential. Considering an arbitrary scalar potential, we derive
a generalised RG equation which sums all leading logarithmic corrections to
the potential. This way, we derive an effective potential and based on it we
analyse the cosmological parameters that are obtained from this modified
potential. In the following, we consider the potential for the so-called
$T$-model [7], which can be represented as

$V=\tanh^{2n}(\phi/(\sqrt{6\alpha M_{Pl}^{2}})),$ (1)

where $\phi$ is the inflaton field, $M_{Pl}$ is Planck’s mass, but $\alpha$ is
a free parameter, $n=1,2$. A qualitative analysis of the behaviour of quantum
corrections to such potentials was carried out in [12], and in this paper we
will be able to check their conclusions about flat asymptotic behaviour of the
one-loop corrected potential.

Based on the obtained numerical solutions of the RG equation, effective
potentials and cosmological parameters, we can understand at under parameters
the effective potential satisfies the observational data.

##  1 Single-field model of slow-roll inflationary scenario: setup and
observables

The starting point in inflationary cosmology with a single field is the
scalar-tensor theory, with the action that can be presented as [4, 13]:

$S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{pl}^{2}}{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right],$
(2)

where $R$ is a scalar curvature, $M_{Pl}=8\pi G$ is Planck’s mass, $\phi$ is
the inflation scalar field, and $V(\phi)$ is self-interaction term. In the
Friedman-Lemetre-Robertson-Walker metric, the equations of motion for the
above model can be written in the form [14]

$3M_{pl}^{2}H^{2}=\frac{\dot{\phi}^{2}}{2}+V,$ (3)
$-2M_{pl}^{2}\dot{H}=\phi^{2},$ (4) $\ddot{\phi}+3H\dot{\phi}+\frac{\partial
V}{\partial\phi}=0.$ (5)

Here $H=\dot{a}/a$ is Hubble’s constant and $a(t)$ is a scale factor.

To describe the evolution of the background, it is useful to introduce
cosmological parameters of the Hubble flow:

$\epsilon_{n}=\frac{d\log\epsilon_{n-1}}{dN_{e}},$ (6)

here $\epsilon_{0}=H_{0}/H$, with $H_{0}$ being initial Hubble’s constant,
$N_{e}$ is the ratio of the scale factor at the end of inflation to the factor
at its beginning: $N_{e}=\ln(a/a_{0})$ (also $N_{e}$ is called the number of
$e$-foldings). Cosmological parameters are responsible for the nature of
inflation and reflect the shape of the inflation potential. Inflation is an
accelerated expansion, and the condition for such an expansion can be written
as $\epsilon_{1}<1$, and the end of inflation corresponds to $\epsilon_{1}=1$.
Under the condition of correctness of the slow roll approximation, we can
consider that $\epsilon_{n}\ll 1$, so that the following estimation of the
cosmological parameter becomes valid:

$\epsilon_{1}\simeq\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}=\frac{1}{2M_{pl}}\left(\frac{d\phi}{dN_{e}}\right)^{2}.$
(7)

The condition $\epsilon=1$ with the help of (7) can be rewritten in terms of
the Klein-Gordon-Fock equation [4]:

$\frac{1}{M_{pl}}\left(\frac{d\phi}{dN_{e}}\right)^{2}=\frac{d\ln V}{dN_{e}}.$
(8)

This equation can be integrated to give an explicit expression for the number
of $e$-folds [4]:

$N_{e}-N_{0}=-\frac{1}{M_{pl}^{2}}\int_{\phi_{0}}^{\phi}\frac{V(x)}{V_{x}(x)}dx.$
(9)

Based on observational evidence, we can also say that the lower limit for
expansion of the Universe should be a number of 50 to 60 $e$-folds. The
observable characteristics of inflation are the spectral indices: CMB tilt of
scalar perturbations [7, 4]

$n_{s}=1-2\epsilon_{1}+\epsilon_{2},$ (10)

the tilt of tensor perturbation [4]:

$n_{t}=-2\epsilon_{1}.$ (11)

The CMB tensor-to-scalar ratio is [4]:

$r=16\epsilon_{1}.$ (12)

According to the relic radiation data obtained from the BICEP/PLANCK [15, 16],
the observed values for these quantities are as follows:

$r<0.036,\leavevmode\nobreak\ n_{s}=0.9649\pm 0.0042.$ (13)

The $n_{s}$-$r$ plot generally allows one to discriminate models and compare
them with observational data.

