SideReferences Sidebar Task Abstractions Author’s submitted version A Problem
Space for Designing Visualizations Michael Gleicher1, Maria Riveiro2, Tatiana
von Landesberger3, Oliver Deussen4, Remco Chang5, and Christina Gillman6
Visualization researchers and visualization professionals seek appropriate
abstractions of visualization requirements that permit considering
visualization solutions independently from specific problems. Abstractions can
help us design, analyze, organize, and evaluate the things we create. The
literature has many task structures (taxonomies, typologies, etc.), design
spaces, and related “frameworks” that provide abstractions of the problems a
visualization is meant to address. In this viewpoint, we introduce a different
one, a _problem space_ that complements existing frameworks by focusing on the
needs that a visualization is meant to solve. We believe it provides a
valuable conceptual tool for designing and discussing visualizations.
††thanks: Author’s submitted version. An article with the same content was
approved for publication in the Visualization Viewpoints Department of IEEE
Computer Graphics and Applications magazine. This preprint will be updated
upon publication of that article.

## Introduction

A problem space is a conceptual structure (i.e., a type of abstraction) that
helps organize thinking about the problem to be addressed, sometimes referred
to as needs (e.g., Olsen [7]).

The design and product development literature provides various schemes that
encourage consideration of the problem to be addressed independently from the
solution that will address it; this separation is the core tenet of a problem
space. Proponents argue that keeping these two separate has advantages that
include focusing on solving correct problems, allowing more creative searches
for solutions free of preconceived ideas, and better articulation of needs to
enable communication and checking. In this viewpoint, we bring the concept of
a problem space to the design of visualizations (the creation of anything from
specific charts to analysis systems) by providing an abstract framework for
considering visualization problems.

We articulate an abstract problem space, i.e., a form of abstraction, to
provide a conceptual tool for organizing thinking about the needs for a
“visualization,” independent of its ultimate design. Arguably, the premise
that the solution will be a visualization is conflating problem and solution;
we take a broad view of what a visualization may be, as it could be that the
right solution to a problem may not require any visualization. It might be
better to call this a problem space of data analysis and communication
problems. We believe our work complements existing abstractions for
visualization design, and provide context in the sidebar.

Table 1: The six axes of the problem space provide aspects to consider in design. Who |  Who has the problem? Who will use the visualization?
---|---
Why |  Why do they need a visualization? What is the objective?
What |  What is the data?
When |  When in the analysis will they use the visualization?
Where |  Where will they use the visualization? What is the context?
How |  How do they expect to be helped?

Our abstract problem space provides a set of different aspects, listed in
Table 1, that should be considered in devising a visualization solution.
Briefly, a visualization is intended to help someone (the _who_) do something
(the _why_) with the data (the _what_) in some phase of the analysis process
(the _when_) and in some context (the _where_) using some method (the _how_).

Note that _how_ refers to user’s expectations of a solution, which is an
aspect to be considered in design, not the ultimate solution, which should not
be part of the problem space. This list of questions serves as the dimensions
of our problem space. It provides a checklist to remind us of factors that we
need to consider when designing a visualization. It provides a vocabulary to
articulate the problems in a consistent way, allowing us to communicate
problem needs and to later assess how a design considers them. It provides a
way to organize aspects so that similar problems, and their solutions, can be
identified.

We consider the problem space using the metaphor of mathematical spaces. Each
aspect forms an “axis” - a generally independent direction in the space, such
that any specific “problem” (a scenario that we might want to create a
visualization for) is a point along that axis, and therefore a point in the
high-dimensional problem space. Each axis represents a range of responses;
they may be a continuous range (truly an axis), they may be multi-faceted with
multiple dimensions that provide a space of responses, or they may be
simplified to a set of discrete categories.

We have developed the problem space based on our experience creating
visualization solutions. It provides a checklist of aspects to consider in
visualization design, and serves as a pedagogical tool in teaching designers
about factors they may need to consider.

## Background

Our problem space is motivated by the work of Schulz et al. [11] who applied
the idea of spaces to the abstraction of task in visualization. However, like
Rind et al. [9], we observe that across the rich literature on “task” in
visualization there is substantial variance in what the term may mean (see
sidebar). Across the literature, “task” may be used as a term for anything
ranging from a specific thing that a viewer is trying to do to a more general
context of what will be done and why. We believe that all of these are useful
concepts, and try to use the problem space to capture the broadest sense, and
use particular aspects of the space to capture the more focused notions of
task. Our problem space seeks to be broader (e.g., including many notions of
task), but also differently limiting (i.e., focusing on problems at the
exclusion of their solution).

Schulz et al. [11] applied the “5 Ws And How framework,” common in journalism
and general information seeking [13] to create a “Design Space of
Visualization Tasks.” In our experience applying this work to our own design
and teaching, we find that their focus on providing a formalism for task was
less appropriate for providing general guidance on the range of aspects one
must consider when designing a visualization. We also apply the 5Ws and H as
our framework, but with a broad focus on the problem without trying to
presuppose the solutions. We include aspects we feel allow us to describe
problems in ways that are independent of the ultimate solutions.

The 5Ws and How framework provides information seekers six questions to ask
(or answer, in the case of journalistic writing): Who, What, When, Where, Why
and How [13]. While the 5Ws and How are commonly referred to in journalism to
help writers organize key information, the idea has historical roots that date
back (at least) to Aristotle’s rhetoric. Intriguingly Aristotle argued that
ignorance of any of these elements can lead to faulty reasoning. We believe
that Aristotle’s advice applies to visualization designers: each question is
something designers should consider. An insight of the work by Schulz et al.
[11] was to define the questions in a Visualization specific manner, a task we
set out to do here to create our problem space.

## The Axes of the Problem Space

The major axes (aspects) of our problem space consist of the Five Ws and How,
introduced in Table 1 and detailed in this section. Admittedly, we are forcing
these to map to the 5Ws and How, which may feel a little contrived. However,
we believe that the history and mnemonic power of the framework make it worth
sticking to this set of words. The six aspects provide a set of considerations
for design.

To help make our abstractions concrete, show the usability, usefulness and the
generalizability of our problem space definition, we provide example use
cases. We selected the use cases across data types and application domains.
The first use case, used as a running example in this section, considers a
keyhole surgery planning with the goal to remove a tumor from a patient’s
brain [4], which can be found in Figure 1. Two other use cases are provided in
a later section. We acknowledge that the examples apply the framework _post-
hoc_ to recent projects that were completed prior to the articulation of the
framework. Our experience with these, and other projects, were an important
basis for developing the framework. Details of the use cases can be found in
their original publications, we only describe aspects related to problem
dimensions.

Figure 1: Running example from Gillmann et al. [4] to demonstrate the use of
the presented problem space for designing visualizations. The tool presents a
visualization to assist keyhole surgery planning. It is composed of a volume
view (a), a surgery tunnel view (b), a control panel (c) and a control graph
to enhance specific areas in the human brain (d).

### Who has the problem?

Generally, our focus is on the intended audience of the visualization; the
people who have some need to see something in the data. Many aspects of the
audience might serve to influence the solution. For example, the level of
analytic sophistication of the expected viewer, their level of motivation,
their spatial reasoning abilities (e.g., Ottley et al. [8]), or their
background knowledge. Considerations such as visual acuity, or accessibility
(e.g., supporting viewers with color vision deficiency) are also part of the
_Who_ aspect.

Design may also consider that problems have multiple stakeholders. While the
focus on visualizations is often the ultimate viewers/users, other
stakeholders involved in the problem may need to be considered in the choice
of a solution. There are always the developers and maintainers to consider,
and in some cases even the “users” might involve a complex set of analysts,
decision makers, writers, and readers.

In the running example, the who is the clinician who aims to process a keyhole
surgery. For communication reasons, the who might also include the patient
that will be treated by the keyhole surgery. In terms of design, this led to a
multi-view system that covers different aspects of the keyhole surgery in
terms of different types of tissue that are interfered with by the procedure,
as shown in Figure 1. The views were chosen to be familiar to the clinicians,
who are medical experts.

### What is the data being worked with?

We assume that in a “data analysis or communication problem” there is data to
be visualized (although obtaining the appropriate data may be part of the
problem). The data impact the solutions in many ways and at many levels.

Many different aspects of the data can influence the solution. The low level
details (i.e., the “form” or type of the data) have obvious implications for
the choice of solutions. However, there is an increasing acknowledgment that
higher level factors in the data, such as its semantics, distribution, value
ranges, important relationships, etc. need to be considered in design [12].
Issues such as quantity and time-dependence of data can influence solutions:
e.g., data that is very large, or changing rapidly requires design
consideration.

The input or provided form of the data may differ from its internal
representation, or even how the user may think of it. Data may be transformed
as part of the analysis process, or thought of in different ways. Any of these
considerations may impact design.

In the running example, the what is a Computed Tomography (CT) scan of the
patient’s head. In addition, the tumor, that needs to be removed is known by
the clinician and indicated in the CT scan. Further, potential surgery paths
that can be used to process the procedure are part of the available data. In
terms of design, this led to a dedicated view for different surgery tunnels
that are classified to be suitable, as shown in Figure 1 b).

