# SpaceML: Distributed Open-source Research with Citizen Scientists for the
Advancement of Space Technology for NASA

Anirudh Koul
Equal Contribution Siddha Ganju 11footnotemark: 1
Meher Kasam 11footnotemark: 1
James Parr

###### Abstract

Traditionally, academic labs conduct open-ended research with the primary
focus on discoveries with long-term value, rather than direct products that
can be deployed in the real world. On the other hand, research in the industry
is driven by its expected commercial return on investment, and hence focuses
on a real world product with short-term timelines. In both cases, opportunity
is selective, often available to researchers with advanced educational
backgrounds. Research often happens behind closed doors and may be kept
confidential until either its publication or product release, exacerbating the
problem of AI reproducibility and slowing down future research by others in
the field. As many research organizations tend to exclusively focus on
specific areas, opportunities for interdisciplinary research reduce.
Undertaking long-term bold research in unexplored fields with non-commercial
yet great public value is hard due to factors including the high upfront risk,
budgetary constraints, and a lack of availability of data and experts in niche
fields. Only a few companies or well-funded research labs can afford to do
such long-term research. With research organizations focused on an exploding
array of fields and resources spread thin, opportunities for the maturation of
interdisciplinary research reduce. Apart from these exigencies, there is also
a need to engage citizen scientists through open-source contributors to play
an active part in the research dialogue. We present a short case study of
SpaceML, an extension of the Frontier Development Lab, an AI accelerator for
NASA. SpaceML distributes open-source research and invites volunteer citizen
scientists to partake in development and deployment of high social value
products at the intersection of space and AI.

## Introduction

Increasingly accessible satellite data, a booming global space industry, and
an abundance of statistical and newer AI technologies opens up the potential
to accelerate interdisciplinary scientific research. This comes with the
challenges of scaling traditional scientific processing workflows and, more
importantly, maintaining the quality of deployed AI, which is plagued by
issues of reproducibility, management of knowledge transfer, and lack of
expertise. NASA Science Mission Directorate has declared [1] a commitment to
open science, with an emphasis on continual monitoring and updating of
deployed systems, improved engagement of the scientific community with citizen
scientists, and data access to the wider research community for robust
validation of published research results. SpaceML is one such international AI
accelerator that applies startup and agile software development principles to
research, thereby reducing time to success significantly.

The SpaceML 111http://spaceml.org/ open-source research program takes eligible
projects from the Frontier Development Lab (FDL) 222FDL is an AI accelerator
supported by both NASA and the European Space Agency, along with a few other
corporate and government partners.[2] and enables continued development of
those projects beyond pure research, to develop and deploy solutions usable in
the real world, eventually handing them over to stakeholders. In addition to
the core goals of the project, extensions to the project for “nice to have”
capabilities are available as mini-projects to open-source contributors.
Mentorship support from industry experts and cloud computing resources are
made available to the contributors where necessary. SpaceML’s first iteration
in 2020 converted many high school students to citizen scientists with unique
stories, like a citizen scientist from Nigeria, who improved FDL’s advanced
AI-powered meteor detector pipeline [3, 4] for the NASA CAMS [5] project along
with an interactive interface in his local cyber-cafe.

FDL and SpaceML undertake interdisciplinary research with researchers and
experts from different organizations, yet operate as one team. This ensures a
smooth handover and efficient transfer of knowledge and capabilities, reducing
transitional failures, ultimately resulting in robust and reproducible work.

## Key Features of SpaceML

We focus on how SpaceML utilizes cloud computing technologies, open-source
challenges, and volunteer citizen scientists to develop industry-level AI
technologies measured by TRL4ML [6].

Open-source Challenges: As a first step, eligible FDL challenges are brought
under the umbrella of SpaceML. FDL researchers create a set of miniature
extension projects on a scale of easy to hard. Most of these mini-projects
have a very narrow focus and are meant to be built by a competent developer
with three weeks of effort. The mini-projects are well defined to include
availability and access of data, a predefined technology stack, reference
material (research papers, web articles, tutorial videos, etc.), and potential
solution ideas. Global calls for SpaceML volunteer open-source contributions
invite people from different backgrounds and different countries for the
positions of researchers and domain leads. Access to experts in various fields
is made possible through FDL and SpaceML public-private partnerships. Once the
contributions reach a certain level of maturity, new volunteer mentors through
the FDL and SpaceML network join in to help boost the project further. Regular
weekly and gated expert reviews provide direct and actionable feedback, and
serve as key decision points to measure the correctness, impact, and relevance
of research.

Cloud Compute: To reduce the barrier to entry, FDL researchers package
accessible code and instructions on GitHub, Google Colab (with free GPUs) so
that the new contributors can run it with relative ease. Colab notebooks allow
you to combine executable code and rich text in a single document, along with
images, HTML, LaTeX thus enabling code execution and instructions or
information to co-occur. Colab has absolute zero configuration or setup
required allowing the open-source contributors to get up and running quickly.
Colab also offers free access to advanced computing hardware like GPUs or
TPUs, which significantly accelerate development. The GitHub repository also
contains a video explaining the overall vision of each project and
descriptions of project-specific beginner challenges, along with detailed
guidance on how to get started. The beginner challenges, akin to an interview,
allow competent and motivated contributors to move further in the challenge.
At the same time, for people who are not yet familiar with the technical stack
(science or AI), this serves as an educational opportunity to level up with
the help of the detailed guides. Beginner guides can include “How to get
started in AI”, “How to use GitHub effectively”, “How to access satellite
imagery data”, “Code Quality standards”, “Citation guide”, and, “Code
Reproducibility standards”. Contributors who have shown enough progress are
then provided with Google Cloud credits to scale up their AI training and
storage needs.

Mentorship: FDL researchers, who worked on the original project, are now the
stakeholders of the open-source project, and switch their roles to mentor
while providing weekly mentorship. Throughout the process of improving the
project, the contributors eventually become an expert in the niche area of
research. Once the contributions reach some maturity, new mentors through the
FDL network join in to level up the project further. For example, an
experienced industry engineer can perform code reviews, while an experienced
scientist can review the experiments for the correctness and suggest other
ways to amplify the value of the contributions. SpaceML provides access to
experienced and often well-known professionals as volunteer mentors to the
public who might not be accessible otherwise. To make efficient use of time
for both the mentors and mentees, the mentees need to earn the time of mentors
in the form of completed work, like through the early challenge or the weekly
goals.

Enhancing STEAM education: Distributed open-source research, as spearheaded by
SpaceML can work given (1) a specific focus for each contributor (2) a support
structure of diverse experts and mentors (3) a plan detailing how all the
individual contributions work together in the final deployment. Motivated by
research with high public value that has a path towards deployment, high
school and early graduate school students from a wide variety of socioeconomic
backgrounds and geographies volunteered in the first iteration of SpaceML.

## Impact of SpaceML

In our pilot study, SpaceML contributors were high school students, with
participants from Nigeria, Mexico, Korea, Germany, and the United States with
no prior AI background and worked on tasks like self-supervised learning using
SimCLR, multi-resolution image search, cost-efficient data labeling, balancing
imbalanced data without labels, and generating synthetic patches for missing
satellite imagery.

SpaceML has also boosted the monitoring and improvement of the AI model and
web tool for the TRL-9 NASA CAMS project (The CAMS project, established in
California in 2010, uses hundreds of low light CCTV cameras to capture the
meteor activity in the night sky. After each night, the resident scientist
performs the triangulation of tracklets captured by two or more cameras and
computes the meteor’s trajectory, orbit, and light curve. Each solution is
then manually classified as a meteor or not a meteor like planes, birds,
clouds etc. CAMS became a part of FDL in 2017 in order to automate the data
processing pipeline and filtration of meteors.)

As part of FDL 2017, beyond the huge efficiency improvements from automation,
an AI pipeline with human-level accuracy, and a new web tool [7] that allowed
global access to the previous night’s activity, the FDL team’s contribution
resulted in raising awareness and bringing in new citizen scientists who
established stations in Brazil. The automation of the data processing allowed
researchers to manage the camera stations better, which led to the detection
of the highest number of meteors in a single night (including 3003 Geminids &
1154 sporadic meteors) in December 2017. Subsequently, increased funding led
to a four-fold global expansion of the camera network in the Southern
Hemisphere.

As the CAMS project grew globally, requirements to add features to the
application arose. During SpaceML 2020 a team of two students and industry
mentors volunteered through open-source contributions to improve
functionality. In 2020, the quick turnaround time of discovery from the CAMS
web tool ultimately led to the an almost monthly reporting of a new and
unusual meteor showers in the sky, including the discovery of multiple meteor
showers, such as the gamma Piscis Austrinids and the sigma Phoenicids. One of
those showers helped better define the orbit of parent comet Grigg-Mellish,
which was observed poorly in 1907 [8].

## Conclusion

We present SpaceML, an open-source interdisciplinary science and AI research
program. SpaceML releases challenges to the public inviting a diverse range of
contributors. The released challenges are made accessible by packaging small
projects within Google Colab (with free GPUs). Contributors who show promise
are provided a personalized educational plan to build a foundation in the area
of research and then receive relevant mentorship on the project from experts.
Starting with freely available compute resources (eg Google Colab)
contributors who show enough progress are provided cloud credits with high
performance compute to allow for additional training and scaling. SpaceML has
a focused aim of converting research into product reaching the highest NASA
Technology Readiness Level (TRL 9). The opportunities afforded by SpaceML have
brought in cohorts of citizen scientists and created opportunities for joint
AI and science research. The first batch of SpaceML researchers were high
school students who entered with no prior AI background, pursued research in
advanced topics like self-supervised learning on unlabelled satellite imagery,
balancing unlabelled data, multi resolution search on satellite data, some
eventually beat the state-of-the-art for certain techniques. The high school
students also presented and/or published their work work in peer review
conferences. Several of the high students who were participants in SpaceML
have expressed interest in pursuing research in earth sciences, climate change
and related STEAM fields as potential career options.