In this paper we consider the T-model potential, which we will rewrite as
follows [17, 6]

$V(\phi)=g\tanh^{2n}\left(\phi\omega/\sqrt{6\alpha}\right),$ (14)

where $\omega=M_{Pl}^{-1}$, $g$ is an appropriate inflation scale, and
$\alpha$ we put equal to unity. In the following, we will restrict ourselves
to two cases of T-model: $n=1$ (for convenience, we will call it the
$T^{2}$-model) and $n=2$ ($T^{4}$-model). According to the above (8), above
for cosmological parameters, one can easily find the exact analytical formula
for the dependence of $\phi$ on $N_{e}$ for T-models [4]:

$\phi^{(n)}(N_{e})={\frac{\sqrt{6}}{2\omega}}\operatorname{arccosh}\left(\sqrt{1+4/3n^{2}}+4/3\leavevmode\nobreak\
n\Delta N\right),$ (15)

where $\Delta N=N_{f}-N_{e}$, and $N_{f}$ can be found from the inflation
ending condition $\epsilon_{1}=1$. Due to the simple form of the $\phi(N)$
dependence the corresponding spectral parameters of the Hubble flow can be
obtained. Thus, the T-model is quite convenient, since all parameters in it
are represented in analytical form. In a general form, the functions and
spectral parameters for T-models in arbitrary form are presented in ref. [4].

##  2 Effective potential in scalar field theory with
arbitrary potential

The effective potential is defined as part of the effective action without
derivatives. The direct way to find the effective potential $V_{eff}(\phi)$ by
perturbation theory is to calculate the sum of one-particle irreducible vacuum
diagrams obtained using Feynman’s rules derived from the shifted action
$S[\phi+\widehat{\phi}]$, where $\phi$ is the classical field obeying the
equation of motion and $\widehat{\phi}(x)$ is the quantum field over which
integration is performed [18]. This means that one has to consider the 1PI
vacuum diagrams with propagators containing an infinite number of insertions
$v_{2}(\phi)\equiv\frac{d^{2}V(\phi)}{d\phi^{2}}$, which act like a mass
depending on the field $\phi$: $m^{2}(\phi)=gv_{2}(\phi)$. The vertices are
also obtained from the expansion of the potential $V(\phi+\widehat{\phi})$ by
the quantum field $\widehat{\phi}$. After that, the effective potential is
constructed as a perturbation expansion by the coupling constant $g$

$V_{eff}=g\sum_{n=0}^{\infty}(-g)^{n}V_{n},$ (16)

where $V_{0}=V$ is the initial classical potential.

We choose a dimensional regularisation to control UV divergence in loop
integrals, taking the dimension of spacetime $D=4-2\epsilon$. Proceeding in
this way, in the one-loop case, we obtain the quantum correction [19]:

$V_{1}=\frac{1}{16\pi^{2}}\frac{1}{4}\frac{v_{2}^{2}}{\epsilon}\left(\frac{\mu^{2}}{m^{2}}\right)^{\epsilon}\to\frac{1}{16\pi^{2}}\frac{v_{2}^{2}}{4}\left(\frac{1}{\epsilon}+\log\frac{\mu^{2}}{m^{2}}\right),\
\ \ m^{2}=gv_{2}(\phi).$ (17)

Further, the singular part $\sim 1/\epsilon$ is removed by introducing of the
UV counterterms, and the finite part $\sim\log(gv_{2}(\phi))$ gives the
contribution to the effective potential. Note that the coefficient of the pole
$1/\epsilon$ exactly coincides with the coefficient of the logarithm. This
property is also preserved in higher loops: the coefficient of the leading
pole $1/\epsilon^{n}$ coincides with the coefficient of the leading logarithm
$\log^{n}(gv_{2}(\phi))$. Hence, the task is to find the coefficients of the
leading poles, which, in turn, due to the special features of the ${\cal
R}$-operation is determined by one-loop diagrams [19].

The next step is to obtain recurrence relations connecting the leading
divergences in subsequent loops. They obviously follow from the local
structure of $\mathcal{R}$-operations [19]. Denoting the singular part of the
effective potential (coefficient at the leading pole $1/\epsilon^{n}$) in the
n-th order of perturbation theory by $\Delta V_{n}$, one can obtain the
following recurrence relation [19]

$n\Delta V_{n}=\frac{1}{4}\sum_{k=0}^{n-1}D_{2}\Delta V_{k}D_{2}\Delta
V_{n-1-k},\ \ \,n\geq 1,\ \ \Delta V_{0}=V_{0},$ (18)

where $D_{2}$ is the second derivative by the field $\phi$. This recurrence
relation allows one to compute all leading divergences $\Delta V_{n}$ for an
arbitrary potential $V$ in an algebraic way without computing diagrams.