### Why is the user working with the data?

Here we consider the goal of the visualization. Typically, this is _what 111We
use _what_ for the object of exploration, so this question is “ _Why?_ is the
viewer trying to do with the data?”_ but the nature of this goal may change as
multiple stakeholders are involved (e.g., it may be what the author intends
for the audience to achieve with the visualization).

The _Why_ most closely aligns with the common notion of task, with the caveat
that many discussions of task do not separate it from the other aspects we
describe (e.g., Schulz et al. [11] who include all 5Ws and H as part of task).
Rind et al. [9] distinguish perspectives on task as “why” and “how” with the
terms _objectives_ and _actions_. Their use of Roth’s concept of objectives
[10] aligns with our goal of separating the problem from the solution.

Many challenges of discussing tasks abstractly, such as the need to consider
multiple levels of abstraction [3], must be considered. The question _Why?_
can provide a useful tool in traversing the levels of abstractions and
separating problem and solution: if a description seems too specific, the
question “Why?” often leads to a useful generalization. Conversely, the
question “How?” can be used to refine an overly general goal, for example, to
break it into subgoals or give a more concrete target. However, this question
might also invite conflating the solution with the problem goal. We can apply
many of the existing works on task descriptions (see sidebar) to articulate
tasks. However, within the framework, we look to focus on the problem, without
presupposing a solution, and to separate the _Why_ as much as possible from
the other aspects. We also acknowledge that there may be multiple tasks
involved in a particular scenario.

In the running example, the why can be phrased as the task to determine the
safest and most suitable surgery path for the planned keyhole surgery. In
order to remove the tumor of the patient, healthy tissue needs to be
interfered with to reach the tumor. Here, the why expresses the task of
determining the path that destroys as little healthy tissue as possible while
being able to reach and remove the tumor. Therefore, the presented
visualization allows the clinician to determine which areas should be avoided
while choosing their path. This can be done by the user using the control
panel, the control graph as shown in Figure 1.

The example highlights the importance of choosing an appropriate level of
abstraction for a task in design. In principle, the surgeon most likely has
“higher level” goals (such as promoting the health of their patient), and
lower level goals (such as making good use of their time or interpreting the
available imagery). While many different tasks may be identified, a designer
should focus on the ones that are likely to inform the design and evaluation.

### When in the process does the problem occur?

In this aspect, we acknowledge that “data work” is a process that may include
gathering information, defining questions, performing analysis, drawing
conclusions (or acting on the analysis), communicating or justifying
rationales, or other steps. The process varies, but is often iterative and not
always sequential. Understanding where along this process the visualization
may be used can inform design: curating data or performing analysis, or making
decisions, may benefit from different choices than communicating an identified
result or justifying a decision.

Different phases of data usage may suggest different choices for visualization
design. For example, a visualization used for the initial examination of the
data may prefer more overview and exploration than one designed to communicate
specific findings later in the process.

In the running example, the question when can be answered with: before the
keyhole surgery. As described, the when can change. In our example, this could
mean that there could be a different visualization assisting the surgeon
during the surgery to ensure that the planned surgery path is followed or even
another visualization after the surgery to examine if the procedure was
successful. In the presented example all visual components are designed to
target the timeframe before the procedure and allow the clinician to explore
different options.

### Where does the problem occur?

For this axis we consider the context of the problem. _Where_ can include the
literal place (e.g., the user is in the lab vs. on a bus), the user’s activity
(e.g., is there attention focused on the visualization?), the social
environment (e.g., will the user be sharing the visualization?), the display
they will be using (e.g., a large monitor vs. a cell phone vs. a printed
page), or the computing environment (how much computation is available at
viewing time). The context should also include where the visualization is
used, e.g., is it placed within a newspaper article, a scientific paper, or
web page, or is it a stand-alone system that viewers access explicitly?
Context might also include the scenario: is the stakeholder making a business
decision, evaluating scientific data, etc.? Aspects such as the amount of time
available, pressure (high stakes), and the need to justify choices later
influence user needs and therefore the ultimate design that addresses them.

In the running example, the _where_ is the computer included in the office of
a surgeon in a dedicated planning session (i.e., before the surgery, but after
imaging and image analysis). The scenario has the user spending focused time
with the tool as part of an explicit planning and analysis activity. The
clinical context also raises data handling and privacy issues. The available
hardware must also be considered as clinics often have older workstations.

### How will the problem be addressed?

In general, the goal of the problem space is to separate the “how” from the
other aspects that specify the problem to be solved. However, it can also be
valuable to acknowledge that sometimes these two are not completely separable:
the stakeholders and/or designers may have pre-conceived notions of what the
solution should be. These should be acknowledged so they can be properly
considered as either design constraints, or as biases that might be
counteracted. For example, a potential user may demand a familiar visual
representation, or an implementer may want to show of an existing tool,
whether or not a different solution might better address the problem needs. In
the running example, the design combined views familiar to the physicians (the
CT scans) with more novel displays. The novelty in a solution can come from
the use of existing components to address a problem in a novel manner.

SIDEBAR: TASK ABSTRACTIONS

There is a strong tradition in the field of visualization of developing ways
to characterize the needs for visualization in a manner that abstracts the
details of particular scenarios. A common approach is to organize scenarios
according to _task_ , roughly the purpose of the visualization, although the
precise definition of task varies widely in the literature
rindTaskCubeThreedimensional2016side. There are many different types of
schemes for organizing tasks, such as taxonomies, typologies, classifications,
and categorizations. While there are technical distinctions between these
different terms (see Brehmer2013side), many researchers use the terms
interchangeably, and the subtle distinctions are not relevant to this
discussion so we refer to them collectively. Early task classification schemes
include those by Wehrend1990, Roth1990, and Zhou1998. Later, Shneiderman1996
discussed the importance of separating tasks from the type of data. Amar2005
provided a general set of basic tasks that continues to serve for organizing
work on understanding what viewers do with visualizations (e.g.,
quadriSurveyPerceptionBasedVisualization2021). Other key examples include
Brehmer2013side, who introduce the importance of considering tasks at multiple
levels, and Schulz2013side, who introduce the importance of considering
multiple facets of tasks. These works already discussed the wide number of
available task schemes, which have continued to proliferate. The works
mentioned in the previous paragraph provide general abstractions that apply to
all visualizations. However, many task abstractions focus on specialized
subsets. For example, leeTaskTaxonomyGraph2006 provide a task taxonomy for
graph data, kerracherTaskTaxonomyTemporal2015 for temporal graphs,
Sarikaya2018 for Scatterplots, and pandeyStateoftheArtSurveyTasks2022 for
trees. rothEmpiricallyDerivedTaxonomyInteraction2013side builds a taxonomy for
the uses of interaction. Two recent works provide important “meta” discussions
of the literature. Kerracher2017side describe the processes for constructing
task classifications and suggest approaches for evaluating them.
rindTaskCubeThreedimensional2016side provide an organizational structure for
the different task classification schemes, defining a three-dimensional space
that different schemes fit into. Our work builds on this existing literature
on task abstraction by recognizing it as one of many aspects that must be
considered in designing a visualization.  plainnat ref-sidebar

Figure 2: Second example use case from Baumgartl et al. [2] to demonstrate the
use of the presented problem space for designing visualizations. Visual
interface for tracing infection control consisting of several views: epidemic
curve, transmission pathway view, contact view and patient timeline view.

## Additional Examples

This section describes two use cases: the tracing of infection pathways in
hospitals [2] with the goal to analyze and monitor infection spreading of
multi-resistant pathogens, and the exploration of news articles [14, 1, 6]
aiming at finding newsworthy information in large document sets.

### Infection Tracing Use Case

#### Who:

The exploration of and analysis of infection pathways is conducted by
infection control, hygienists, infection control data management,
epidemiologists and infection control managers. They are experts in infection
control with little knowledge of novel visualizations. The design tried to
support their expertise by integrating familiar views with novel elements.
They commonly used two types of visualizations: 1) line-charts that show the
progress in numbers of infections and 2) manually created patient timelines.
Based on this knowledge and expectations, the visual design leveraged and
included these two views in the final visualizations. The other views
complemented the designs based on common knowledge of node-link diagrams and a
storyline that combined several aspects in one view (see Figure 2).

#### What:

The data for infection control was complex and large. It consisted of a
combination of two datasets: patient locations over time and patient infection
status over time. The data comprised hundreds of thousands of patients over
several years at a second frequency. This led to a design that focused on one
focus patient and her first- and second-level contacts with a multi-scale time
resolution.

#### Why:

The infection control expert aims at 1) detecting an outbreak, 2) identifying
the outbreak pathway (via contacts of patients), 3) determining the location
of the outbreak, 4) quantifying the duration of the outbreak and 5)
identifying potentially infected patients. These goals (intents, tasks) need
to be solved using the visualization. Therefore the views are designed so that
they enable reaching the analytic goals. Especially, the epidemic curve
supports task 1, the transmission pathway view supports tasks 2-4, the patient
timeline view supports tasks 3 and 4, the contact view supports task 5 (see
Figure 2).