## References

  * [1] Strategic Data Management Working Group, “Science mission directorate: Strategy for data management and computing for groundbreaking science 2019-2024,” 2019.
  * [2] S. Ganju, A. Koul, A. Lavin, J. Veitch-Michaelis, M. Kasam, and J. Parr, “Learnings from Frontier Development Lab and SpaceML - AI Accelerators for NASA and ESA,” 2020.
  * [3] S. Zoghbi, M. Cicco, A. P. Stapper, A. J. Ordonez, J. Collison, P. S. Gural, S. Ganju, J.-L. Galache, and P. Jenniskens, “Searching for long-period comets with deep learning tools,” in _Deep Learning for Physical Science Workshop, NeurIPS_ , 2017.
  * [4] P. Jenniskens, J. Baggaley, I. Crumpton, P. Aldous, P. Pokorny, D. Janches, P. S. Gural, D. Samuels, J. Albers, A. Howell, C. Johannink, M. Breukers, M. Odeh, N. Moskovitz, J. Collison, and S. Ganju, “A survey of southern hemisphere meteor showers,” _Planetary and Space Science_ , vol. 154, pp. 21 – 29, 2018. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0032063317304002
  * [5] P. Jenniskens, P. Gural, L. Dynneson, B. Grigsby, K. Newman, M. Borden, M. Koop, and D. Holman, “Cams: Cameras for allsky meteor surveillance to establish minor meteor showers,” _Icarus_ , vol. 216, no. 1, pp. 40 – 61, 2011. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0019103511003290
  * [6] A. Lavin and G. Renard, “Technology readiness levels for machine learning systems,” in _ICML Workshop on Challenges in Deploying Machine Learning Systems_ , 2020.
  * [7] M. De Cicco, S. Zoghbi, A. P. Stapper, A. J. Ordoñez, J. Collison, P. S. Gural, S. Ganju, J.-L. Galache, and P. Jenniskens, “Artificial intelligence techniques for automating the CAMS processing pipeline to direct the search for long-period comets,” in _Proceedings of the International Meteor Conference_ , Jan. 2018, pp. 65–70.
  * [8] P. Jenniskens, E. Lyytinen, and J. Baggaley, “An outburst of delta pavonids and the orbit of parent comet c/1907 g1 (grigg-mellish),” _Planetary and Space Science_ , vol. 189, p. 104979, 2020. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0032063319304052


# Two New $\mu-\tau$ Deviated Neutrino Mass Matrix Textures

Pralay Chakraborty<EMAIL_ADDRESS>Physics Department, Gauhati
University, India  Tanmay Dev<EMAIL_ADDRESS>Physics Department,
Gauhati University, India  Subhankar Roy<EMAIL_ADDRESS>Physics
Department, Gauhati University, India.

###### Abstract

We propose two new neutrino mass matrix textures deviating from $\mu-\tau$
symmetry and explore their phenomenological implications. Both textures
predict $\theta_{23}$ and $\delta$ within sharp bounds while considering both
normal and inverted ordering of neutrino masses. We try to realize these
proposed textures within the framework of the seesaw mechanism, in association
with the $SU(2)_{L}\times A_{4}\times Z_{3}$ symmetry group.

##  I Introduction

In the year 1957, Bruno Pontecorvo proposed the idea of neutrino oscillation,
suggesting that neutrinos change their identity as they move through space
[1]. This phenomenon arises from the fact that the three flavours of neutrinos
($\nu_{l=e,\mu,\tau}$) are admixtures of three mass eigenstates
$(\nu_{i=1,2,3})$ with corresponding mass values ($m_{i=1,2,3}$). The
phenomenon of neutrino oscillation predicts that neutrinos must have non-zero
mass, which contradicts with the predictions of the Standard Model (SM) [2, 3,
4]. To understand the origin of neutrino mass, it is necessary to explore
theories that go beyond the SM. The neutrino mass matrix ($M_{\nu}$)
originates from the Yukawa Lagrangian, formulated within the framework of the
see-saw mechanism [5, 6, 7]. The $M_{\nu}$ contains twelve parameters: three
mass eigenvalues ($m_{i=1,2,3}$), three mixing angles
($\theta_{12},\theta_{13},\theta_{23}$), three CP phases
($\delta,\alpha,\beta$), and three unphysical phases
($\phi_{1},\phi_{2},\phi_{3}$). In the literature, various constraints, such
as texture zeroes [8], vanishing minors [9], hybrid textures [10] and
$\mu-\tau$ symmetry [11, 12] have been applied to $M_{\nu}$ so that some
physical parameters can be predicted. It is worth noting that experiments have
ruled out the $\mu-\tau$ symmetry due to its prediction of $\theta_{13}$ being
zero. Despite this, the $\mu-\tau$ symmetry along with its predictions has
been well-established, the theorists have made several attempts to explore
deviations from the exact $\mu-\tau$ symmetry, in the form of $\mu-\tau$ anti-
symmetry [13], $\mu-\tau$ reflection symmetry [14], $\mu-\tau$ mixed symmetry
[15, 16], etc. In that light, we propose two new $\mu-\tau$ deviated neutrino
mass matrix textures as shown in the following,

$\displaystyle M^{1}_{\nu}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}A&-B&-iB/2\\\ -B&B&D\\\ -iB/2&D&2B\\\
\end{bmatrix},,$ (1) $\displaystyle M^{2}_{\nu}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}A&B&(iB)^{*}\\\ B&2B^{*}&D\\\ (iB)^{*}&D&-2B\\\
\end{bmatrix},$ (2)

where, $A,\,B$ and $D$ appearing in both the textures are complex parameters.

The plan of the paper is outlined as follows: In section (II), we give the
formalism of our work. The section (III) is devoted to the numerical analysis
and possible symmetry realization. In section (IV), we conclude.

##  II Formalism

In addition to $M_{\nu}$, the neutrino physics phenomenology deals with an
important quantity known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
[17], $V$, which is a $3\times 3$ unitary matrix. In general, the PMNS matrix
is defined in a basis where the charged lepton mass matrix is diagonal
(flavour basis). The Particle Data Group has adopted a parametrization scheme
for $V$ as shown in the following,

$V=P_{\phi}.U.P_{M},$ (3)

where, $P_{\phi}=diag(e^{i\phi_{1}},e^{i\phi_{2}},e^{i\phi_{3}})$ and
$P_{M}=diag(e^{i\alpha},e^{i\beta},1)$. The matrix $U$ is depicted as shown
below,

$\displaystyle U$ $\displaystyle=$ $\displaystyle\begin{bmatrix}1&0&0\\\
0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{bmatrix}\times\begin{bmatrix}c_{13}&0&s_{13}\,e^{-i\delta}\\\
0&1&0\\\
-s_{13}e^{i\delta}&0&c_{13}\end{bmatrix}\times\begin{bmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{bmatrix},$ (4)

where, $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$.

As discussed in Section I, $M_{\nu}$ shelters information about all physical
parameters. To extract this information, we need to diagonalize $M_{\nu}$:
$M^{d}=V^{T}M_{\nu}V$. It is to be noted that the unphysical phases
($\phi_{i=1,2,3}$) can be removed from $V$ by redefining the charged lepton
fields in terms of these phases. However, in the present work, we stick to the
general form of $M_{\nu}$ containing the unphysical phases. We distinguish
$M_{\nu}$ from $\tilde{M_{\nu}}$ that excludes the unphysical phases as shown
in the following,

$M_{\nu}=P^{*}_{\phi}\tilde{M_{\nu}}P^{*}_{\phi},$ (5)

where $P_{\phi}$ carries the unphysical phases.

The constrained relations appearing in $M_{\nu}^{1}$ as shown in Eq. (1) hold
good provided the following conditions for the unphysical phases are
satisfied,

$\displaystyle\phi_{2}-\phi_{3}$ $\displaystyle=$ $\displaystyle
Arg[\tilde{(M^{1}_{\nu})}_{12}]-Arg[\tilde{(M^{1}_{\nu})}_{13}]+\frac{\pi}{2},$
(6) $\displaystyle\phi_{2}-\phi_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(Arg[\tilde{(M^{1}_{\nu})}_{22}]-Arg[\tilde{(M^{1}_{\nu})}_{33}\right),$
(7) $\displaystyle\phi_{1}-\phi_{2}$ $\displaystyle=$ $\displaystyle
Arg[\tilde{(M^{1}_{\nu})}_{12}]-Arg[\tilde{(M^{1}_{\nu})}_{22}-\frac{\pi}{2}.$
(8)

On the other hand, for $M_{\nu}^{2}$ as appearing in Eq. (2), the conditions
for the unphysical phases are given by,

$\displaystyle 2\phi_{1}+\phi_{2}+\phi_{3}$ $\displaystyle=$ $\displaystyle
Arg[\tilde{(M^{2}_{\nu})}_{12}]+Arg[\tilde{(M^{2}_{\nu})}_{13}]+\frac{\pi}{2},$
(9) $\displaystyle\phi_{2}+\phi_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(Arg[\tilde{(M^{2}_{\nu})}_{22}]+Arg[\tilde{(M^{2}_{\nu})}_{33}-\frac{\pi}{2}\right),$
(10) $\displaystyle\phi_{1}+\phi_{2}-2\phi_{3}$ $\displaystyle=$
$\displaystyle
Arg[\tilde{(M^{2}_{\nu})}_{12}]-Arg[\tilde{(M^{2}_{\nu})}_{33}-\frac{\pi}{2}.$
(11)

It is to be noted that the choice between sticking to $M_{\nu}$ or
$\tilde{M_{\nu}}$ is subjective. However, keeping the said phases intact in
$M_{\nu}$ provides some phenomenological prospects. The importance of these
unphysical phases is highlighted in the Ref. [14]. It is evident from Eq. (5)
that $\tilde{M_{\nu}}$ does not involve the unphysical phases. In this regard,
constraints applied to $\tilde{M_{\nu}}$ ensure that predictions of physical
parameters remain unaffected by these phases. However, the same is not true if
we are dealing with $M_{\nu}$. In most studies, model builders prefer to stick
to $\tilde{M_{\nu}}$, along with the fact that $\phi_{1}$, $\phi_{2}$, and
$\phi_{3}$ are removed by redefining the charged lepton fields. However, there
are no such theoretical impositions that actually dictate the removal of the
said phases. In the present work, we try to formulate constraints on the mass
matrix elements of $M_{\nu}$ rather than $\tilde{M_{\nu}}$.

##  III Proposed Texture Viability

In this section, we discuss the phenomenological implications of the proposed
textures $M^{1}_{\nu}$ and $M^{2}_{\nu}$ in the light of experimental
observations. In addition, we shall try to realize the proposed textures from
group theoretical framework.

###  III.1 Numerical analysis

The proposed texture $M^{1}_{\nu}$ deals with three constrained relations as
shown in the following,

$\displaystyle(M^{1}_{\nu})_{12}$ $\displaystyle=$ $\displaystyle
2i(M^{1}_{\nu})_{13},$ (12) $\displaystyle(M^{1}_{\nu})_{22}$ $\displaystyle=$
$\displaystyle\frac{(M^{1}_{\nu})_{33}}{2},$ (13)
$\displaystyle(M^{1}_{\nu})_{22}$ $\displaystyle=$
$\displaystyle-(M^{1}_{\nu})_{12}.$ (14)

The fourth relation $(M^{1}_{\nu})_{33}=-2(M^{1}_{\nu})_{12}$ is not
independent and can be derived from the above three.