To sum up the leading divergences (or, what is the same, the leading
logarithms in the finite part), it is convenient to pass to the differential
equation for the sum of series

$\Sigma(z,\phi)=\sum_{n=0}^{\infty}(-z)^{n}\Delta V_{n}(\phi),$ (19)

where $z=g/\epsilon$. Indeed, multiplying Eq.(18) by the factor $(-z)^{n}$ and
summing over n from $n=2$ to $\infty$, we obtain the differential equation for
the function $\Sigma(z,\phi)$,

$\frac{d\Sigma}{dz}=-\frac{1}{4}(D_{2}\Sigma)^{2},\ \
\Sigma(0,\phi)=V_{0}(\phi).$ (20)

This is the desired generalised RG equation for the effective potential in the
approximation of leading logarithms. For renormalizable, interaction it
reduces to the usual RG equation [19]. To receive an effective potential in
the solution of this equation, one should replace the pole by $\epsilon$ with
the corresponding logarithm:

$V_{eff}(g,\phi)=\Sigma(z,\phi)|_{z\to-\frac{g}{16\pi^{2}}\log{gv_{2}/\mu^{2}}}.$
(21)

Recall that here $v_{2}(\phi)\equiv\frac{d^{2}V_{0}(\phi)}{d\phi^{2}}$.

Note that Eq.(20) is a partial derivative equation, and the function
$\Sigma(z,\phi)$ depends on two variables: $z$ and $\phi$. In general, it is
usually more convenient to reduce this function to dimensionless variables. In
some special cases, (20) can be reduced to an ordinary differential equation,
but still one has to use numerical methods to solve it [19].

##  3 Inflation and effective potentials

###  3.1 $T^{2}$-model

Consider the theory with the potential (14) for $n=1$, which corresponds to
the $T^{2}$-model:

$V=g\tanh^{2}\left(\frac{\phi\omega}{\sqrt{6}}\right).$ (22)

Further, to simplify equation (20) it is convenient to represent the function
$\Sigma$ in dimensionless variables $x=z\omega^{4}$ and
$y=\tanh^{2}(\phi\omega/\sqrt{6})$, because the loop decomposition of this
function can be represented as polynomials on tangents. Then

$\Sigma(z\omega^{4},\tanh^{2}(\phi\omega/\sqrt{6}))\equiv S(x,y).$

The obtained function is in some way an analogue of an arbitrary function
$F(\tanh(\frac{\phi}{\sqrt{6}}))$ in the theory of conformal chaotic inflation
[7]. But in our case this function is restricted by the RG equation (20) and
it contains an additional regularisation parameter $\mu^{2}$.

Renaming the variables and functions in the original equation (20), the
generalised RG-equation for such a potential can be written as:

$\frac{\partial}{\partial
x}S(x,y)=-\frac{1}{4}\left(\omega^{2}\frac{\partial^{2}}{\partial\phi^{2}}S(x,y(\phi))\right)^{2}.$
(23)

Such change of variables considerably simplifies the form of the equation, so
that for the $T^{2}$-model the generalised RG-equation, after the
transformation from the derivative over $\phi$ to the derivative over $y$, has
the following form:

$\frac{1}{36}(y-1)^{2}\left((1-3y)S_{y}-2(y-1)yS_{yy}\right)^{2}=-S_{x}.$ (24)

For brevity, the lower indices are introduced here to denote the corresponding
derivative of the function $S$. The initial and boundary conditions for the
equation can be defined as:

$S(0,y)=y,\leavevmode\nobreak\ S(x,1)=1,\leavevmode\nobreak\ S_{x}(x,1)=0,$
(25)

because the potential in the limit $g=0$ must coincide with the classical
potential, as stated in (20) and as $\phi\rightarrow\pm\infty$ must reach a
constant. The exact solution of Eq. (24) cannot be represented analytically:
it is a complex two-dimensional surface, and to find an effective potential,
we have to perform a substitution
$z\to-\frac{g}{16\pi^{2}}\log{gv_{2}/\mu^{2}}$, and in the case of the
$T^{2}$-model [19]

$v_{2}=-\frac{1}{3}\omega^{2}\left(\cosh\left(\sqrt{\frac{2}{3}}\omega\phi\right)-2\right)\text{sech}^{4}\left(\frac{\omega\phi}{\sqrt{6}}\right).$
(26)

That is, this substitution is equivalent to the choice of a particular curve
on a two-dimensional surface given by the solution of Eq. (24). The numerical
solution of the equation can be carried out using Euler’s backward
differentiation method [20] or built-in methods in Wolfram Mathematica.