#### When:

The data is analyzed after an infection control expert has a hypothesis about
a potential outbreak. It is a retrospective analysis to detect, confirm and
reconstruct an outbreak in a hospital. It occurs in the analytical phase
before reporting to management and health control authorities and before
taking any of the infection control measures.

#### Where:

The infection control experts in the office at the hospital with standard
hardware (common desktop PC and screen). This needs to be considered and
tested in the visualization, however, no responsive visualizations, e.g. for
smartphones, were needed.

#### How will the problem be addressed?

The infection control experts had experience with existing visualizations,
including 1) line-charts that show the progress in numbers of infections and
2) manually created patient timelines. They expected to see the epidemic curve
with the exactly defined axes and data and the patient timeline. To leverage
their knowledge and expectations, the visual design leveraged and included
these two views in the final visualizations. The other views complemented the
designs based on common knowledge of node-link diagrams and a storyline that
combined several aspects in one view (see Figure 2).

### Journalism – Exploration of a Document collection – Use Case

#### Who:

The goal was to support journalists in exploring collections of documents.
These users often work under time pressure, and typically were unwilling to
invest time in learning new visualizations. Therefore, the visual design could
not include complex visualizations but leveraged popular dashboard design with
bar charts and a node-link diagram. The need for fast analysis under time
pressure was supported by an interactive set of filters and degree-of-interest
based data exploration. Moreover, journalists need to work in asynchronous
teams with the need to share exploration results with colleagues to continue
the exploration. This is supported in the visualization by provenance tracking
and sharing functions (see Figure 3).

#### What:

Journalists needed to explore large collections of documents (millions), from
which named entities and their relationships were extracted. For example,
WikiLeaks PlusD (Cablegate) documents has 1.4 million entities that are
connected by 163 million edges. Showing all the data at once is infeasible so
linked views with filtering, details on demand and degree-of-interest based
data exploration were developed and used in the solution.

#### Why:

Journalists aim at identifying newsworthy information in large document sets.
The visualization allows them to browse and filter document collections by
time and named entities, explore links between entities, and read interesting
documents in close reading mode. Moreover, it supports data curation and
browsing history (see Figure 3). In this way, interesting pieces of
information can be found and shared.

#### When:

The data exploration is done during journalistic investigation, in particular,
in the early phase of investigation, when interesting facts need to be
identified. After their identification using the visualization, these
interesting pieces of information need to be checked and confirmed so that
they can finally be published. The design focused on the early phases of the
journalistic process (e.e., investigation), but required creating results that
could be used in later phases (e.g., using the facts to craft the story).

#### Where:

The journalists in our use case worked in the office with standard hardware
(common desktop PC and screen). Thus responsive visualizations, e.g. for
smartphones, were not needed.

#### How will the problem be addressed?

The journalists posed constraints on the solution, such as expectations to be
able to closely read the articles. Summary views and distant reading were not
sufficient.

Figure 3: Third running example to demonstrate the use of the presented
problem space for designing visualizations. Visual dashboard for investigative
journalism based on large document collections. The linked views comprise the
distribution and filtering of documents by time and entity and a node-link
diagram of entity connections that can be explored using a degree-of-interest
function. The exploration process has provenance features in a view on the top
[6]

.

## Is the Problem Space Useful or Good?

To justify our problem space, we consider a range of desirable properties in
such abstractions. These goals are motivated by Kerracher and Kennedy [5] who
discuss the evaluation of task abstractions:

  * •

Relevant (does each axis make a meaningful distinction): Each aspect was
chosen and defined to be relevant to the design and analysis of visualization
solutions.

  * •

Independent (allows us to discuss problems independently of solutions): We
have tried to frame each axis as something that can be discussed in a
solution-agnostic manner.

  * •

Thorough (does it cover all aspects?): we do not assert that our framework is
fully thorough, there may be aspects that do not fit neatly into one of our
axes. However, we believe that the framework covers an expansive enough range
of issues that it is still useful.

  * •

Complete (any problem will exist somewhere in our space): Any data problem
will involve each of the aspects we describe. In some cases, it might not be
specific (e.g., there may be a range of users).

  * •

Orthogonal (the aspects are independent of each other): While we have aimed to
make the aspects orthogonal, they are ultimately inter-twined or at least
correlated. For example, different types of users tend to operate in
particular contexts or address different types of tasks. However, we believe
that each provides a different lens on what might be the same situation.

  * •

Precise (any problem can be uniquely positioned in the space): We did not
provide specific formalisms for how to consider each aspect; this limits the
precision of our framework. If precision is important, one can make precise
choices for the axes. In practice, problems often cover ranges on the axes,
such as multiple users or objectives, so precision may not be a valid goal.

## Using the framework

Our primary premise is that all of these aspects of a problem should be
considered as part of a design or evaluation process.

Often, when confronted with a problem we don’t know all of the aspects. This
uncertainty in the specification of the problem, e.g., not being able to
predict who the users will be or where they will be using a visualization, is
quite different from uncertainty in the problem itself (e.g., having to deal
with noisy data). These unknowns may still be useful: they point to aspects
where more clarity might be sought (e.g., better identifying the potential
users) or the lack of clarity may influence the design itself (e.g., not
knowing the viewers makes for a greater need to support a range of them
through inclusive design).

The problem space can serve as a checklist: have the various aspects been
considered? For design, each aspect can influence the choices made in defining
a solution. For example: how can the needs of a certain type of user be met?
how do the challenges of a particular phase of analysis affect the needs? or,
how should the limitations of a display influence the visualization?
Similarly, the evaluation of a solution should consider the problem. For
example, does it work for the actual users, in the appropriate process phase,
and in the context of the real use case? The problem space can also aid as a
checklist for documentation: in describing a solution, authors can make clear
the problem they were intending to solve by checking they have included at
least these aspects.

We believe that a problem space for designing visualizations can provide a
useful conceptual tool that can help designers articulate their problem in a
manner independent of how it will be solved. We believe this will have
benefits in the specification, design (creation of solutions), and evaluation
of visualizations.

## ACKNOWLEDGMENTS

This work has benefitted from Dagstuhl Seminar 22331 ”Visualization and
Decision Making Design Under Uncertainty”, https://www.dagstuhl.de/22331. CG
was supported in part by the Federal Ministry of Education and Research of
Germany and by the Sächsische Staatsministerium für Wissenschaft Kultur und
Tourismus in the program Center of Excellence for AI-research ”Center for
Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig”, project
identification number: ScaDS.AI. MG was supported in part by NSF award
2007436. MR was supported by the Swedish Research Council – project EXPLAIN VR
2018-03622. OD was supported in part by Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Project-ID 251654672 – TRR 161. RC was supported
in part by NSF awards 1940175, 1939945, and 2118201.