The equations arising in (12) and (14) are complex equations and give rise to
six real equations as shown in the following,

$\displaystyle\text{Re}[(M_{\nu}^{1})_{12}]+2\,\text{Im}[(M_{\nu}^{1})_{13}]$
$\displaystyle=$ $\displaystyle 0,$ (15)
$\displaystyle\text{Im}[(M_{\nu}^{1})_{12}]-2\,\text{Re}[(M_{\nu}^{1})_{13}]$
$\displaystyle=$ $\displaystyle 0,$ (16)
$\displaystyle\text{Re}[(M_{\nu}^{1})_{22}]-\frac{1}{2}\,\text{Re}[(M_{\nu}^{1})_{33}]$
$\displaystyle=$ $\displaystyle 0,$ (17)
$\displaystyle\text{Im}[(M_{\nu}^{1})_{22}]-\frac{1}{2}\,\text{Im}[(M_{\nu}^{1})_{33}]$
$\displaystyle=$ $\displaystyle 0,$ (18)
$\displaystyle\text{Im}[(M_{\nu}^{1})_{22}]+\,\text{Re}[(M_{\nu}^{1})_{12}]$
$\displaystyle=$ $\displaystyle 0,$ (19)
$\displaystyle\text{Im}[(M_{\nu}^{1})_{22}]+\,\text{Im}[(M_{\nu}^{1})_{12}]$
$\displaystyle=$ $\displaystyle 0.$ (20)

Needless to mention, the left hand side of above six equations involves twelve
parameters including the unphysical phases. We solve the Eqs. (15)-(20)
numerically and generate the sufficient number of random numbers for the
parameters $\Delta m^{2}_{21}$, $\Delta m^{2}_{31}$, $\theta_{12}$,
$\theta_{13}$, $\theta_{23}$ and $\delta$ and obtain the correlations plots.
For normal ordering, we observe that $\theta_{23}$ lies in the lower octant
and takes minimum and maximum values as $41.83^{\circ}$ and $42.19^{\circ}$
whereas the bound for $\delta$ is constrained to take the minimum and maximum
values as $217.24^{\circ}$ and $221.86^{\circ}$ respectively (see Figs. 1-1).
For Inverted ordering, it is seen that $\theta_{23}$ lies in the upper octant
with minimum and maximum values as $49.34^{\circ}$ and $49.54^{\circ}$ whereas
$\delta$ is constrained to a sharp bound with minimum value $308.07^{\circ}$
and maximum value $315.42^{\circ}$ respectively (see Figs. 2-2). In addition
to this, the obtained bound for the mass squared differences ($\Delta
m^{2}_{21}$ and $\Delta m^{2}_{31}$) and two mixing angles ($\theta_{12}$ and
$\theta_{13}$), considering the normal orderings are consistent with the
experimental observations [18, 19]. On the other hand, the mixing scheme
obtained for inverted ordering aligns well with experimental observations
except for $\theta_{13}$ (see Fig. 2). Hence, for inverted ordering, the
texture $M^{1}_{\nu}$ is ruled out in the light of experimental evidences. For
the above analysis, the choice of input parameters are taken as per Table 1
(for normal ordering) and Table 2 (for inverted ordering).

The texture $M^{2}_{\nu}$ shelters three constrained relations as shown below,

$\displaystyle(M^{2}_{\nu})_{12}$ $\displaystyle=$
$\displaystyle(i(M^{2}_{\nu})_{13})^{*},$ (21)
$\displaystyle(M^{2}_{\nu})_{22}$ $\displaystyle=$
$\displaystyle-((M^{2}_{\nu})_{33})^{*},$ (22)
$\displaystyle(M^{2}_{\nu})_{33}$ $\displaystyle=$
$\displaystyle-2((M^{2}_{\nu})_{12}).$ (23)

For $M^{2}_{\nu}$ as well, there is a corresponding dependent relation:
$(M^{1}_{\nu})_{22}=2(M^{1}_{\nu})_{12}^{*}$. This relation can be derived
from the aforementioned three relations.

As before, the three complex relations appearing in Eqs.(21) and (23) give
rise to six real equations as shown below,

$\displaystyle\text{Re}[(M_{\nu}^{2})_{12}]+\text{Im}[(M_{\nu}^{2})_{13}]$
$\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle\text{Im}[(M_{\nu}^{2})_{12}]+\text{Re}[(M_{\nu}^{2})_{13}]$
$\displaystyle=$ $\displaystyle 0,$ (25)
$\displaystyle\text{Re}[(M_{\nu}^{2})_{22}]+\text{Re}[(M_{\nu}^{2})_{33}]$
$\displaystyle=$ $\displaystyle 0,$ (26)
$\displaystyle\text{Im}[(M_{\nu}^{2})_{22}]-\text{Im}[(M_{\nu}^{2})_{33}]$
$\displaystyle=$ $\displaystyle 0,$ (27)
$\displaystyle\text{Re}[(M_{\nu}^{2})_{33}]+2\,\text{Re}[(M_{\nu}^{2})_{12}]$
$\displaystyle=$ $\displaystyle 0,$ (28)
$\displaystyle\text{Im}[(M_{\nu}^{2})_{33}]+2\,\text{Im}[(M_{\nu}^{2})_{12}]$
$\displaystyle=$ $\displaystyle 0.$ (29)

Following the similar analysis, we generate the correlations plots for the
parameters $\Delta m^{2}_{21}$, $\Delta m^{2}_{31}$, $\theta_{12}$,
$\theta_{13}$, $\theta_{23}$ and $\delta$, considering both normal and
inverted orderings. It is evident from the graphical visualization that for
normal ordering, $\theta_{23}$ lies in the lower octant with minimum value
$44.87^{\circ}$ and maximum value $44.89^{\circ}$ respectively. A constrained
bound for $\delta$ is obtained with minimum and maximum value as
$174.77^{\circ}$ and $179.28^{\circ}$ respectively (see Figs. 3-3). For
inverted ordering, it is found that $\theta_{23}$ lies in the lower octant and
takes the minimum and maximum value as $44.77^{\circ}$ and $44.98^{\circ}$
while $\delta$ is constrained to a sharp bound with minimum value
$336.87^{\circ}$ and maximum value $340.93^{\circ}$ respectively(see Figs.
4-4). In addition, the prediction of the mass squared differences and mixing
angles obtained for both the orderings are consistent with experimental data
[18, 19]. The choice of the input parameters are listed in Table 3 (normal
ordering) and Table 4 (inverted ordering). We highlight that in the present
work, the sum of three neutrino masses is consistent with cosmological data
i.e., $\leq 0.12\,eV$ [20].

The justification of the Majorana nature of neutrinos is subjected to the
discovery of neutrinoless double beta decay [21, 22]. The effective Majorana
neutrino mass, denoted as $m_{\beta\beta}=|\sum_{k=1}^{3}{U^{2}_{1k}m_{k}}|$,
is a observational parameter in neutrinoless double beta decay, where,
$m_{1}$, $m_{2}$ and $m_{3}$ are the three mass eigenvalues, $U_{11}$,$U_{12}$
and $U_{13}$ are the elements of the PMNS matrix that carry the information of
the Majorana phases. In recent years, several experiments have provided the
upper bounds of $m_{\beta\beta}$: SuperNEMO(Se82) as $67-131$ meV, GERDA(Ge76)
as $104-228$ meV, EXO-200(Xe136) as $111-477$ meV, CUORE(Te130) as $75-350$
meV and KamLAND-Zen(Xe136) as $61-165$ meV [23, 24, 25, 26, 27, 28, 29, 30].
In this regard, we predict the parameter $m_{\beta\beta}$ from the proposed
textures (see Figs. 5-5), in the light of both normal and inverted orderings.

###  III.2 Texture Feasibility from Symmetry

To understand the proposed texture from symmetry framework, we try to
incorporate see-saw mechanism in association with $A_{4}$ group. In this
regard, we extend the field content of the SM by introducing three right-
handed neutrinos ($\nu_{e_{R}}$, $\nu_{\mu_{R}}$, $\nu_{\tau_{R}}$), a scalar
singlet ($\zeta$), and two scalar triplets ($\phi,\Delta$). The transformation
properties of the field content are summarised in Table 5. The
$SU(2)_{L}\times A_{4}\times Z_{3}$ invariant Lagrangian is constructed in the
following way,

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$ $\displaystyle
y_{e}(\bar{D}_{l_{L}}H)_{1}e_{R_{1}}+y_{\mu}(\bar{D}_{l_{L}}H)_{1^{\prime}}\mu_{R_{1^{\prime\prime}}}+y_{\tau}(\bar{D}_{l_{L}}H)_{1^{\prime\prime}}\tau_{R_{1^{\prime}}}+y_{1}(\bar{D}_{l_{L}}\Phi)_{1}\nu_{eR_{1}}$
(30) $\displaystyle+y_{2}(\bar{D}_{l_{L}}\Phi)_{1^{\prime\prime}}\nu_{\mu
R_{1^{\prime}}}+y_{3}(\bar{D}_{l_{L}}\Phi)_{1^{\prime}}\nu_{\tau_{1^{\prime\prime}}}+\frac{1}{2}y_{R_{1}}(\overline{\nu^{c}}_{e_{R}}\nu_{e_{R}})_{1}\zeta+\frac{1}{2}y_{R_{2}}$
$\displaystyle[(\overline{\nu^{c}}_{\mu_{R}}\nu_{\tau_{R}})+(\overline{\nu^{c}}_{\tau_{R}}\nu_{\mu_{R}})]\zeta+y_{T_{2}}(\bar{D}_{l_{L}}D_{l_{L}}^{c})_{3_{S}}\Delta_{3}+h.c.$

The additional group $Z_{3}$ is incorporated to cut short some undesired
terms. The vacuum expectation values (vev) for the scalar fields are chosen
as: $\langle H\rangle_{0}=v_{H}(1,1,1)^{T}$ and
$\langle\Phi\rangle_{0}=v_{\Phi}(1,1,1)^{T},\langle\zeta\rangle_{0}=v_{\zeta}$.
The scalar potential in support of the chosen vacuum alignments can be found
in appendix B. We derive the charged lepton mass matrix($M_{L}$) in the
following form,

$M_{l}=v_{H}\begin{bmatrix}y_{e}&y_{\mu}&y_{\tau}\\\
y_{e}&\omega\,y_{\mu}&\omega^{2}\,y_{\tau}\\\
y_{e}&\omega^{2}\,y_{\mu}&\omega\,y_{\tau}\\\ \end{bmatrix},$ (31)

We diagonalize $M_{l}$ as $M_{l}^{diag}=U_{l_{L}}^{\dagger}M_{l}U_{l_{R}}$,
where $M^{diag}_{l}=\sqrt{3}\,diag(y_{e},y_{\mu},y_{\tau})$. The $U_{l_{L}}$
and $U_{l_{R}}$ are expressed as,

$U_{l_{L}}=\frac{1}{\sqrt{3}}\begin{bmatrix}e^{i\eta}&e^{i\kappa}&e^{i\xi}\\\
e^{i\eta}&\omega\,e^{i\kappa}&\omega^{2}\,e^{i\xi}\\\
e^{i\eta}&\omega^{2}\,e^{i\kappa}&\omega\,e^{i\xi}\\\ \end{bmatrix},\quad\quad
U_{l_{R}}=diag\begin{bmatrix}e^{i\eta},&e^{i\kappa},&\,e^{i\xi}\end{bmatrix},$
(32)

The arbitrary phases $\eta$, $\kappa$ and $\xi$ in $U_{l_{L}}$ and $U_{l_{R}}$
are added with the motivation that the choice of eigenvectors is not unique.