The initial potential has flat asymptotics, so it is often used in models of
chaotic inflation, as it was indicated earlier due to the attractor properties
of solutions (solutions of Friedmann equations do not depend on initial
conditions and shrink into a stable limit cycle in the phase space). The
solutions of the generalised RG-equation have the same asymptotics but can
modify the behaviour of the potential in inflationary cosmological models
leading to different slow roll scenarios and creating a different dynamical
picture of solutions. More specifically, one can study the form of the total
quantum effective potential at different parameters $g,\mu^{2}$. And since
they are related, it is possible to restrict them by modifying one of them,
say, fixing 111The scale of inflation is considered to be equal to about
$10^{-5}M_{Pl}$ $g\leavevmode\nobreak\ \sim 1$ , and changing $\mu$ to satisfy
the conditions of convergence. The convergence conditions of the original
series for the potential (19) are represented by the following expressions
[19]:

$\log
g\frac{\omega^{2}(2-\cosh(\phi\omega))}{\mu^{2}\cosh^{4}(\phi\omega)}>1,\leavevmode\nobreak\
\frac{g\omega^{4}}{16\pi^{2}}<1,$ (27)

they set the bounds of applicability of the approximation of the leading logs
for the effective potential.

Figure 1: Comparison of the classical potential (red dashed line), the one-
loop correction (green solid line), and the RG effective potential (blue solid
line) at $g\sim 1,\leavevmode\nobreak\ \mu^{2}\sim 10^{-2}$ (upper plot) and
$g\sim 1,\leavevmode\nobreak\ \mu^{2}\sim 10$ (lower plot)

We compare the behaviour of the resummed potential with a one-loop
contribution which has the following form

$V_{1loop}(\phi)=V(\phi)+\frac{g^{2}}{16\pi^{2}}\frac{1}{4}v_{2}^{2}\log\left(\frac{gv_{2}}{\mu^{2}}\right).$
(28)

In fact, the $T^{2}$-model ($T^{4}$-model either) is called an attractor
because the phase portrait of the Friedman equations (3-5) represents a limit
cycle (or attractor), i.e., the general dynamics of this system does not
depend on the initial conditions. So introducing additional corrections to
this potential can affect the dynamical properties, and there can appear the
Hopf bifurcation picture [21].

From Fig.1 one can see that the one-loop potential strongly modifies the
classical part of $V(\phi)$ causing a maximum (or minimum at $\mu^{2}>1$) at
$\phi=0$, while the all-loop quantum effective potential smoothes the
perturbation from the first loop contribution. One can conclude from this
observation that the quantum effective potential is more stable in contrast to
the one-loop-corrected potential. We also note that for small $\mu$, a
situation may occur where the contributions of all loops are unable to flatten
the maximum near $\phi=0$.

The relevant potentials at different $\mu$ are shown in Figure 2. Spontaneous
symmetry breaking with a maximum near zero is observed and, starting from
certain extremely small $\mu^{2}\sim 10^{-80}$, additional minima can arise
which correspond to a false vacuum.

Figure 2: Comparison of the behaviour of different potentials with all loop
corrections at $g\sim 1$. Blue line — $\mu^{2}\sim 0.6$, green $\mu^{2}\sim
10^{-2}$, purple $\mu^{2}\sim 10^{-10}$, orange $\mu^{2}\sim 10^{-16}$, red
$\mu^{2}\sim 10^{-36}$

The varying behaviour of the potential can find its application in slow
rolling. It is convenient to depict the whole set of cosmological parameters
to demonstrate how inflation changes for different values of $\mu^{2}$. Using
the formulas from the first chapter and numerical calculations, we plot the
value of $\epsilon_{1}$ as a function of $\phi$. Based on these plots, one can
conclude that the occurrence of extra minima for some values of $\mu^{2}$ does
not strongly affect the slow roll behaviour, although, as one can see, the
smaller $\mu^{2}$, the shorter is the inflation (e.g., for the classical
$T^{2}$-model the inflation ends at $\phi_{end}\omega\simeq 1.208$, while for
$\mu^{2}\sim 10^{-3}$ the inflation ends at $\phi_{end}\omega\simeq 1.061$).
Thus, the slow rolling is realised within the observational data for rather
wide bounds on $\mu^{2}$.