## References

  * Ballweg et al. [2016] K. Ballweg, F. Zouhar, P. Wilhelmi-Dworski, U. Fahrer, A. Panchenko, S. Yimam, C. Biemann, M. Regneri, and H. Ulrich.  new/s/leak–a tool for visual exploration of large text document collections in the journalistic domain.  In _Visualization in Practice, VIS Workshop_ , 2016.
  * Baumgartl et al. [2021] T. Baumgartl, M. Petzold, M. Wunderlich, M. Hohn, D. Archambault, M. Lieser, A. Dalpke, S. Scheithauer, M. Marschollek, V. M. Eichel, N. T. Mutters, Highmed Consortium, and T. Von Landesberger.  In Search of Patient Zero: Visual Analytics of Pathogen Transmission Pathways in Hospitals.  _IEEE Transactions on Visualization and Computer Graphics_ , 27(2):711–721, February 2021.  ISSN 1077-2626, 1941-0506, 2160-9306.
  * Brehmer and Munzner [2013] M. Brehmer and T. Munzner.  A Multi-Level Typology of Abstract Visualization Tasks.  _IEEE Transactions on Visualization and Computer Graphics_ , 19(12):2376–2385, December 2013.  ISSN 1077-2626.  10.1109/TVCG.2013.124.
  * Gillmann et al. [2018] C. Gillmann, R. G.C. Maack, T. Post, T. Wischgoll, and H. Hagen.  An uncertainty-aware workflow for keyhole surgery planning using hierarchical image semantics.  _Visual Informatics_ , 2(1):26 – 36, 2018.  ISSN 2468-502X.  Proceedings of PacificVAST.
  * Kerracher and Kennedy [2017] N. Kerracher and J. Kennedy.  Constructing and evaluating visualisation task classifications: Process and considerations.  In _Computer Graphics Forum_ , volume 36, pages 47–59. Wiley Online Library, 2017.
  * Müller et al. [2017] Martin Müller, Kathrin Ballweg, Tatiana Von Landesberger, Seid Yimam, Uli Fahrer, Chris Biemann, Marcel Rosenbach, Michaela Regneri, and H. Ulrich.  Guidance for Multi-Type Entity Graphs from Text Collections.  page 5 pages. The Eurographics Association, 2017.  ISBN 9783038680420.  10.2312/EUROVA.20171111.
  * Olsen [2015] D. Olsen.  Chapter 2: Problem Space versus Solution Space.  In _The Lean Product Playbook_ , chapter 2, pages 13–22. John Wiley & Sons, Ltd, 2015.  ISBN 978-1-119-15482-2.
  * Ottley et al. [2016] E. M. Ottley, A.and Peck, L. T. Harrison, D. Afergan, C. Ziemkiewicz, H. A. Taylor, P. K. J. Han, and R. Chang.  Improving Bayesian Reasoning: The Effects of Phrasing, Visualization, and Spatial Ability.  _IEEE Transactions on Visualization and Computer Graphics_ , 22(1):529–538, January 2016.  ISSN 1941-0506.  10.1109/TVCG.2015.2467758.
  * Rind et al. [2016] A. Rind, W. Aigner, M. Wagner, S. Miksch, and T. Lammarsch.  Task Cube: A three-dimensional conceptual space of user tasks in visualization design and evaluation.  _Information Visualization_ , 15(4):288–300, 2016\.
  * Roth [2013] R. E. Roth.  An Empirically-Derived Taxonomy of Interaction Primitives for Interactive Cartography and Geovisualization.  _IEEE Transactions on Visualization and Computer Graphics_ , 19(12):2356–2365, December 2013.  ISSN 1941-0506.
  * Schulz et al. [2013] H.-J. Schulz, T. Nocke, M. Heitzler, and H. Schumann.  A Design Space of Visualization Tasks.  _IEEE Transactions on Visualization and Computer Graphics_ , 19(12):2366–2375, December 2013.  ISSN 1077-2626.
  * Walny et al. [2020] Jagoda Walny, Christian Frisson, Mieka West, Doris Kosminsky, Soren Knudsen, Sheelagh Carpendale, and Wesley Willett.  Data Changes Everything: Challenges and Opportunities in Data Visualization Design Handoff.  _IEEE Transactions on Visualization and Computer Graphics_ , 26(1):12–22, January 2020.  ISSN 1077-2626, 1941-0506, 2160-9306.  10.1109/TVCG.2019.2934538.
  * Wikipedia [referenced 9/12/2022] Wikipedia.  Five Ws.  https://en.wikipedia.org/wiki/Five_Ws, referenced 9/12/2022.
  * Yimam et al. [2016] S. M. Yimam, H. Ulrich, T. von Landesberger, M. Rosenbach, M. Regneri, A. Panchenko, F. Lehmann, U. Fahrer, C. Biemann, and K. Ballweg.  new/s/leak–information extraction and visualization for investigative data journalists.  In _Proceedings of ACL System Demonstrations_ , pages 163–168, 2016\.


# The apolar inner product and a Bombieri type inequality for polynomials

J. M. Aldaz, A. Bravo and H. Render  J.M. Aldaz: Instituto de Ciencias
Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad
Autónoma de Madrid, Cantoblanco 28049, Madrid, Spain<EMAIL_ADDRESS>A. Bravo: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and
Departamento de Matemáticas, Universidad Autónoma de Madrid, Cantoblanco
28049, Madrid, Spain<EMAIL_ADDRESS>H. Render: School of Mathematical
Sciences, University College Dublin, Dublin 4, Ireland<EMAIL_ADDRESS>
###### Abstract.

We consider inequalities of Bombieri type for polynomials that need not be
homogeneous, using the apolar inner product.

2020 Mathematics Subject Classification: _Primary: 26D05_ , _Secondary: 15A63_

Key words and phrases: _Apolar inner product, Bombieri’s inequality_

##  1\. Introduction

The apolar inner product $\langle\cdot,\cdot\rangle_{a}$ on spaces of
polynomials originates within the Theory of Invariants, from the XIX century,
and has since found several other applications. For instance, it is essential
in the proof of the Fischer decomposition of polynomials, which in turn helps
to find polynomial solutions to certain differential equations (in this
regard, among the many references that could be given we mention [15], [7],
[8], [13], [10], [1])).

Related to the apolar inner product is the celebrated Bombieri’s inequality
for homogeneous polynomials $f$ and $g$ (cf. for instance [3], [4], [5], [6],
[9], [12], [16]) which for the apolar norm states that

$\|fg\|_{a}\geq\|f\|_{a}\|g\|_{a}.$

By induction, the extension to the case of $n$ homogeneous polynomials is
immediate:

(1) $\|f_{1}\cdots f_{n}\|_{a}\geq\|f_{1}\|_{a}\cdots\|f_{n}\|_{a}.$

As is well known, the inequality $\|fg\|_{a}\geq\|f\|_{a}\|g\|_{a}$ still
holds when only one of the polynomials is homogeneous, cf. [16] for instance,
or the final part of the next section. Hence, it is natural to explore what
happens when both $f$ and $g$ are non-homogeneous. This paper, which studies
the preceding question, is organized as follows. The relevant definitions and
standard results appear in Section 2. A simple example given in Section 3,
shows that Bombieri’s inequality does not hold with constant 1 if both
polynomials are non-homogeneous. But by reduction to the homogeneous case and
an application of Bombieri’s inequality, we show that letting $n_{i}$ be the
degree of $f_{i}$, the inequality

$[(n_{1}+\cdots+n_{s})!]^{1/2}\|f_{1}\cdots
f_{n}\|_{a}\geq\|f_{1}\|_{a}\cdots\|f_{n}\|_{a}$

holds. It may well be that a direct proof will lead to better bounds, but we
have not found such a proof from first principles. Section 4 presents some
special cases where better bounds do hold. Finally, in Section 5 we consider
powers of a single polynomial $P$.

##  2\. Notation, definitions and background results

Denote by $\mathbb{C}[z_{1},\dots,z_{d}]$ the vector space of all polynomials
in $z=\left(z_{1},\dots,z_{d}\right)$ with complex coefficients, and by
$\mathbb{C}[z_{1},\dots,z_{d}]_{\leq n}$ the subspace of
$\mathbb{C}[z_{1},\dots,z_{d}]$ consisting of polynomials with degree bounded
by $n$. Let $P$ and $Q$ be polynomials of degree $N$ and $M$, respectively
given by

$P\left(z\right)=\sum_{\alpha\in\mathbb{N}^{d},\left|\alpha\right|\leq
N}c_{\alpha}z^{\alpha}\text{ and
}Q\left(z\right)=\sum_{\alpha\in\mathbb{N}^{d},\left|\alpha\right|\leq
M}d_{\alpha}z^{\alpha},$

where the standard notation for multi-indices is used: for
$\alpha=\left(\alpha_{1},\dots,\alpha_{d}\right)\in\mathbb{N}^{d}$, we write
$z^{\alpha}=z_{1}^{\alpha_{1}}\cdots z_{d}^{\alpha_{d}},$
$\alpha!=\alpha_{1}!\cdots\alpha_{d}!$, and
$\left|\alpha\right|=\alpha_{1}+\cdots+\alpha_{d}$. We call the largest value
of $\left|\alpha\right|$ such that $c_{\alpha}\neq 0$ the total degree of $P$,
or just its degree.

Given $P\left(z\right)$, we denote by $P^{\ast}\left(z\right)$ the polynomial
obtained from $P\left(z\right)$ by conjugating its coefficients, and by
$P\left(D\right)$ the linear differential operator obtained from
$P\left(z\right)$ by replacing each variable $z_{j}$ with the differential
operator $\frac{\partial}{\partial z_{j}}$, for $j=1,\dots,d$. The _apolar
inner product_ $\langle\cdot,\cdot\rangle_{a}$ on
$\mathbb{C}[z_{1},\dots,z_{d}]$ is defined by

(2) $\left\langle
P,Q\right\rangle_{a}:=\left(Q^{*}\left(D\right)P\right)\;(0)=\sum_{\alpha\in\mathbb{N}^{d}}\alpha!c_{\alpha}\overline{d_{\alpha}},$

and the associated apolar norm, by $\left\|f\right\|_{a}=\sqrt{\left\langle
f,f\right\rangle_{a}}.$ Note that for monomials $z^{\alpha}$ and $z^{\beta}$,
definition (2) means that $\left\langle
z^{\alpha},z^{\beta}\right\rangle_{a}=0$ if $\alpha\neq\beta$, and
$\left\langle z^{\alpha},z^{\alpha}\right\rangle_{a}=\alpha!$. Furthermore,
from these relations definition (2) is recovered by sesquilinearity.

Bombieri’s inner product $\left\langle\cdot,\cdot\right\rangle_{b}$ is the
apolar inner product with a different normalization: $\left\langle
z^{\alpha},z^{\alpha}\right\rangle_{b}=\alpha!/|\alpha|!$. Thus,
$\left\langle\cdot,\cdot\right\rangle_{b}$ depends on the total degree
$|\alpha|$ of the monomials, or more generally, the total degree of the
homogeneous polynomials.