The Dirac neutrino mass matrix($M_{D}$) and the right-handed neutrino mass
matrix($M_{R}$) are obtained as shown in the following,

$M_{D}=v_{\Phi}\begin{bmatrix}y_{1}&y_{2}&y_{3}\\\
y_{1}&\omega\,y_{2}&\omega^{2}\,y_{3}\\\
y_{1}&\omega^{2}\,y_{2}&\omega\,y_{3}\\\ \end{bmatrix},\quad\quad
M_{R}=\begin{bmatrix}A&0&0\\\ 0&0&B\\\ 0&B&0\\\ \end{bmatrix},$ (33)

where, $\omega=e^{2i\pi/3}$, $A=\frac{1}{2}y_{R_{1}}v_{\zeta}$ and
$B=\frac{1}{2}y_{R_{2}}v_{\zeta}$.

The Type-I seesaw contribution to the neutrino mass matrix is taken as
$M_{T_{1}}=-M_{D}M_{R}^{-1}M_{D}^{T}$. The Type-II seesaw contribution with
the specific choice of vev
$\langle\Delta\rangle_{0}=v_{\Delta}(\omega^{2},0,1)^{T}$ comes as shown
below,

$M_{T_{2}}=y_{T_{2}}v_{\Delta}\begin{bmatrix}0&1&0\\\ 1&0&\omega^{2}\\\
0&\omega^{2}&0\\\ \end{bmatrix}.$ (34)

The neutrino mass matrix is constructed by taking the contribution from Type-I
and Type-II seesaw mechanisms. Now, to move to a basis where the charged
lepton mass matrix is diagonal, we incorporate a certain transformation
$(U_{l_{L}}^{T}M_{\nu_{s}}U_{l_{L}})$ and obtain the effective neutrino mass
matrix as shown below,

$M^{1}_{\nu}=\begin{bmatrix}\frac{2r}{3}-\frac{3y_{1}v_{\Phi}^{2}}{2A}-\frac{3\sqrt{3}iy_{1}v_{\Phi}^{2}}{2A}&-\frac{2r}{3}&\frac{ir}{3}\\\
-\frac{2r}{3}&\frac{2r}{3}&\frac{3\sqrt{3}iy_{2}y_{3}v_{\Phi}^{2}}{2B}-i(\frac{r}{3}+\frac{3y_{2}y_{3}v_{\Phi}^{2}}{2B})\\\
\frac{ir}{3}&\frac{3\sqrt{3}iy_{2}y_{3}v_{\Phi}^{2}}{2B}-i(\frac{r}{3}+\frac{3y_{2}y_{3}v_{\Phi}^{2}}{2B})&\frac{4r}{3}\\\
\end{bmatrix},$ (35)

where, $r=y_{T_{2}}v_{\Delta}$. It is to be highlighted that to obtain the
texture $M^{1}_{\nu}$, we choose $\eta=\frac{\pi}{6}$, $\kappa=\frac{3\pi}{6}$
and $\xi=\frac{2\pi}{6}$.

The proposed texture $M^{2}_{\nu}$ can be derived from the same Lagrangian
appearing in Eq. (30) under the specific choice of vev
$\langle\Delta\rangle_{0}=v_{\Delta}(0,1,0)^{T}$ and by setting
$\eta=\frac{\pi}{2}$, $\kappa=\frac{\pi}{6}$ and $\xi=\frac{\pi}{3}$. We give
the product rules under $A_{4}$ in appendix C.

##  IV Summary

To summarize, we propose two new neutrino mass matrix textures $M_{\nu}^{1}$
and $M_{\nu}^{2}$ deviating from $\mu-\tau$ symmetry and study their
phenomenological implications. The mixing schemes provided by both textures
are consistent with experimental observations. As a notable feature, both
textures predict $\theta_{23}$ in the lower octant for normal ordering of
neutrino masses. On the other hand, for inverted ordering, $M_{\nu}^{1}$
predicts $\theta_{23}$ in the lower octant and $M_{\nu}^{2}$ is ruled out in
the light of experimental observations. In addition, the Dirac CP phase
$\delta$ is constrained to a sharp bound. We aim to realize these proposed
textures within the framework of the $SU(2)_{L}\times A_{4}\times Z_{3}$
symmetry group in light of the seesaw mechanism.

As a future prospect, we aim to study the stability of the proposed textures
under renormalization group running. In addition, the present work can be
extended to study the baryon asymmetry based upon the proposed textures.

##  V Acknowledgement

The research work of PC is supported by Innovation in Science Pursuit for
Inspired Research (INSPIRE), Department of Science and Technology, Government
of India, New Delhi vide grant No. IF190651.

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##  Appendix A Tables and Figures

Parameters |  Input values
---|---
$m_{1}/eV$ |  0.02-0.027
$\alpha/^{\circ}$ |  109-109.5
$\beta/^{\circ}$ |  266.5-267.5
$\phi_{1}/^{\circ}$ |  39.9-40
$\phi_{2}/^{\circ}$ |  0.5-1
$\phi_{3}/^{\circ}$ |  0.5-1
Table 1: Numerical values of the input parameters for texture $M^{1}_{\nu}$ (Normal Ordering) Parameters |  Input values
---|---
$m_{3}/eV$ |  0.012-0.015
$\alpha/^{\circ}$ |  276-276.4
$\beta/^{\circ}$ |  108.2-108.5
$\phi_{1}/^{\circ}$ |  74-74.4
$\phi_{2}/^{\circ}$ |  0.3-0.6
$\phi_{3}/^{\circ}$ |  0.3-0.6
Table 2: Numerical values of the input parameters for texture $M^{1}_{\nu}$ (Inverted Ordering) Parameters |  Input values
---|---
$m_{1}/eV$ |  0.022-0.027
$\alpha/^{\circ}$ |  81-81.5
$\beta/^{\circ}$ |  76-76.3
$\phi_{1}/^{\circ}$ |  14.5-15.5
$\phi_{2}/^{\circ}$ |  53.2-53.5
$\phi_{3}/^{\circ}$ |  17-17.2
Table 3:  Numerical values of the input parameters for texture $M^{2}_{\nu}$ (Normal Ordering) Parameters |  Input values
---|---
$m_{3}/eV$ |  0.01-0.014
$\alpha/^{\circ}$ |  66.5-67.5
$\beta/^{\circ}$ |  47.5-48
$\phi_{1}/^{\circ}$ |  20-20.5
$\phi_{2}/^{\circ}$ |  80-80.5
$\phi_{3}/^{\circ}$ |  97-97.5
Table 4:  Numerical values of the input parameters for texture $M^{2}_{\nu}$ (Inverted Ordering) Fields |  $D_{l_{L}}$ |  $l_{R}$ |  $\nu_{l_{R}}$ |  $H$ |  $\Phi$ |  $\zeta$ |  $\Delta$
---|---|---|---|---|---|---|---
$SU(2)_{L}$ |  2 |  1 |  1 |  2 |  2 |  1 |  3
$A_{4}$ |  3 |  (1, 1′, 1′′) |  (1, 1′, 1′′) |  3 |  3 |  1 |  3
$Z_{3}$ |  1 |  ($\omega^{2}$, $\omega^{2}$, $\omega^{2}$) |  ($\omega$, $\omega$, $\omega$) |  $\omega$ |  $\omega^{2}$ |  $\omega$ |  1
Table 5: The transformation properties of various fields under
$SU(2)_{L}\times A_{4}\times Z_{3}$.

Figure 1: The correlation plots between (a) $\theta_{23}$ and $\delta$. (b)
$\theta_{12}$ and $\theta_{13}$. (c) $\Delta m^{2}_{31}$ and $\Delta
m^{2}_{21}$ for normal ordering of neutrino masses. The blue and green strips
represent the $3\sigma$ bounds for the respective parameters.

Figure 2: The correlation plots between (a) $\theta_{23}$ and $\delta$. (b)
$\theta_{12}$ and $\theta_{13}$. (c) $\Delta m^{2}_{31}$ and $\Delta
m^{2}_{21}$ for inverted ordering of neutrino masses. The blue and green
strips represent the $3\sigma$ bounds for the respective parameters.

Figure 3: The correlation plots between (a) $\theta_{23}$ and $\delta$. (b)
$\theta_{12}$ and $\theta_{13}$. (c) $\Delta m^{2}_{31}$ and $\Delta
m^{2}_{21}$ for normal ordering of neutrino masses. The blue and green strips
represent the $3\sigma$ bounds for the respective parameters.

Figure 4: The correlation plots between (a) $\theta_{23}$ and $\delta$. (b)
$\theta_{12}$ and $\theta_{13}$. (c) $\Delta m^{2}_{31}$ and $\Delta
m^{2}_{21}$ for inverted ordering of neutrino masses. The blue and green
strips represent the $3\sigma$ bounds for the respective parameters.

Figure 5:  (a) Shows the plot of effective Majorana neutrino mass
$m_{\beta\beta}$ against the lightest neutrino mass for the texture
$M_{\nu}^{1}$. (b) Presents the plot of effective Majorana neutrino mass
$m_{\beta\beta}$ against the lightest neutrino mass for the texture
$M_{\nu}^{2}$. The red and blue data points stand for normal and inverted
orderings respectively.