(a)

(b)

Figure 3: Plots for the Hubble flow parameter $\epsilon_{1}$ In Fig. 3(a) the
blue line — $\mu^{2}\sim 10^{-2}$, green line — $\mu^{2}\sim 10^{-8}$, yellow
line — $\mu^{2}\sim 10^{-10}$ , at the 3(b) purple line $\mu^{2}\sim
10^{-16}$, orange one $\mu^{2}\sim 10^{-56}$. The red dashed line corresponds
to the classical potential and the grey line indicates the end of inflation

Interestingly, for the all-loop effective potential with $\mu^{2}\leq
10^{-5}$, the situation arises when slow-roll inflation can become eternal
(for the one-loop potential perpetual inflation occurs already with
$\mu^{2}\leq 10^{-1}$) [22, 23]. This is due to spontaneous symmetry breaking
and the fact that the maximum of the effective potential is less than unity in
our notation. That is, the behaviour of the potential deteriorates when
$\mu^{2}$ decreases, since false vacua emerge. The existence of false vacua
leads to the problem of phase transitions in the early Universe [24], although
on the other hand, these features can change the power spectrum of
gravitational waves at certain scales, which may be useful for explaining
primordial black holes production [25]. The question of how exactly tunnelling
through potential barriers can occur was studied in the literature [24, 26],
and is beyond the scope of our consideration (as well as the analysis of the
postinflationary reheating phase in these models). A qualitative study of the
reheating phase for corrected $T$-model potential was made in [12].

Figure 4: Classical potential $\tanh^{4}$ (red dashed line), single-loop
(green solid line) and full effective (blue solid line) potentials at
$\mu^{2}\sim 10^{-80}$, the purple solid line corresponds to the quantum
effective potential at $\mu^{2}\sim 10^{-80}$

###  3.2 $T^{4}$-model

Similar to the case of the $T^{2}$-model, we can consider $T^{4}$-model with
the potential

$V=g\tanh^{4}(\frac{\phi\omega}{\sqrt{6}}).$ (29)

For this potential, one can also write down a generalised RG equation with the
same initial and boundary conditions and then construct the effective
potential. The initial differential equation (20) can be written in the form

$\frac{\partial}{\partial
z}\Sigma(z\omega^{4},\tanh^{4}(\phi\omega/\sqrt{6}))=-\frac{1}{4}\left(\frac{\partial^{2}}{\partial\phi^{2}}\Sigma(z\omega^{4},\tanh^{4}(\phi\omega/\sqrt{6}))\right)^{2}$
(30)

After changing the variables to dimensionless $x=z\omega^{4}$,
$y=\tanh^{4}(\phi\omega/\sqrt{6})$ and the desired function as
$\Sigma(z\omega^{4},\tanh^{4}(\phi\omega/\sqrt{6}))=S(x,y)$ the differential
generalised RG-equation, after passing from the derivative of $\phi$ to the
derivative on $y$, will have the following form:

$\frac{1}{9}\left(y^{1/2}-1\right)^{2}y\left(\left(5y^{1/2}-3\right)S_{y}+4y\left(y^{1/2}-1\right)S_{yy}\right)^{2}=-S_{x},$
(31)

with the boundary and initial conditions in the same form as for the
$T^{2}$-model:

$S(0,y)=y,\leavevmode\nobreak\ S(x,1)=1,\leavevmode\nobreak\ S_{x}(x,1)=0.$
(32)

Equation (31) looks more complicated than in the case of the $T^{2}$ model but
it is numerically analysable. To find an effective potential from (31), we
still need to substitute $z\to-\frac{g}{16\pi^{2}}\log{gv_{2}/\mu^{2}}$, and
in the case of the $T^{4}$ model

$v_{2}=-\frac{2}{3}\omega^{2}\left(\cosh\left(\sqrt{\frac{2}{3}}\omega\phi\right)-4\right)\tanh^{2}\left(\frac{\omega\phi}{\sqrt{6}}\right)\text{sech}^{4}\left(\frac{\omega\phi}{\sqrt{6}}\right).$
(33)

Figure 4 shows a comparison of the classical potential, the one-loop-corrected
potential, with the all loop effective potential for various $\mu^{2}$. Here
we also show spontaneous symmetry breaking for which the contribution from the
first loop is responsible, and still the quantum effective potential is
characterised by a smoother behaviour. Unlike the previous case the effective
potential for the $T^{4}$-model turns to zero at the minimum, though from a
certain value of $\mu^{2}$ there is also symmetry breaking leading to the
appearance of the maximum of the effective potential at $\phi=0$, which has
the same meaning of instability as in the case of the $T^{2}$-model. Also as
can be seen from the figure 4, the $T^{4}$ potential is much more stable to
quantum corrections and almost does not change in a large range of values of
the $\mu^{2}$ parameter. Thus, for estimations of the all-loop effective
potential, one can use the classical $T^{4}$ potential with a one-loop quantum
correction.