It is well known and easy to see that Bombieri’s inequality extends to the
case were one of the polynomials is arbitrary and the other is homogeneous.
Let $f_{m}$ be homogeneous of total degree $m$ and let $P$ be a non-zero
polynomial of total degree $k$, say

$P\left(z\right)=P_{k}\left(z\right)+\cdots+P_{0}\left(z\right),$

where for each $j=0,\dots,k,$ $P_{j}\left(z\right)$ is a homogeneous
polynomial of total degree $j$. We call $P_{k}$ the principal part of $P$. By
the Pythagorean Theorem and Bombieri’s inequality for the homogeneous case we
have

$\left\|P\cdot f_{m}\right\|_{a}^{2}=\left\|\sum_{j=0}^{k}P_{j}\cdot
f_{m}\right\|_{a}^{2}=\sum_{j=0}^{k}\left\|P_{j}\cdot
f_{m}\right\|_{a}^{2}\geq\sum_{j=0}^{k}\left\|P_{j}\right\|_{a}^{2}\left\|f_{m}\right\|_{a}^{2}=\left\|P\right\|_{a}^{2}\left\|f_{m}\right\|_{a}^{2}.$

##  3\. A Bombieri type theorem

In general, Bombieri’s inequality with constant 1 can fail: consider for
instance

$(x-1)(x+1)=x^{2}-1.$

Then $\|(x-1)\|_{a}^{2}\|(x+1)\|_{a}^{2}=4$, while $\|x^{2}-1\|_{a}^{2}=3$.

Thus, Bombieri’s inequality does not extend to the general case without some
modification.

######  Theorem 1.

Let $P^{(1)},\dots,P^{(s)}\in\mathbb{C}[z_{1},\dots,z_{d}]$ be nonzero
polynomials of total degrees $n_{1},\dots,n_{s}$ respectively. Then

$(n_{1}+\cdots+n_{s})!\left\|P^{(1)}\cdots
P^{(s)}\right\|_{a}^{2}\geq\left\|P^{(1)}\right\|_{a}^{2}\cdots\left\|P^{(s)}\right\|_{a}^{2}.$

In order to prove the result, we introduce the following notation. Given $P$
of degree $k$, say

$P\left(z\right)=P_{k}\left(z\right)+\cdots+P_{0}\left(z\right),$

we select a variable $w$ different from $z_{1},\dots,z_{d}$, and define the
one variable homogenization of $P$ as

$P_{hom}\left(z,w\right)=P_{k}\left(z\right)+wP_{k-1}\left(z\right)+\cdots+w^{k}P_{0}\left(z\right).$

Thus, $P_{hom}\in\mathbb{C}[z_{1},\dots,z_{d},w]$ is a homogeneous polynomial
of degree $k$. The abbreviation $\left(z,w\right)$ obviously stands for the
vector $\left(z_{1},\dots,z_{d},w\right)$.

The next Lemma states that the one variable homogenization commutes with
taking products.

######  Lemma 2.

Let $P^{(1)},\dots,P^{(s)}\in\mathbb{C}[z_{1},\dots,z_{d}]$. Then
$P^{(1)}_{hom}\cdots P^{(s)}_{hom}=\left(P^{(1)}\cdots P^{(s)}\right)_{hom}.$

###### Proof.

By induction it is enough to prove the case $s=2$. Let $P,Q$ be polynomials
with degrees $m$ and $n$ respectively, say

$P\left(z\right)=\sum_{\alpha\in\mathbb{N}^{d},\left|\alpha\right|\leq
m}c_{\alpha}z^{\alpha}\text{ and
}Q\left(z\right)=\sum_{\alpha\in\mathbb{N}^{d},\left|\alpha\right|\leq
n}d_{\alpha}z^{\alpha}.$

Select $\gamma\in\mathbb{N}^{d}$ with $0\leq|\gamma|\leq m+n$, and let
$\ell_{\gamma}z^{\gamma}$ be the term of $PQ$ corresponding to $\gamma$. Let
$\leq$ also denote the standard partial order on $\mathbb{N}^{d}$, that is,
$\alpha\leq\beta$ if and only if for all coordinates (so $1\leq i\leq d$) we
have $\alpha_{i}\leq\beta_{i}$. Then

(3)
$\ell_{\gamma}z^{\gamma}=\sum_{\alpha\leq\gamma}c_{\alpha}z^{\alpha}d_{\gamma-\alpha}z^{\gamma-\alpha}.$

With the natural notation, the term of $(PQ)_{hom}$ corresponding to the
vector $(\gamma,m+n-|\gamma|)\in\mathbb{N}^{d+1}$ is thus

(4)
$\ell_{\gamma}z^{\gamma}w^{m+n-|\gamma|}=\sum_{\alpha\leq\gamma}c_{\alpha}z^{\alpha}d_{\gamma-\alpha}z^{\gamma-\alpha}w^{m+n-|\gamma|}=\sum_{\alpha\leq\gamma}c_{\alpha}z^{\alpha}w^{m-|\alpha|}d_{\gamma-\alpha}z^{\gamma-\alpha}w^{n-|\gamma|+|\alpha|},$

which coincides with the term of $P_{hom}Q_{hom}$ corresponding to the vector
$(\gamma,m+n-|\gamma|)$. ∎

######  Lemma 3.

Let $P\in\mathbb{C}[z_{1},\dots,z_{d}]$ be a polynomial of total degree
$|\alpha|$. Then

$\|P\|_{a}^{2}\leq\|P_{hom}\|_{a}^{2}\leq|\alpha|!\|P\|_{a}^{2}.$

###### Proof.

The first inequality is obvious, since all terms of $P_{hom}$ have degree at
least as large as those of $P$ and the coefficients are the same. For the
second inequality, writing $P=P_{k}+\cdots+P_{0}$ (where each $P_{j}$ is
homogeneous of degree $j$, and $P_{k}\neq 0$, so the total degree is
$|\alpha|=k$) we have

$\|P_{hom}\|_{a}^{2}=\left\langle
P_{hom},P_{hom}\right\rangle_{a}=\left\langle\sum_{j=0}^{k}w^{k-j}P_{j},\sum_{j=0}^{k}w^{k-j}P_{j}\right\rangle_{a}=\sum_{j=0}^{k}\left\langle
w^{k-j}P_{j},w^{k-j}P_{j}\right\rangle_{a}$ $=\sum_{j=0}^{k}(k-j)!\left\langle
P_{j},P_{j}\right\rangle_{a}\leq k!\sum_{j=0}^{k}\left\langle
P_{j},P_{j}\right\rangle_{a}=k!\|P\|_{a}^{2}.$

∎

Proof of Theorem 1. Let
$P^{(1)},\dots,P^{(s)}\in\mathbb{C}[z_{1},\dots,z_{d}]$ be nonzero polynomials
of total degrees $n_{1},\dots,n_{s}$ respectively. Then by the preceding
lemmas and Bombieri’s inequality (1) for several polynomials, we have

$(n_{1}+\cdots+n_{s})!\left\|P^{(1)}\cdots
P^{(s)}\right\|_{a}^{2}\geq\left\|\left(P^{(1)}\cdots
P^{(s)}\right)_{hom}\right\|_{a}^{2}$ $=\left\|P^{(1)}_{hom}\cdots
P^{(s)}_{hom}\right\|_{a}^{2}\geq\left\|P^{(1)}_{hom}\right\|_{a}^{2}\cdots\left\|P^{(s)}_{hom}\right\|_{a}^{2}\geq\left\|P^{(1)}\right\|_{a}^{2}\cdots\left\|P^{(s)}\right\|_{a}^{2}.$

∎

Since the large constants come from the homogenization of the polynomials,
having the additional information that the coefficients of the low order terms
are zero yields the following theorem, with the same proof.

######  Theorem 4.

Let $P,Q\in\mathbb{C}[z_{1},\dots,z_{d}]$ be nonzero polynomials of total
degrees $m$ and $n$ respectively, such that for some $0\leq j\leq m$ and some
$0\leq i\leq n$, we have
$P\left(z\right)=P_{m}\left(z\right)+\cdots+P_{m-j}\left(z\right)$ and
$Q\left(z\right)=Q_{n}\left(z\right)+\cdots+Q_{n-i}\left(z\right)$. Then

$(j+i)!\left\|P\cdot
Q\right\|_{a}^{2}\geq\left\|P\right\|_{a}^{2}\left\|Q\right\|_{a}^{2}.$

##  4\. Obtaining better bounds in some special cases

Next we present some immediate improvements for certain special cases. If all
the coefficients of $P$ and of $Q$ are non-negative, then cancellation cannot
occur and thus Bombieri’s inequality holds.