##  Appendix B The Scalar Potential

The $SU(2)_{L}\times A_{4}\times Z_{3}$ invariant scalar potential can be
constructed as shown in the following,

$\displaystyle V$ $\displaystyle=$ $\displaystyle
V(H)+V(\Phi)+V(\Delta)+V(\zeta)+V(H,\Phi)+V(H,\Delta)+V(H,\zeta)+V(\Phi,\Delta)$
(36) $\displaystyle+V(\Phi,\zeta)+V(\Delta,\zeta),$

where,

$\displaystyle V(H)$ $\displaystyle=$
$\displaystyle\mu^{2}_{H}(H^{\dagger}H)+\lambda^{H}_{1}(H^{\dagger}H)(H^{\dagger}H)+\lambda^{H}_{2}(H^{\dagger}H)_{1^{\prime}}(H^{\dagger}H)_{1^{\prime\prime}}+\lambda^{H}_{3}(H^{\dagger}H)_{3_{s}}$
(37)
$\displaystyle(H^{\dagger}H)_{3_{s}}+\lambda^{H}_{4}(H^{\dagger}H)_{3_{s}}(H^{\dagger}H)_{3_{a}}+\lambda^{H}_{5}(H^{\dagger}H)_{3_{a}}(H^{\dagger}H)_{3_{a}},$
$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\Phi}(\Phi^{\dagger}\Phi)+\lambda^{\Phi}_{1}(\Phi^{\dagger}\Phi)(\Phi^{\dagger}\Phi)+\lambda^{\Phi}_{2}(\Phi^{\dagger}\Phi)_{1^{\prime}}(\Phi^{\dagger}\Phi)_{1^{\prime\prime}}+\lambda^{\Phi}_{3}(\Phi^{\dagger}\Phi)_{3_{s}}(\Phi^{\dagger}\Phi)_{3_{s}}$
(38)
$\displaystyle+\lambda^{\Phi}_{4}(\Phi^{\dagger}\Phi)_{3_{s}}(\Phi^{\dagger}\Phi)_{3_{a}}+\lambda^{\Phi}_{5}(\Phi^{\dagger}\Phi)_{3_{a}}(\Phi^{\dagger}\Phi)_{3_{a}},$

$\displaystyle V(\Delta)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\Delta}Tr(\Delta^{\dagger}\Delta)+\lambda^{\Delta}_{1}Tr(\Delta^{\dagger}\Delta)Tr(\Delta^{\dagger}\Delta)+\lambda^{\Delta}_{2}Tr(\Delta^{\dagger}\Delta)_{1^{\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime\prime}}$
(39)
$\displaystyle+\lambda^{\Delta}_{3}Tr(\Delta^{\dagger}\Delta)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{s}}+\lambda^{\Delta}_{4}Tr(\Delta^{\dagger}\Delta)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}$
$\displaystyle+\lambda^{\Delta}_{5}Tr(\Delta^{\dagger}\Delta)_{3_{a}}Tr(\Delta^{\dagger}\Delta)_{3_{a}},$
$\displaystyle V(\zeta)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\zeta}(\zeta^{\dagger}\zeta)+\lambda^{H}(\zeta^{\dagger}\zeta)(\zeta^{\dagger}\zeta),$
(40) $\displaystyle V(H,\Phi)$ $\displaystyle=$
$\displaystyle\lambda^{H\Phi}_{1}(H^{\dagger}H)(\Phi^{\dagger}\Phi)+\lambda^{H\Phi}_{2}[(H^{\dagger}H)_{1^{\prime}}(\Phi^{\dagger}\Phi)_{1^{\prime\prime}}+(H^{\dagger}H)_{1^{\prime\prime}}(\Phi^{\dagger}\Phi)_{1^{\prime}}]$
(41)
$\displaystyle+\lambda^{H\Phi}_{3}(H^{\dagger}H)_{3_{s}}(\Phi^{\dagger}\Phi)_{3_{s}}+\lambda^{H\Phi}_{4}[(H^{\dagger}H)_{3_{s}}(\Phi^{\dagger}\Phi)_{3_{a}}+(H^{\dagger}H)_{3_{a}}(\Phi^{\dagger}\Phi)_{3_{s}}]$
$\displaystyle+\lambda^{H\Phi}_{5}(H^{\dagger}H)_{3_{a}}(\Phi^{\dagger}\Phi)_{3_{a}},$
$\displaystyle V(H,\Delta)$ $\displaystyle=$
$\displaystyle\lambda^{H\Delta}_{1}(H^{\dagger}H)Tr(\Delta^{\dagger}\Delta)+\lambda^{H\Delta}_{2}[(H^{\dagger}H)_{1^{\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime\prime}}+(H^{\dagger}H)_{1^{\prime\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime}}]$
(42)
$\displaystyle+\lambda^{H\Delta}_{3}(H^{\dagger}H)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{s}}+\lambda^{H\Delta}_{4}[(H^{\dagger}H)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}+(H^{\dagger}H)_{3_{a}}$
$\displaystyle
Tr(\Delta^{\dagger}\Delta)_{3_{s}}]+\lambda^{H\Delta}_{5}(H^{\dagger}H)_{3_{a}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}+\mu_{1}(H^{T}i\sigma_{2}\Delta^{\dagger}H),$
$\displaystyle V(H,\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{H\zeta}_{1}(H^{\dagger}H)(\zeta^{\dagger}\zeta),$ (43)
$\displaystyle V(\Phi,\Delta)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\Delta}_{1}(\Phi^{\dagger}\Phi)Tr(\Delta^{\dagger}\Delta)+\lambda^{\Phi\Delta}_{2}[(\Phi^{\dagger}\Phi)_{1^{\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime\prime}}+(\Phi^{\dagger}\Phi)_{1^{\prime\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime}}]$
(44)
$\displaystyle+\lambda^{\Phi\Delta}_{3}(\Phi^{\dagger}\Phi)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{s}}+\lambda^{\Phi\Delta}_{4}[(\Phi^{\dagger}\Phi)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}+(\Phi^{\dagger}\Phi)_{3_{a}}$
$\displaystyle
Tr(\Delta^{\dagger}\Delta)_{3_{s}}]+\lambda^{\Phi\Delta}_{5}(\Phi^{\dagger}\Phi)_{3_{a}}Tr(\Delta^{\dagger}\Delta)_{3_{a}},$
$\displaystyle V(\Phi,\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\zeta}_{1}(\Phi^{\dagger}\Phi)(\zeta^{\dagger}\zeta),$
(45) $\displaystyle V(\Delta,\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta\zeta}_{1}Tr(\Delta^{\dagger}\Delta)(\zeta^{\dagger}\zeta),$
(46)

Based on the minimization of the scalar potential, the following equations
hold good for the chosen vacuum expectation values (VEVs) $\langle
H\rangle_{\circ}=v_{H}\,(1,1,1)^{T}$,
$\langle\Phi\rangle_{\circ}=v_{\Phi}\,(1,1,1)^{T}$,
$\langle\Delta\rangle_{\circ}=v_{\Delta}\,(0,1,0)^{T}$ and
$\langle\zeta\rangle_{\circ}=v_{\zeta}$ :

$\displaystyle\frac{dV}{dH_{1}}$ $\displaystyle=$ $\displaystyle
v_{H}(\mu_{H}^{2}+(\lambda_{1}^{H\Delta}-\lambda_{2}^{H\Delta})v_{\Delta}^{2}+2\mu_{1}v_{\Delta}+\lambda_{1}^{H\zeta}v_{\zeta}^{2}+\left(6\lambda_{1}^{H}+8\lambda_{3}^{H}\right)$
(48) $\displaystyle v_{H}^{2}+3\lambda_{1}^{\Phi
H}v_{\Phi}^{2}+4\lambda_{3}^{\Phi H}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{dH_{2}}$ $\displaystyle=$ $\displaystyle
v_{H}(\mu_{H}^{2}+(6\lambda_{1}^{H}+8\lambda_{3}^{H})v_{H}^{2}+(\lambda_{1}^{H\Delta}+2\lambda_{2}^{H\Delta})v_{\Delta}^{2}+\lambda_{1}^{H\zeta}v_{\zeta}^{2}+3\lambda_{1}^{\Phi
H}v_{\Phi}^{2}$ (49) $\displaystyle+4\lambda_{3}^{\Phi H}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{dH_{3}}$ $\displaystyle=$ $\displaystyle
v_{H}(\mu_{H}^{2}+(6\lambda_{1}^{H}+8\lambda_{3}^{H})v_{H}^{2}+(\lambda_{1}^{H\Delta}-\lambda_{2}^{H\Delta})v_{\Delta}^{2}+\lambda_{1}^{H\zeta}v_{\zeta}^{2}+3\lambda_{1}^{\Phi
H}v_{\Phi}^{2}$ (50) $\displaystyle+4\lambda_{3}^{\Phi H}v_{\Phi}^{2})=0,$

$\displaystyle\frac{dV}{d\Phi_{1}}$ $\displaystyle=$ $\displaystyle
v_{\Phi}(3v_{H}^{2}\lambda_{1}^{\Phi H}+4v_{H}^{2}\lambda_{3}^{\Phi
H}+\mu_{\Phi}^{2}+\lambda_{1}^{\Phi\Delta}v_{\Delta}^{2}-\lambda_{2}^{\Phi\Delta}v_{\Delta}^{2}+\lambda_{1}^{\Phi\zeta}v_{\zeta}^{2}+6\lambda_{1}^{\Phi}v_{\Phi}^{2}$
(51) $\displaystyle+8\lambda_{3}^{\Phi}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{d\Phi_{2}}$ $\displaystyle=$ $\displaystyle
v_{\Phi}(3v_{H}^{2}\lambda_{1}^{\Phi H}+4v_{H}^{2}\lambda_{3}^{\Phi
H}+\mu_{\Phi}^{2}+\lambda_{1}^{\Phi\Delta}v_{\Delta}^{2}+2\lambda_{2}^{\Phi\Delta}v_{\Delta}^{2}+\lambda_{1}^{\Phi\zeta}v_{\zeta}^{2}+6\lambda_{1}^{\Phi}v_{\Phi}^{2}$
(52) $\displaystyle+8\lambda_{3}^{\Phi}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{d\Phi_{3}}$ $\displaystyle=$ $\displaystyle
v_{\Phi}(3v_{H}^{2}\lambda_{1}^{\Phi H}+4v_{H}^{2}\lambda_{3}^{\Phi
H}+\mu_{\Phi}^{2}+\lambda_{1}^{\Phi\Delta}v_{\Delta}^{2}-\lambda_{2}^{\Phi\Delta}v_{\Delta}^{2}+\lambda_{1}^{\Phi\zeta}v_{\zeta}^{2}+6\lambda_{1}^{\Phi}v_{\Phi}^{2}$
(53) $\displaystyle+8\lambda_{3}^{\Phi}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{d\Delta_{1}}$ $\displaystyle=$ $\displaystyle
2v_{H}^{2}((\lambda_{3}^{H\Delta}-\lambda_{4}^{\Delta})v_{\Delta}+\mu_{1})+2(\lambda_{3}^{\Phi\Delta}-\lambda_{4}^{\Phi\Delta})v_{\Delta}v_{\Phi}^{2}+3v_{1}\lambda_{1}^{\Phi\Delta}v_{\Phi}^{2}=0,$
(54) $\displaystyle\frac{dV}{d\Delta_{2}}$ $\displaystyle=$ $\displaystyle
v_{\Phi}^{2}(3\lambda_{1}^{H\Delta}v_{\Delta}+2\mu_{1})+v_{\Delta}(\mu_{\Delta}^{2}+3v_{H}^{2}\lambda_{1}^{\Phi\Delta}+(2\lambda_{1}^{\Delta}+i(\sqrt{3}+i)\lambda_{2}^{\Delta})v_{\Delta}^{2}$
(55) $\displaystyle+\lambda_{1}^{\Delta\zeta}v_{\zeta}^{2})=0,$
$\displaystyle\frac{dV}{d\Delta_{3}}$ $\displaystyle=$ $\displaystyle
2(v_{H}^{2}((\lambda_{3}^{H\Delta}+\lambda_{4}^{H\Delta})v_{\Delta}+\mu_{1})+(\lambda_{3}^{\Phi\Delta}+\lambda_{4}^{\Phi\Delta})v_{\Delta}v_{\Phi}^{2})=0,$
(56) $\displaystyle\frac{dV}{d\zeta}$ $\displaystyle=$ $\displaystyle
2v_{\zeta}(\mu_{\zeta}^{2}+3v_{H}^{2}\lambda_{1}^{H\zeta}+\lambda_{1}^{\Delta\zeta}v_{\Delta}^{2}+2\lambda^{\zeta}v_{\zeta}^{2}+3\lambda_{1}^{\Phi\zeta}v_{\Phi}^{2})=0.$
(57)

In a similar way, the chosen VEVs $\langle
H\rangle_{\circ}=v_{H}\,(1,1,1)^{T}$,
$\langle\Phi\rangle_{\circ}=v_{\Phi}\,(1,1,1)^{T}$,
$\langle\Delta\rangle_{\circ}=v_{\Delta}\,(\omega^{2},0,1)^{T}$ and
$\langle\zeta\rangle_{\circ}=v_{\zeta}$ can be justified based on the
minimization conditions of the scalar potential.