The Fig. 5 shows the cosmological parameter $\epsilon_{1}$ for several
effective potentials. It can be seen that irrespective of the value of the
parameter $\mu^{2}$, the slow rolling is completed (if we do not speak about
too small values of the parameter $\mu^{2}$, where the occurrence of false
vacua occurs).

Equations for the effective potentials of $T^{2n}$-models can be derived, but
it turns out that they are not fundamentally different from the case of the
full RG-summed potential of the $T^{4}$-model.

Figure 5: Plots for the cosmological parameter $\epsilon_{1}$. The red dashed
line corresponds to the classical potential, the blue line all-loop effective
potential — $\mu^{2}\sim 10^{-2}$, the green $\mu^{2}\sim 10^{-10}$, the
purple $\mu^{2}\sim 10^{-16}$, the orange $\mu^{2}\sim 10^{-56}$. The grey
line represents the end of inflation

## Conclusion

In this paper, we have constructed quantum effective potentials and used them
to study inflationary cosmology with slow roll. We derived the generalised RG
equation for the effective potential in the leading logarithmic approximation
and applied it to evaluate the potentials of the $T^{2}$ and $T^{4}$-models,
which are often used in modern cosmology of slow roll inflation. We found that
while the one-loop correction strongly affects the potential breaking its
original symmetry, the contribution of higher loops smoothes the behaviour of
the potential. However, unlike the $\phi^{4}$-case [8], we found that the
effective potentials preserve spontaneous symmetry breaking when summing all
the leading corrections (1). This may lead to effects related to the decay of
the metastable vacuum.

As shown in [12], quantum one-loop corrections do not deviate the asymptotic
values of the effective potential at large fields and can change the form of
the potential only at small values of the inflaton field. However, taking into
account all leading logarithms by means of the RG-formalism (or generalised
RG) may lead to a significant change of the asymptotics [8, 19]. In the case
of $T$-models of $\alpha$-attractors, we have found that the asymptotics of
the potential remains unchanged when all-loop corrections are taken into
account, thus confirming the conclusions of [12].

In this work, we also constructed and studied the Hubble flow parameters
reflecting the basic properties of slow-roll inflation. It was found that for
many values of the dimensional regularization parameter $\mu$ the inflation
can become eternal. In some region of parameters the all-loop quantum
effective potential only slightly differs from the classical one and leads to
a similar slow-roll behavior. Indeed, our calculations of cosmological
inflation parameters fit the diagram of the $n_{s}-r_{s}$ plot. In Fig. 6, we
combined the data for the indices $n_{s}$ and $r_{s}$ for the effective
potentials of both models in the purple region. The diagram reveals that all
effective potentials are consistent with the observational data [16]. The
purple region on the plot corresponds to all allowed values of the parameter
$\mu^{2}$ for the potential. It can be seen that the effective potentials
based on $T^{2}$ or $T^{4}$ do not differ significant but are still
distinguishable. The $T^{2}$ and $T^{4}$ with one-loop corrections can also
enter this region, but a fine tuning of the $\mu$ parameter is required (for
example, $\mu^{2}\simeq 2$ for the $T^{2}$-model in our units).

The approach used in this paper allows one to calculate the effective
potentials for other cosmological models of inflation. It can be used to
constrain or to generalize some inflationary models in the slow-roll
approximation.

Figure 6: Comparison in the $n_{s}$ versus $r$ plot. The dark purple region
corresponds to the full effective potentials for $T^{2}$ in the limits
$10^{2}<\mu^{2}<10^{-3}$ and $T^{4}$ in the limits $10^{10}<\mu^{2}<10^{-14}$
in the interval from 50 to 60 $e$-folds. The bright green line corresponds to
the classical $T^{2}$-model. Data source is taken from [16]

## Acknowledgments

The authors are grateful to A.Starobinsky and S. Ketov for the statement of
the problem and for D.Gorbunov for numerous useful discussions of inflationary
scenarios. Financial support from the Russia Scientific Foundation Grant #
21-12-00129 is cordially acknowledged.

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