######  Theorem 5.

Let $P,Q\in\mathbb{C}[z_{1},\dots,z_{d}]$ be nonzero polynomials with real,
non-negative coefficients. Then

$\left\|P\cdot Q\right\|_{a}\geq\left\|P\right\|_{a}\left\|Q\right\|_{a}.$

###### Proof.

Let $P$ be a polynomial of total degree $m$, say

$P\left(z\right)=P_{m}\left(z\right)+\cdots+P_{0}\left(z\right),$

where for each $j=0,\dots,m,$ $P_{j}\left(z\right)$ is a homogeneous
polynomial of total degree $j$, and likewise, write

$Q\left(z\right)=Q_{n}\left(z\right)+\cdots+Q_{0}\left(z\right).$

Since the coefficients of $P$ and $Q$ are non-negative, it follows from the
definition of the apolar inner product that for all $j,k,r,s$ we always have
$\left\langle P_{j}Q_{k},P_{r}Q_{s}\right\rangle_{a}\geq 0$. Thus

(5) $\left\langle
PQ,PQ\right\rangle_{a}=\left\langle\sum_{j=0}^{m}\sum_{k=0}^{n}P_{j}Q_{k},\sum_{j=0}^{m}\sum_{k=0}^{n}P_{j}Q_{k}\right\rangle_{a}=\sum_{j=0}^{m}\sum_{k=0}^{n}\sum_{r=0}^{m}\sum_{s=0}^{n}\left\langle
P_{j}Q_{k},P_{r}Q_{s}\right\rangle_{a}$ (6)
$\geq\sum_{j=0}^{m}\sum_{k=0}^{n}\left\langle
P_{j}Q_{k},P_{j}Q_{k}\right\rangle_{a}=\sum_{j=0}^{m}\sum_{k=0}^{n}\left\|P_{j}Q_{k}\right\|_{a}^{2}\geq\sum_{j=0}^{m}\sum_{k=0}^{n}\left\|P_{j}\right\|_{a}^{2}\left\|Q_{k}\right\|_{a}^{2}$
(7)
$=\sum_{j=0}^{m}\left\|P_{j}\right\|_{a}^{2}\sum_{k=0}^{n}\left\|Q_{k}\right\|_{a}^{2}=\left\|P\right\|_{a}^{2}\left\|Q\right\|_{a}^{2}.$

∎

In order to try to lower the constant $(m+n)!$ from Theorem 1 when dealing
with two polynomials, one can search for homogenizations that raise the norms
of the homogenized polynomials by a smaller amount.

Given $P$ of degree $k$, say

$P\left(z\right)=P_{k}\left(z\right)+\cdots+P_{0}\left(z\right),$

we select variables $w_{1},\dots,w_{k}$ different from $z_{1},\dots,z_{d}$
(but we do not assume that all the variables $w_{1},\dots,w_{k}$ are different
among themselves), and define a many variables homogenization of $P$ as

$P_{hom}\left(z,w_{1},\dots,w_{k}\right)=P_{k}\left(z\right)+w_{1}P_{k-1}\left(z\right)+w_{1}w_{2}P_{k-2}\left(z\right)+\cdots+w_{1}\cdots
w_{k}P_{0}\left(z\right).$

Suppose all the variables appearing in $P$ and $Q$ are raised to an even
degree. Then a two variables homogenization leads to an improvement of the
constant in Theorem 1. Let us denote by $2\mathbb{N}$ the set of even natural
numbers.

######  Theorem 6.

Let $m,n\in 2\mathbb{N}\setminus\\{0\\}$, and let
$P,Q\in\mathbb{C}[z_{1},\dots,z_{d}]$ be nonzero polynomials of the form

$P\left(z\right)=\sum_{\alpha\in(2\mathbb{N})^{d},\left|\alpha\right|\leq
m}c_{\alpha}z^{\alpha}\text{ and
}Q\left(z\right)=\sum_{\alpha\in(2\mathbb{N})^{d},\left|\alpha\right|\leq
n}d_{\alpha}z^{\alpha}.$

Then

$[\left(m+n\right)/2]!\left\|P\cdot
Q\right\|_{a}\geq\left\|P\right\|_{a}\left\|Q\right\|_{a}.$

###### Proof.

We use the following two variable homogenization: write $P$ as

$P\left(z\right)=P_{m}\left(z\right)+\cdots+P_{2}(z)+P_{0}\left(z\right),$

where for each even $j=0,\dots,m,$ the polynomial $P_{j}\left(z\right)$ is
homogeneous of degree $j$, and set

$P_{hom}\left(z,w_{1},w_{2}\right)=P_{m}\left(z\right)+\cdots+(w_{1}w_{2})^{(m-2)/2}P_{2}(z)+(w_{1}w_{2})^{m/2}P_{0}\left(z\right),$

doing likewise with $Q$ and $PQ$. By arguing as in Lemma 2, we see that this
homogenization also commutes with products, so $P_{hom}Q_{hom}=(PQ)_{hom}$.
And as in Lemma 3, we see that

$\|P_{hom}\|_{a}^{2}=\left\langle
P_{hom},P_{hom}\right\rangle_{a}=\left\langle\sum_{j=0}^{m/2}(w_{1}w_{2})^{m/2-j}P_{2j},\sum_{j=0}^{m/2}(w_{1}w_{2})^{m/2-j}P_{2j}\right\rangle_{a}$
$=\sum_{j=0}^{m/2}\left\langle(w_{1}w_{2})^{m/2-j}P_{2j},(w_{1}w_{2})^{m/2-j}P_{2j}\right\rangle_{a}=\sum_{j=0}^{m/2}((m/2-j)!)^{2}\left\langle
P_{2j},P_{2j}\right\rangle_{a}$ $\leq((m/2)!)^{2}\sum_{j=0}^{m/2}\left\langle
P_{2j},P_{2j}\right\rangle_{a}=((m/2)!)^{2}\|P\|_{a}^{2}.$

And now we argue as in the proof of Theorem 1. ∎

For some purposes, a well known integral representation of the apolar inner
product due to V. Bargmann, cf. [2], turns out to be useful (see also [11] and
[12]).

######  Theorem 7.

Let $P$ and $Q$ be polynomials in $d$ complex variables. Then

(8) $\left\langle
P,Q\right\rangle_{a}=\frac{1}{\pi^{d}}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}P\left(x+iy\right)\overline{Q\left(x+iy\right)}e^{-\left|x\right|^{2}-\left|y\right|^{2}}dxdy<\infty,$

where $dxdy$ is Lebesgue measure on $\mathbb{R}^{2d}$.

The next equality result for the homogeneous case is due to B. Reznick, cf.
[14, Main Theorem]. In fact, Reznick’s result is a characterization. We
present the “if” part and note that the “only if” direction does not hold when
both polynomials are non-homogeneous.

######  Theorem 8.

Let $P,Q\in\mathbb{C}[z_{1},\dots,z_{d}]$, and suppose there is a linear
change of variables $T:\mathbb{C}^{d}\to\mathbb{C}$ which makes $P$ and $Q$
depend on disjoint sets of variables. If the absolute value of the Jacobian
determinant of $T$ takes the constant value $1$, then

$\left\|P\cdot Q\right\|_{a}=\left\|P\right\|_{a}\left\|Q\right\|_{a}.$

###### Proof.

Use the change of variables formula and the integral representation from
Theorem 7, integrating first with respect to the variables that appear in $P$
after $T$ is applied, and then with respect to the remaining variables. ∎

######  Example 9.

It is possible to have equality for non-constant polynomials of one variable,
so the “only if” part of Reznick’s characterization (stating that equality
occurs only if after a suitable transformation, the polynomials depend on
dijoint sets of variables) does not extend to the non-homogeneous case.

For $0\leq t\leq 1$ define $f(t):=\left\|(-1+x)\cdot(t+x)\right\|_{a}^{2}$, so
$f(0)=f(1)=3$, and set
$g(t):=\left\|(-1+x)\right\|_{a}^{2}\left\|(t+x)\right\|_{a}^{2}$, so $g(0)=2$
and $g(1)=4$. Since both $f$ and $g$ are continuous functions of $t$, there
exists a $c\in(0,1)$ such that $f(c)=g(c)$.

If $Q$ is obtained from $P$ by deleting some terms, then
$\left\|Q\right\|_{a}\leq\left\|P\right\|_{a}$ by orthogonality. It would be
useful to have some analogous result when taking products. More precisely, it
is natural to ask whether the following “monotonicity property” holds: let
$P,Q\in\mathbb{C}[z_{1},\dots,z_{d}]$ be nonzero polynomials such that $P$ has
degree $n$, and suppose $P_{n+1}$ is homogeneous of degree $n+1$. Is it true
that

$\left\|(P_{n+1}+P)\cdot Q\right\|_{a}\geq\left\|P\cdot Q\right\|_{a}?$

The answer is no, as the following example shows. Let $P(x)=1$, $P_{1}(x)=tx$
and $Q(x)=1-tx$. Then

$\left\|\left(1+tx\right)\left(1-tx\right)\right\|_{a}^{2}=\left\|1-t^{2}x^{2}\right\|_{a}^{2}=1+2t^{4},$

while

$\left\|1\cdot\left(1-tx\right)\right\|_{a}^{2}=1+t^{2}.$

Taking $0<t<1/2$ we get $\left\|(P_{1}+P)\cdot Q\right\|_{a}<\left\|P\cdot
Q\right\|_{a}.$

##  5\. Iterated products of a polynomial

It follows from Bombieri’s inequality (1) for several polynomials that if $P$
is _homogeneous_ , then for $s\geq 1$, $s\in\mathbb{N}$, we have
$\left\|P^{s}\right\|_{a}\geq\left\|P\right\|_{a}^{s}.$ An immediate
consequence of Bargmann’s integral representation is that this inequality
holds for $P$ arbitrary.