##  Appendix C Product Rules of $A_{4}$

The discrete group $A_{4}$ is a subgroup of $SU(3)$ which has 12 elements. The
group has four irreducible representation out of which three are one
dimensional and one three dimensional. In S-basis, the product of two triplets
gives,

$3\times 3=1+1^{\prime}+1^{\prime\prime}+3_{S}+3_{A},$ (58)

where,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle(a_{1}\bar{b_{1}}+a_{2}\bar{b_{2}}+a_{3}\bar{b_{3}})$ (59)
$\displaystyle 1^{\prime}$ $\displaystyle=$
$\displaystyle(a_{1}\bar{b_{1}}+\omega^{2}a_{2}\bar{b_{2}}+\omega
a_{3}\bar{b_{3}})$ (60) $\displaystyle 1^{\prime\prime}$ $\displaystyle=$
$\displaystyle(a_{1}\bar{b_{1}}+\omega
a_{2}\bar{b_{2}}+\omega^{2}a_{3}\bar{b_{3}})$ (61) $\displaystyle(3\times
3)_{S}$ $\displaystyle=$ $\displaystyle\begin{bmatrix}a_{2}b_{3}+a_{3}b_{2}\\\
a_{3}b_{1}+a_{1}b_{3}\\\ a_{1}b_{2}+a_{2}b_{1}\\\ \end{bmatrix},$ (62)
$\displaystyle(3\times 3)_{A}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}a_{2}b_{3}-a_{3}b_{2}\\\ a_{3}b_{1}-a_{1}b_{3}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ \end{bmatrix}.$ (63)

The trivial singlet can be obtained from the following singlet product rules,

$1\times 1=1,\,\,1^{\prime}\times
1^{\prime\prime}=1,\,\,1^{\prime\prime}\times 1^{\prime}=1.$ (64)


EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-LHCb-DP-2019-002

A comprehensive real-time analysis model at the LHCb experiment

An evolved real-time data processing strategy is proposed for high-energy
physics experiments, and its implementation at the LHCb experiment is
presented. The reduced event model allows not only the signal candidate firing
the trigger to be persisted, as previously available, but also an arbitrary
set of other reconstructed or raw objects from the event. This allows for
higher trigger rates for a given output data bandwidth, when compared to the
traditional model of saving the full raw detector data for each trigger,
whilst accommodating inclusive triggers and preserving data mining
capabilities. The gains in physics reach and savings in computing resources
already made possible by the model are discussed, along with the prospects of
employing it more widely for Run 3 of the Large Hadron Collider.

Published in JINST 14 P04006

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

Experimental tests of the Standard Model must become ever more precise if
small effects due to new physics are to be observed. To meet this challenge,
at the Large Hadron Collider (LHC) both the centre-of-mass energy of the
colliding proton-proton beams and the instantaneous luminosity delivered to
the experiments are periodically increased. The corresponding increase in
signal production rate must be balanced against the availability of
computational resources required to store the data for offline analysis. The
disk space required is given by the product of the running time of the
experiment and the trigger output bandwidth defined as

$\text{Bandwidth}\,[$\mathrm{MB}\text{\,}{\mathrm{s}}^{-1}$]\propto\text{Trigger
output rate}\,[$\mathrm{kHz}$]\times\text{Average event
size}\,[$\mathrm{kB}$].$

When the output rate of any given trigger is dominated by events containing
signal processes, tightening the selection further to reduce the output rate
is undesirable. The size on disk of the full raw detector information cannot
be decreased beyond the standard zero-suppression and compression techniques.
Therefore, reduced event formats must be employed instead, wherein a subset of
the high-level reconstructed event information computed in the final software
trigger stage is recorded and sent to permanent storage. As the rate of signal
processes increases, so must the reliance on reduced event formats.

The CMS, LHCb, and ATLAS experiments utilised their own versions of reduced
formats during Run 2 (2015–2018) [1, 2, 3]. Typically, the reduced format
contains information pertaining only to the reconstructed physics objects
which passed the relevant set of trigger selections, as well as some event
summary information. For a jet trigger, for example, the objects may be the
momentum vector and particle multiplicity value of the highest energy jet in
an event. For a heavy flavour decay trigger it may be a set of four momenta
and decay vertex positions. Such an approach provides a maximal reduction in
persisted event size whilst still allowing for the analysis performed in the
trigger to be continued offline. This allows for higher trigger rates for a
given output bandwidth, extending the physics reach of an experiment within
limited computational resources. However, it also restricts the utility of an
event for broader analysis and data mining. Such a reduction is then
unsuitable for inclusive triggers, which constitute a large fraction of
trigger output rates, as well as for analyses in which other information may
be required later, such as performing flavour tagging or isolation studies,
the exact input to which is not well-defined at the time of the trigger
decision.

The LHCb experiment has pioneered the widespread usage of a reduced event
format since the beginning of Run 2. This has been driven by the desire to
continue the rich charm physics programme started in Run 1 (2010–2012) in
spite of the large charm production rate in the detector acceptance, being 25
times larger than that for beauty [4, 5]. During Run 2, almost all events
selected by charm triggers at LHCb were persisted in a reduced format,
enabling a broader physics programme which would otherwise not be possible
within the available resources. In Run 3 (2021–2023) the instantaneous
luminosity delivered to the LHCb experiment will increase by a factor of five.
The rate of events processed by the trigger that contain charm or beauty
hadrons will scale not only by this factor, but also by a factor of two due to
the implementation of a higher-efficiency full software trigger [6, 7].
Coupled with the larger raw data size produced by the upgraded detector, it is
necessary for some fraction of the beauty programme to migrate to reduced
event formats in order to fit within available computational resources. Given
the strong reliance the beauty programme has on inclusive triggers [8, 9], the
style of reduced event formats previously described are not sufficient, and so
this model must be extended if the charm and beauty programmes are to be
sustained into the future.

A significant evolution of the reduced event model is proposed here, in which
additional reconstructed objects in the event are selected and persisted after
the trigger decision has been made. When implemented with sufficient
flexibility, this allows for fine-grained tuning between trigger output
bandwidth and event utility offline. Since mid-2017, such a scheme has been
adopted in the LHCb trigger, resulting in substantial bandwidth savings
without a loss of physics reach. The reduction in bandwidth has allowed for
the introduction of new trigger selections, increasing the physics scope of
the experiment. It is foreseen that the reduced event format will be adopted
for a majority of the LHCb physics programme in Run 3 [10]. The latest
evolution, presented here, now caters to all use cases.

The rest of this paper is organised as follows. The LHCb detector and trigger
system is described in Section 2, and the need for a fully flexible reduced
event format in Run 2 is motivated quantitatively. A technical overview of the
new format is given in Section 3. The benefits already achieved in Run 2 and
the prospects for Run 3 are presented in Sections 4 and 5. A summary is given
in Section 6.

##  2 The LHCb detector and trigger strategy

The LHCb detector is a forward-arm spectrometer designed to measure the
production and decay properties of charm and beauty hadrons with high
precision [11, 12]. Such objects are predominantly produced at small angles
with respect to the proton beam axis [13]. A tracking system is used to
reconstruct the trajectories of charged particles. It consists of a silicon
vertex detector surrounding the interaction region (VELO), a silicon strip
detector located upstream of a dipole magnet (TT), and three tracking stations
downstream of the magnet. The latter each consist of a silicon strip detector
in the high-intensity region close to the beamline (IT) and a straw-tube
tracker in the regions further from the beamline (OT). Neutral particles are
identified with a calorimeter system made of a scintillating pad detector
(SPD), an electromagnetic calorimeter (ECAL) preceded by a pre-shower detector
(PS), and a hadronic calorimeter (HCAL). Charged particle identification is
provided by combining information from the ring-imaging Cherenkov detectors
(RICH1 and RICH2), wire chambers used to detect muons, and the calorimeter
system. Shower counters located in high-rapidity regions either side of the
main detector can be used to veto events with particles produced at a very low
angle to the beam, mainly used for studies of central exclusive production
[14]. The LHC provides proton-proton collisions to the experiment at a rate of
$30\text{\,}\mathrm{MHz}$ during most run periods. The detector also operates
during special physics runs, such as heavy ion collisions with lead and xenon
nuclei, and in a fixed target mode where a noble gas is injected in the
interaction region [15, 16].

During proton-proton operation of the LHC, an interesting physics signal
process occurs at a rate of around $10\text{\,}\mathrm{Hz}$. To filter out the
large majority of collisions that do not contain interesting information, and
to fit the experimental output rate into the available computing resources, a
three-stage trigger system is employed [9], illustrated in Fig. 1.

Figure 1:  An overview of the LHCb trigger scheme in Run 2 [9]. The green
boxes represent trigger stages, the blue ellipses represent storage units, and
arrows between elements represent data flow, labelled with approximate flow
rates. Events that reach a terminal storage unit are kept for offline
analysis.

In total, it reduces the event rate by three orders of magnitude, with the
output being sent to permanent storage. The level-0 hardware trigger (L0) uses
information from the calorimeter and muon systems to compute a decision using
field-programmable gate arrays within a fixed latency of
$4\text{\,}\mathrm{\SIUnitSymbolMicro s}$. Events are selected by the L0 at a
rate of about $1\text{\,}\mathrm{MHz}$, and are sent to an Event Filter Farm
(EFF) where they are processed by the two-stage High Level Trigger (HLT) on
commodity processors. The first stage, HLT1, uses tracking and calorimetry
information to perform a partial reconstruction of charged particles, and
writes the raw information for each passing event to a
$10\text{\,}\mathrm{PB}$ disk buffer at a rate of around
$110\text{\,}\mathrm{kHz}$. Asynchronously to HLT1, the data in the buffer are
used to compute alignment and calibration constants which, if significantly
different from previous runs, are saved to a conditions database [17]. The
second stage of the software trigger, HLT2, performs a full event
reconstruction using the latest alignment and calibration constants and all
available detector information. Selections in HLT2 span a spectrum from
_inclusive_ selections, which require the presence of a heavy flavour decay
signature such as a displaced multi-body vertex or a high transverse momentum
lepton, and _exclusive_ selections, which fully reconstruct signal decays.
Events are written to offline storage from HLT2 at a rate of around
$12.5\text{\,}\mathrm{kHz}$.

Within the HLT, an event comprises a set of so-called sub-detector raw banks,
each containing the zero-suppressed readout of a given sub-detector. Each
trigger stage adds a set of trigger raw banks to the event which summarise how
the event passed that stage. Selected events are persisted to a set of
_streams_ in permanent storage. During this _streaming_ , different raw banks
may be kept or removed depending on the stream the event is sent to. Each
trigger selection in HLT2 is associated to a particular stream, and an event
is sent to a stream if it passes at least one associated selection. An event
may be sent to multiple streams, and different sets of raw banks for that
event can be saved in each stream. The LHCb physics trigger rate is
distributed over three streams:

  1. 1.

Around half of all triggered events enter the _full_ stream, which contains
the full set of sub-detector and trigger raw banks, with the trigger
reconstruction being discarded. Events in the full stream are reconstructed by
a separate application offline, followed by a set of physics selections that
reconstruct thousands of decay chains for direct analysis. This scheme is used
in many other experiments, where it is common for the trigger reconstruction
to be of poorer quality to that of the offline reconstruction in order to fit
within the timing constraints of their online systems.

  2. 2.

Around a third of triggered events are sent to the _Turbo_ stream, where a
reduced event format is persisted in a dedicated raw bank, described in detail
in Section 3.

  3. 3.

The remaining trigger rate is captured by the _TurCal_ calibration stream,
where both the reduced and full formats are kept.

The motivation for creating the reduced format used in the Turbo and TurCal
streams now follows.

###  2.1 Real-time analysis

Here, ‘real time’ is defined as the interval between a collision occurring and
the point at which the corresponding event must be either discarded forever or
sent offline for permanent storage. In most high-energy physics experiments,
this interval cannot exceed the time between collisions as their trigger
systems are synchronous. A multi-stage trigger increases the available event
processing time from one stage to the next; however each stage requires more
computing power than the previous one. The addition of a disk buffer between
two stages increases the effective computation time allowed in the latter
stage as that stage can then also operate during periods when data are not
being taken. The buffer also permits the execution of tasks that provide
additional input to the latter trigger stage, such as the running of alignment
and calibration algorithms.

During Run 1, LHCb installed a disk buffer such that $20\text{\,}\mathrm{\char
37\relax}$ of the L0 output was deferred to disk. This increased the effective
processing power of the software trigger by allowing it to run both during
beam operation and when the LHC was in downtime. Between Run 1 and Run 2, the
buffer size was increased to $10\text{\,}\mathrm{PB}$, with events being
buffered between the two software trigger stages. As the buffer is so large,
‘real time’ can be up to two weeks during nominal data-taking [9]. The
possibility to buffer the data for such a long period has allowed for the
execution of the aforementioned real-time detector alignment and calibration
on the data in the buffer. The increase in computing power has permitted the
implementation of the full offline reconstruction in HLT2. With these
substantial additions, the HLT2 reconstruction is of equal quality to what is
achieved in the offline processing, such that full physics selections are
performed in real time in the trigger without a loss of precision. This
permits trigger selections to be much closer or identical to those applied in
the offline analysis, as there are no resolution effects between online and
offline to be accounted for. Such a scheme reduces experimental systematic
uncertainties and saves money by making tighter trigger selections acceptable,
therefore reducing the rate and output bandwidth.

With an offline-quality reconstruction in the final trigger stage (HLT2), it
is no longer necessary to run another reconstruction offline. Instead, the
objects created by trigger selections are written out to the permanent storage
directly. Physics measurements are performed on these objects. This technique
almost eliminates processing requirements offline and reduces output
bandwidth, if the relevant subset of the reconstruction is smaller than the
raw event. Offline analysis of trigger-level information has been used at the
CMS experiment [1], called ‘scouting’, and at the ATLAS experiment [3], called
‘trigger-object level analysis’. Although the method has extended their
physics reach, in both cases only object summaries are available for analysis,
rather than offline-equivalent constructs, and the quality of the
reconstruction is worse than that achieved offline. Since 2015, the LHCb
experiment has employed a persistence model called ‘Turbo’. It allows the
offline-equivalent information computed in HLT2 to be saved, sacrificing
neither physics performance nor analyst convenience, as existing tools are
used to process the data.

Until recently, the reduced event formats employed by the LHC experiments have
described only the objects that enter into some trigger selection, such as a
jet or a charm hadron and its decay products. These formats are best used to
reduce the output bandwidth of events captured by exclusive trigger
selections. In contrast, an inclusive trigger selection, which does not
necessarily consider all the information on the physics process of interest in
its decision, cannot be accommodated. In order to be able to cater for such
use cases, an advanced reduced persistency model has recently been developed
as an evolution of the Turbo model, and is described in the following Section.

##  3 The Turbo data processing model

The Turbo data processing model has evolved considerably over the course of
Run 2. The initial prototype, established in 2015 and described in detail in
Ref. [2], saved the set of objects associated to individual reconstructed
decay cascades extracted from events fully reconstructed in the trigger. This
exploits the event topology characteristic of hadron colliders wherein only a
small subset of objects produced in the collision are relevant for analysis
offline. In a hard proton-proton scatter at LHCb, on average 40 tracks are
associated to the resulting primary vertex, whereas only 2–6 tracks are
required to reconstruct a typical heavy flavour decay. Significant savings in
persisted event size are then possible by discarding reconstructed objects not
needed in the offline analysis. In this section, three new developments to the
Turbo processing model are described, which together allow for all LHCb
analyses to be accommodated.

In order to perform physics analyses with the output of the trigger
reconstruction, decay candidates must appear in the same format used by
existing analysis tools. Containers of physics object classes are serialised
per event into raw banks, as illustrated in Fig. 2, in order to conform to the
trigger output format. This format is optimised for simple event-by-event
concatenation, rather than the heavily compressed format used offline.

Figure 2:  Data flow in saving and restoring trigger objects from online
(left) to offline (right) [2].

The initial prototype of the Turbo model [2] serialised only the candidates
that enter the trigger decision. To allow for additional objects to be
persisted, the serialisation framework used in the offline infrastructure was
adapted to work within the online system. This increases the compatibility
between analyses using the online and offline reconstructions, and requires
that only one serialisation framework be maintained. Furthermore, trigger
selection configurations which use the Turbo model were extended to allow for
a set of ‘additional selections’ to be run _after_ the trigger decision has
been made. These allow for any reconstructed object to be captured. At the end
of each event processing, the set of all C++ physics objects selected by all
trigger lines using the Turbo model is copied to a common location in memory.
The instances in the copied location are then compressed and serialised into
raw banks [18], suitable for transfer within and out of the online system.

A dedicated application, called Tesla [2], runs offline to transform the HLT2
output into a format that is ready for analysis. This involves a file
conversion from serialised raw data to a more compressed format used by the
offline storage, as well as the computation and storage of information
necessary for luminosity determination. Tesla also ensures that additional
information calculated in the online reconstruction is accessible to standard
analysis tools, for example event-level sub-detector occupancies and
information calculated using the whole reconstructed event. When processing
simulated events, the application also matches reconstructed objects to true
simulated objects and stores the information as relations tables. In
comparison with the traditional model of an additional offline reconstruction,
the processing cost of running Tesla is negligible. In principle, some or even
all parts of the work done by Tesla can be moved into the trigger itself in
the LHCb upgrade, such as if compressed object containers were written
directly out of HLT2 instead of encoding them into raw banks.

With the serialisation and file preparation frameworks in place, different
levels of granularity on what physics objects to select are now available. In
the following, the resulting flexibility is explained, such that all
measurements can take advantage of the real-time model.

###  3.1 Standard Turbo model

A majority of trigger lines based on the Turbo model define exclusive
selections where the full decay is completely specified, and no additional
objects from the event are required for subsequent analysis. In this model,
the objects saved are:

  * •

The reconstructed decay chain that fired the line, which comprises:

    * –

The set of all tracks and neutral objects, calorimeter and PID information
relating to those objects, and decay vertices that form the candidate.

    * –

The tracking detector clusters associated to the candidate tracks, such that
the tracks can be re-fitted offline.

  * •

All of the reconstructed primary vertices in the event, which are necessary to
perform PV mis-association studies offline.

Other reconstructed objects in the event as well as the raw data are not kept
for offline processing. That allows for a significant reduction of the event
size and hence also trigger output bandwidth. Offline disk space and CPU
processing time are also saved as the offline reconstruction step is omitted.
This model has been operational since the beginning of data-taking in 2015,
and enabled the first LHCb Run 2 measurements to be presented 18 days after
the data were collected [4, 5].

###  3.2 Complete reconstruction persistence

In order to use the Turbo model for inclusive triggers, the ability to store
all the reconstructed objects in the event was introduced in the beginning of
2016. It is made available on a per-selection basis by a user flag. When
enabled, the full event reconstruction as performed in HLT2 is persisted in
addition to the information described in Section 3.1. In comparison to saving
the raw event, this approach reduces disk space usage and requires no further
processing offline. However, the information needed to re-run the
reconstruction offline is discarded. This technique permits high-rate
inclusive triggers that would otherwise not be feasible [19].

Persisting the whole reconstructed event is expensive in terms of event size
in comparison with only saving the trigger candidate, as shown in Table 1.

Table 1:  Average event sizes for trigger lines requesting varying levels of information persistence, measured on data collected in 2018.  Persistence method | Average event size ($\mathrm{kB}$)
---|---
Turbo | $7$
Selective persistence | $16$
Complete persistence | $48$
Raw event | $69$

In most cases, only a small fraction of the full reconstructed event is
required in an offline analysis. Therefore, a middle-ground between persisting
only the candidate or the whole reconstructed event is introduced.

###  3.3 Selective reconstruction persistence

Selective persistence allows for explicit specification of which information
is stored on top of the trigger candidate itself. This permits a significant
event size reduction without the usual sacrifice of allowing for exploratory
analysis offline. At the beginning of 2017, trigger lines were augmented with
additional selections that are executed after the trigger decision has been
computed. A typical additional selection captures objects in the event that
are somehow related to the trigger candidate. However, any selections are
possible.