######  Theorem 10.

Let $P\in\mathbb{C}[z_{1},\dots,z_{d}]$. Then for every $s\geq 1$,
$s\in\mathbb{N}$, we have

$\left\|P^{s}\right\|_{a}\geq\left\|P\right\|_{a}^{s}.$

###### Proof.

Apply Jensen’s inequality to the integral representation from Theorem 7, using
the convex function $\varphi(t):=|t|^{s}$, to conclude that
$\left\|P^{s}\right\|_{a}^{2}\geq\left\|P\right\|_{a}^{2s}.$ ∎

Note that if $\left\|P\right\|_{a}<1$ then
$\lim_{s\to\infty}\left\|P\right\|_{a}^{s}=0$. We show next that
$\left\|P^{s}\right\|_{a}$ behaves very differently, provided $P$ is not
constant.

######  Theorem 11.

Let $P$ be a polynomial of degree $k\geq 1.$ Then

$\lim_{s\rightarrow\infty}\sqrt[s]{\left\|P^{s}\right\|_{a}}=\infty.$

###### Proof.

Denote by $\gamma$ the gaussian probability measure on $\mathbb{R}^{2d}$ from
Bargmann’s representation (8). Using Theorem 7 we obtain

$\left\|P^{s}\right\|_{a}^{2}=\left\|P\right\|_{L^{2s}(\mathbb{R}^{2d},\gamma)}^{2s},$

so

$\lim_{s\rightarrow\infty}\sqrt[s]{\left\|P^{s}\right\|_{a}}=\lim_{s\rightarrow\infty}\left\|P\right\|_{L^{2s}(\mathbb{R}^{2d},\gamma)}=\left\|P\right\|_{L^{\infty}(\mathbb{R}^{2d},\gamma)}=\infty$

since $P$ is not constant. ∎

Data availability Data sharing is not applicable to this article since no data
sets were generated or analyzed.

Declarations

Conflict of interest The authors declare that they have no competing
interests.

Funding The first and second named authors were partially supported by Grants
PID2019-106870GB-I00 and PID2022-138916NB-I00 (respectively) of the MICINN of
Spain, by ICMAT Severo Ochoa project CEX2019-000904-S (MICINN), and by the
Madrid Government (Comunidad de Madrid - Spain) V PRICIT (Regional Programme
of Research and Technological Innovation), 2022-2024.

## References

  * [1] Aldaz J.M. and Render, H.:A Fischer type decomposition theorem from the apolar inner product. Anal. Math. Phys. 13, 91, 29 pp. (2023). Available at the Math. ArXiv.
  * [2] Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, (1961) 187–214.
  * [3] Beauzamy, B., Bombieri, E., Enflo, P., Montgomery, H.L.: Products of Polynomials in Many Variables, J. Number Theory 36 (1990) 219–245.
  * [4] Beauzamy, B.: Products of polynomials and a priori estimates for coefficients in polynomial decompositions: A sharp result. J. Symbolic Computation 13 (1992), 463-472.
  * [5] Beauzamy, B.: Reconstructing a signal from the knowledge of the norms of its multiples. Numerical Algorithms 12 (1996) 409–414.
  * [6] Beauzamy, B.: Extremal products in Bombieri’s norm. Rend. Istit. Mat. Univ. Trieste, Suppl. Vol. XXVIII (1997), 73–89.
  * [7] Ebenfelt, P. Shapiro, H.S.: The Cauchy–Kowaleskaya theorem and Generalizations. Commun. Partial Differential Equations 20 (1995) 939–960.
  * [8] Ebenfelt, P. Shapiro, H.S.: The mixed Cauchy problem for holomorphic partial differential equations. J. D’Analyse Math. 65 (1996) 237–295.
  * [9] Etayo, U.: A sharp Bombieri inequality, logarithmic energy and well conditioned polynomials. Trans. Amer. Math. Soc. 374 (2021) 5113–5129.
  * [10] Khavinson,D., Lundberg, E.: _Linear holomorphic partial differential equations and classical potential theory._ Amer. Math. Soc. Providence, RI (2018).
  * [11] Newman, D.J., Shapiro, H.S.: Certain Hilbert spaces of entire functions. Bull. Amer. Math. Soc.  72 (1966) 971–977.
  * [12] Pinasco, D.: Lower bounds for norms of products of polynomials via Bombieri inequality. Trans. Amer. Math. Soc. 364 (2012) 3993–4010.
  * [13] Render, H.: Real Bargmann spaces, Fischer decompositions and sets of uniqueness for polyharmonic functions. Duke Math. J. 142 (2008) 313–352.
  * [14] Reznick, B.: An inequality for products of polynomials. Proc. Amer. Math. Soc. 117 no. 4 (1993) 1063–1073.
  * [15] Shapiro, H.S.:_An algebraic theorem of E. Fischer and the Holomorphic Goursat Problem_. Bull. London Math. Soc. 21 (1989) 513–537.
  * [16] Zeilberger, D.: _Chu’s identity implies Bombieri’s 1990 norm-inequality_. Amer. Math. Monthly, 101 (1994) 894–896.


#  One-proton emission from 148-151Lu in the DRHBc+WKB approach

Yang Xiao  School of Space and Environment, Beihang University, Beijing
102206, China  School of Physics, Beihang University, Beijing 102206, China
Si-Zhe Xu  School of Physics, Beihang University, Beijing 102206, China  Ru-
You Zheng  School of Physics, Beihang University, Beijing 102206, China
Xiang-Xiang Sun  Institut für Kernphysik, Institute for Advanced Simulation
and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425
Jülich, Germany  School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 100049, China  Li-Sheng Geng
<EMAIL_ADDRESS>School of Physics, Beihang University, Beijing
102206, China  Peng Huanwu Collaborative Center for Research and Education,
Beihang University, Beijing 100191, China  Beijing Key Laboratory of Advanced
Nuclear Materials and Physics, Beihang University, Beijing, 102206, China
Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province,
China  Shi-Sheng Zhang<EMAIL_ADDRESS>School of Physics, Beihang
University, Beijing 102206, China

###### Abstract

One-proton radioactivity in 149Lu, the latest identified proton emitter, is
studied in the Wentzel-Kramers-Brillouin (WKB) approach with the proton-
nucleus potential extracted from the deformed relativistic Hartree-Bogoliubov
theory in continuum (DRHBc) for the first time. The predicted half-life turns
out to be consistent with the experimental measurement within uncertainties
and (almost) independent of the density functionals in the DRHBc theory. Such
a microscopic self-consistent calculation reveals that 149Lu is oblately
deformed with a quadrupole deformation $\beta_{2}\approx-0.18$, and rules out
the possibility of a prolate quadrupole deformation suggested in the
nonadiabatic quasiparticle model. We also check the validity of this approach
in the description of 150,151Lu and their isomeric states. The deviations of
the predicted half-lives from their experimental counterparts are mostly
smaller than those of the theoretical studies without considering deformation
effects. Furthermore, we predict 148Lu to be a more oblately deformed proton-
emitter with a longer half-life than that of 149Lu, which can be checked in
the future. Our studies show that the DRHBc + WKB approach provides a new
alternative method to evaluate the half-lives of well-deformed proton
emitters.

##  I Introduction

The exact location of the proton dripline and decay patterns of those nuclei
close to the dripline are of great theoretical and experimental interests. The
half-lives of proton-emitters are sensitive to the decay energy, the orbital
angular momentum of the emitted proton, and the spectroscopic information of
the daughter nucleus Delion  _et al._ (2006a). Therefore, it is challenging to
precisely predict their lifetimes and sometimes even the order of magnitudes.

The conception of proton radioactivity was proposed even earlier than the
discovery of proton Marsden and Lantsberry (1915). The first proton emitter
was observed and confirmed in an isomeric state of 53Co in 1970 Cerny  _et
al._ (1970); Jackson  _et al._ (1970); Cerny  _et al._ (1972). Since then,
more than 30 proton emitters have been discovered Woods and Davids (1997);
Blank and Borge (2008); Pfutzner  _et al._ (2012); Auranen  _et al._ (2022).
Up to now, the properties of these proton emitters have been investigated in
both macroscopic and microscopic methods, for reviews see, e.g., Refs. Blank
and Borge (2008); Qi  _et al._ (2019). Lately, a novel ground-state proton
emitter 149Lu was discovered with the shortest directly measured half-life of
$T_{1/2}=450^{+170}_{-100}$ ns Auranen  _et al._ (2022). It is the purpose of
the present work to study the half-life of this nucleus and explore how its
shape, i.e., deformation, affects the result in a microscopic theory.