Figure 3:  A cartoon of the same reconstructed event with varying levels of
object persistence: Turbo (top); selective persistence (middle); and complete
reconstruction persistence (bottom). Solid objects are those persisted in each
case. A trigger selection may also ask for one or more sub-detector raw banks
to also be stored, shown as solid rectangles.

As an example, consider a trigger using the Turbo model that reconstructs and
selects the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ charm decay, as
shown in Fig. 3, where the information specified in Section 3.1 is saved. If
complete reconstruction persistence is enabled for this line, the underlying
reconstructed event will also be stored. With selective persistence,
additional objects are instead specified explicitly, such as all charged pions
that are associated to the same PV as the ${D}^{0}$ and that form a good-
quality excited ${D}^{*\pm}$ candidate. The selection framework allows for
requirements to be made on both the pions themselves and on the
${{D}^{0}}{{\pi}^{\pm}}$ combination, but then only the pions that pass these
cuts are persisted. The ${D}^{*\pm}$ candidates are discarded, since it can be
built again exactly in an offline processing if required. Similar selections
can be added for other extra particles, such as kaons, photons, and hyperons
to support a wide spectrum of charm spectroscopy measurements with a single
trigger selection.

In the LHCb upgrade for Run 3, selective persistence is a key ingredient in
the migration of the physics programme to the real-time analysis model [10].
One use-case is flavour tagging, a determination of the initial flavour of a
beauty or anti-beauty meson. The decision of one or more tagging algorithms
together with the probability of the assigned flavour being wrong is computed
using a set of reconstructed objects in the event, namely objects that are
related to the same PV as the signal decay. This set can be loosely defined
upfront and added as additional selections to trigger lines which reconstruct
the signal beauty hadron decays of interest. As the tagging algorithms undergo
improvements during data-taking, the flavour tagging can be re-run offline
using the information captured by the additional selections. Initial studies
show that the set of reconstructed objects required as input to the tagging
algorithms constitute only around $10\text{\,}\mathrm{\char 37\relax}$ of the
space that would be required for persisting the full reconstruction.

###  3.4 Selective raw persistence

There are some cases in which saving all possible information from an event is
required. One important example is for efficiency measurements on calibration
samples. Today, detector-level efficiencies are determined from control
channels whose trigger lines save the full raw detector information. The high
bandwidth associated with doing this means that the choice and selection of
control channels is severely restricted. This impacts the calibration sample
size and leads to larger uncertainties on the efficiencies. The ability to
study the efficiency of a given aspect of the reconstruction requires saving
detector hits that were not used in the reconstructed object. However, the
efficiency of reconstructing tracks with the OT, for example, would require
saving only the raw banks associated with that sub-detector, and would not
need the banks associated with the calorimeter or the RICH sub-detectors. If a
more fine-grained, per-trigger specification of required raw banks was
possible, the resulting bandwidth savings would allow the calibration samples
to more than double in size, resulting in more precise efficiency
determinations and therefore more accurate physics measurements.

In order to accomplish this, a new algorithm has been developed and deployed.
On initialisation, it determines the list of trigger selections and their
requested raw banks from the trigger configuration, as shown in Fig. 4. On a
per-event basis, the decision of each trigger selection is examined and the
superset of the required raw banks is persisted to the corresponding output
stream. Raw banks not requested by any firing trigger line are discarded.

Figure 4:  Algorithm flow to decide whether a given raw bank is persisted for
the current event. The resulting list of banks persisted for a given event is
the superset of those required by the individual lines.

This algorithm has been running in the LHCb trigger since 2018. With it, each
trigger selection has complete control on additional information in the event
is persisted, from any object created in the reconstruction, to the raw
information created by any sub-detector. Therefore, the Turbo model is now
able to cater for any use case.

##  4 Achievements in Run 2

Since the beginning of Run 2, a considerable fraction of triggered events have
used the Turbo model. In 2018, the number of trigger lines using the Turbo
model was around $50\text{\,}\mathrm{\char 37\relax}$ of the number using the
traditional model, however the Turbo stream bandwidth was
$25\text{\,}\mathrm{\char 37\relax}$ that of the full stream due to the
reduced average event size. Physics measurements using the Turbo model include
charm and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ cross-sections [4,
5], the discovery of new ground-state and excited charm baryons [20, 21],
searches for dark photons [22], and the characterisation of charmonium
production within jets [19]. This Section summarises how the new Turbo model
has increased physics reach whilst reducing the event size since its
introduction in 2016.

As an example of the gains that can and have been made, we consider a subset
of trigger lines using the Turbo model that exclusively reconstruct and select
Cabibbo-favoured decays of ground-state charm hadrons, such as
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ and their charge
conjugates, originating directly from beam-beam interactions. These trigger
selections are intended for calibration studies, as the properties of these
objects and decays are well known, as well as for charm spectroscopy,
searching for and characterising excited charm states that cascade down to the
ground states. In 2016, these trigger lines were enhanced by the addition of
the complete reconstruction persistence, to allow for excited states to be
reconstructed offline, and so the average event size increased from
$7\text{\,}\mathrm{kB}$ in 2015 to $48\text{\,}\mathrm{kB}$.111 The average
size in 2016 across all events using the Turbo model was
$42\text{\,}\mathrm{kB}$, illustrating the dominating rate of these selected
processes  In 2017, the spectroscopy lines moved to the selective
reconstruction persistence, reducing the average event size to
$16\text{\,}\mathrm{kB}$. In turn, the bandwidth used by the Turbo stream
decreased from $139\text{\,}\mathrm{MB}\text{\,}{\mathrm{s}}^{-1}$ to
$79\text{\,}\mathrm{MB}\text{\,}{\mathrm{s}}^{-1}$. The newly-available
bandwidth was then utilised for new inclusive charm baryon trigger lines that
would have otherwise not been possible within the given computational
resources. The selections used to reduce the set of persisted objects from the
spectroscopy lines were aligned with the ones used for spectroscopy offline.
The additional selections are then $100\text{\,}\mathrm{\char 37\relax}$
efficient with respect to the offline selections by definition.

##  5 Prospects for Run 3

The LHCb detector will be upgraded for Run 3, where the instantaneous
luminosity will increase from that in Run 2 by a factor of five. With the
corresponding increase in heavy flavour production rate, in addition to
expected trigger efficiency gains and an increase in the raw event size, the
trigger output bandwidth will increase by an order of magnitude from Run 2.
Due to constraints on offline storage resources, the bandwidth is limited to a
maximum of $10\text{\,}\mathrm{GB}\text{\,}{\mathrm{s}}^{-1}$ [10]. The
breadth of the current experimental physics programme can only survive within
these resources if the fraction of trigger output rate sent to the Turbo
stream increases by over a factor of two. The flexibility of the reduced event
model described here has been designed to reach that goal without an inherent
loss of physics performance or reach. This Section briefly discusses some
relevant techniques.

In principle, reducing the amount of persisted information comes with risk, as
the set of information needed for a given measurement is not always known
upfront, and indeed the analysis itself has often not been conceived.
Information discarded in the trigger can be later required by some unforeseen
analysis offline. However, the factor-five increase in instantaneous
luminosity means a corresponding increase in the average number of visible
beam-beam interactions per bunch crossing. Therefore, a relatively safe
selection is to discard objects which are identified as originating from
primary vertices other than those associated to the signal trigger object.
This information is not relevant to an analysis of the trigger object as it is
unrelated to the signal process, having been produced from independent parton-
parton scatters.222Such a technique is similar in concept to jet grooming, in
that one wishes to keep only the objects associated to the hard scatter which
produced the heavy flavour quark-antiquark pair present in the event. Given a
fully reconstructed signal candidate, its associated primary vertex can be
defined as the one with the smallest impact parameter with respect to the
signal momentum vector. Primary vertices from which associated information
should not be persisted can then be identified using a minimum impact
parameter cut, the exact value of which can be tuned based on the expected
resolution available to distinguish separate primary vertices.

Inclusive trigger selections of heavy flavour decays present a particularly
challenging event size reduction problem as they have very high rates. The
potentially incomplete reconstruction of the signal momentum vector reduces
the accuracy of non-signal primary vertex suppression. Complimentary
selections that reduce the persisted information in inclusively triggered
events include:

  * •

The rejection of tracks that form a very poor quality vertex with the
inclusive candidate, as such tracks would not be used in an offline analysis;
and

  * •

The rejection of objects identified by a multivariate algorithm trained to
distinguish uninteresting objects from those associated to the portion of the
signal process captured by the inclusive selection.

Preliminary studies have shown such techniques can fully capture all signal
objects in an inclusively triggered event with an efficiency of around
$90\text{\,}\mathrm{\char 37\relax}$, compared to saving the full
reconstruction, whilst rejecting over $90\text{\,}\mathrm{\char 37\relax}$ of
the unrelated objects. This reduces the bandwidth of the prototype inclusive
beauty trigger under study from
$7.5\text{\,}\mathrm{GB}\text{\,}{\mathrm{s}}^{-1}$ to
$1\text{\,}\mathrm{GB}\text{\,}{\mathrm{s}}^{-1}$. While promising, further
work is needed to quantify any possible biases which may be associated to the
signal decays which are not completely captured by the selective persistence.

##  6 Summary and conclusions

Traditional reduced event formats allow for a broader physics programme within
available computational resources, but this is usually countered by a poorer
quality reconstruction in the trigger and reduced data mining capabilities.
Since 2015, a real-time alignment and calibration procedure between trigger
stages has allowed the LHCb experiment to deploy and exploit its Turbo model.
With this, offline-quality signal candidates are persisted directly from the
trigger for later analysis. This has been crucial for the charm programme,
which otherwise would have had to significantly compromise its reach and
diversity.

Since 2017, the implementation of the Turbo model has been overhauled and
extended to allow an arbitrary subset of the trigger reconstruction and raw
sub-detector information to be persisted along with the trigger candidate. As
such, the model is now capable of supporting the entirety of the experiment’s
broad research programme, and in particular the parts which rely on inclusive
trigger selections. Given the large increase in instantaneous luminosity and
trigger efficiency foreseen in Run 3, this evolution completes a crucial step
in allowing the continuation of today’s physics measurements into the future.

The updated Turbo model has already provided a $50\text{\,}\mathrm{\char
37\relax}$ reduction in bandwidth in comparison with saving the full
reconstruction, and this saving has been exploited with the addition of new
high-rate trigger selections. Even larger gains should be possible when
applying similar techniques to the remaining set of trigger selections, and
studies are ongoing into these avenues.

LHC

    Large Hadron Collider
PV

    primary vertex
IP

    impact parameter
EFF

    Event Filter Farm

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  *[LHC]: Large Hadron Collider
  *[EFF]: Event Filter Farm
  *[PV]: primary vertex