There are various ways to evaluate the half-lives of proton emitters Woods and
Davids (1997); Davids  _et al._ (1997); Aberg  _et al._ (1997); Maglione  _et
al._ (1998); Guzmán  _et al._ (1999); Balasubramaniam and Arunachalam (2005);
Basu  _et al._ (2005); Delion  _et al._ (2006b); Bhattacharya and Gangopadhyay
(2007); Dong  _et al._ (2009); Zhang  _et al._ (2010); Routray  _et al._
(2011); and and and (2010); Ferreira  _et al._ (2011); Zhao  _et al._ (2014);
Qian and Ren (2016); Lim  _et al._ (2016); Zdeb  _et al._ (2016); Santhosh and
Sukumaran (2017); Chen  _et al._ (2019a, b); Cheng  _et al._ (2020a, b); Oishi
(2023), including perturbative approaches, such as the distorted wave Born
approximation (DWBA) Aberg  _et al._ (1997), and the modified two potential
approach (TPA) Aberg  _et al._ (1997); Qian and Ren (2016); Cheng  _et al._
(2020a, b). Except those, the WKB approach is commonly used to calculate decay
widths because of its simplicity and effectiveness Woods and Davids (1997);
Davids  _et al._ (1997); Aberg  _et al._ (1997); Guzmán  _et al._ (1999);
Balasubramaniam and Arunachalam (2005); Basu  _et al._ (2005); Delion  _et
al._ (2006b); Bhattacharya and Gangopadhyay (2007); Dong  _et al._ (2009);
Zhang  _et al._ (2010); Routray  _et al._ (2011); and and and (2010); Ferreira
_et al._ (2011); Zhao  _et al._ (2014); Lim  _et al._ (2016); Zdeb  _et al._
(2016); Santhosh and Sukumaran (2017); Chen  _et al._ (2019a, b). Despite some
conceptual differences between the WKB and perturbative approaches, the
predicted half-lives based on the same nuclear potentials are rather similar,
as shown in Refs. Aberg  _et al._ (1997); Xu (2022). In particular, we note
that the WKB results agrees well with the exact solutions for the widths of
neutron resonant states of 17O, crucial for nucleosynthesis Xu and Zhang
(2022). Therefore, in the present work, we adopt the WKB approach to study
one-proton emission.

Up to now, the short half-life of 149Lu has only been studied in the
nonadiabatic quasiparticle model Auranen  _et al._ (2022). However, its
deformation cannot yet be completely determined. In Ref. Auranen  _et al._
(2022), it was found that 149Lu could be either the most oblately deformed
proton emitter ever observed with a quadrupole deformation
$\beta_{2}\lesssim-0.17$ or a prolately deformed proton emitter with a
quadrupole deformation $\beta_{2}\approx 0.15$. We note that most previous
theoretical works predict an oblate shape for 149Lu Lalazissis  _et al._
(1999); Möller  _et al._ (2016, 2019); Geng  _et al._ (2004).

One of the most important inputs in theoretical studies of proton emissions is
the nuclear potential. In a phenomenological model, although by adjusting free
parameters of the model, a reasonable explanation of experimental half-lives
can be achieved, the uncertainties might be large because of the non-
uniqueness of the nuclear potential in the case that experimental measurements
of nucleon levels are not available. In that case, the predicting power of
phenomenological model is rather limited. Therefore, microscopic potentials
obtained in a self-consistent way are more reliable, which can account for
various effects, such as deformations, pairing correlations, low-lying
resonances and the continuum. The deformed relativistic Hartree-Bogoliubov
theory in continuum (DRHBc) theory Zhou  _et al._ (2010); Li  _et al._ (2012a)
turns out to be one of the state of the art densify functional theories that
can provide a microscopic nuclear potential.

In our latest work on an unified description of the one-neutron halo, 31Ne,
both its microscopic structure and the related reaction observables, we found
that the DRHBc theory combined with the Glauber model can reproduce prominent
halo evidences, such as the sudden increase of the reaction cross section of
31Ne bombarding 12C, and the narrow momentum distributions of the breakup
interaction Zhang  _et al._ (2022a); Zhong  _et al._ (2022). While these
crucial features cannot be reproduced using Skyrme forces within the non-
relativistic mean field framework together with the Glauber model Horiuchi
_et al._ (2012). From the structure point of view, the DRHBc theory has
successfully described halo nuclei Zhou  _et al._ (2010); Li  _et al._
(2012a); Meng and Zhou (2015); Sun  _et al._ (2018); Zhang  _et al._ (2019);
Sun  _et al._ (2020); Yang  _et al._ (2021); Sun and Zhou (2021); Sun (2021);
Zhang  _et al._ (2023) and shell evolution Sun  _et al._ (2020) with
deformation, pairing, and continuum effects properly taken into account In
_et al._ (2021); He  _et al._ (2021); Zhang  _et al._ (2022b); Pan  _et al._
(2022). These fruitful achievements trigger us to exploit the potential of the
DRHBc theory to determine the deformation of the newly measured 149Lu, and
provide structure inputs for the WKB approach to evaluate the half-life of the
proton emitter 149Lu and those of its neighbouring nuclei.

This article is composed of three sections. Firstly, we briefly review the
main formalism for the DRHBc+WKB approach in Sect. II. In this approach, the
half-lives, deformations, and spectroscopic factors of proton emitter 149Lu
and its neighbouring nuclei are calculated and analyzed in Sect. III. Finally,
we provide a brief summary in Sect. IV.

##  II Framework

###  II.1 The WKB approximation

Inspired by our previous successful studies of the neutron decay widths Xu and
Zhang (2022), we adopt the simple and effective semi-classical WKB
approximation to calculate the proton emission half-life, i.e.,

$\displaystyle T_{1/2}=\frac{\hbar\ln{2}}{S_{F}\it{\Gamma}},$ (1)

where the spectroscopic factor $S_{F}$ for a nucleus with odd atomic number
$Z$ in the independent-quasiparticle approximation can be written as Sorensen
_et al._ (1966)

$\displaystyle S_{F}=u^{2},$ (2)

in which $u^{2}$ denotes the nonoccupation probability of the corresponding
orbital in the daughter nucleus. In this work, the value of $S_{F}$ is
obtained in the DRHBc framework self-consistently. The final decay width
$\it{\Gamma}$ is obtained by averaging $\it{\Gamma}(\theta)$ in all directions
with $\theta$ the orientation angle of the proton with respect to the
symmetric axis of the daughter nucleus Qian and Ren (2016).

In the WKB approximation, $\it{\Gamma}(\theta)$ is approximated as

$\displaystyle{\it{\Gamma}}(\theta)=N\frac{\hbar^{2}}{4\mu}\exp{\left(-2\int_{r_{2}}^{r_{3}}k(r,\theta)\text{d}r\right)},$
(3)

in which $k(r,\theta)$ is related to the decay energy $E$ and potential
barrier $V(r,\theta)$, i.e., $k(r,\theta)=\sqrt{2\mu|E-V(r,\theta)|}$, $\mu$
refers to the reduced mass of the emitted proton and the daughter nucleus, $r$
denotes the distance between the proton and the mass center of the daughter
nucleus, and $r_{i}$ corresponds to the $i$-th turning point at barrier
$V(r,\theta)$ with respect to the decay energy $E$. The quasiclassic bound-
state normalization factor $N$ in Eq.(3) is defined by Buck  _et al._ (1992)

$\displaystyle
N^{-1}(\theta)=\int_{r_{1}}^{r_{2}}\frac{\text{d}r}{k(r,\theta)}\cos^{2}\left(\int_{r_{1}}^{r}\text{d}r^{\prime}k(r^{\prime},\theta)-\frac{\pi}{4}\right).$
(4)

The potential barrier consists of three parts, i.e., nuclear, Coulomb and
centrifugal potentials. It reads

$\displaystyle
V(r,\theta)=V_{N}(r,\theta)+V_{C}(r,\theta)+\frac{\hbar^{2}}{2\mu}\frac{l(l+1)}{r^{2}},$
(5)

where $V_{N/C}(r,\theta)$ refers to the nuclear/Coulomb potential
respectively, which is self-consistently calculated in the DRHBc theory.

###  II.2 The DRHBc theory

The potential $V(r,\theta)$ is crucial to calculate the half-life of a proton
emitter. As stated above, we choose the DRHBc theory to provide the potential
in a self-consistent way, in which the deformed RHB equations are solved with
the Dirac Woods–Saxon basis Zhou  _et al._ (2003) and the blocking effects are
taken into account accordingly Li  _et al._ (2012b). We briefly introduce the
DRHBc theory in what follows. One can find more details in Refs. Li  _et al._
(2012a); Zhang  _et al._ (2020) and references cited therein.

With the mean-field and no-sea approximations, the RHB equations for the
nucleons treat the mean field and pairing correlations self-consistently,
which read Kucharek and Ring (1991)

$\displaystyle\left(\begin{array}[]{cc}h_{D}-\lambda_{\tau}&\Delta\\\
-\Delta^{*}&-h_{D}^{*}+\lambda_{\tau}\end{array}\right)\left(\begin{array}[]{c}U_{k}\\\
V_{k}\end{array}\right)=E_{k}\left(\begin{array}[]{c}U_{k}\\\
V_{k}\end{array}\right),$ (12)

where $h_{D}$ is the Dirac Hamiltonian, $\lambda_{\tau}$ is the Fermi energy
for protons/neutrons ($\tau=p/n$), $\Delta$ is the pairing potential, $E_{k}$
and $\left(U_{k},V_{k}\right)^{T}$ refer to the quasiparticle energy and wave
function, respectively. In coordinate space, $h_{D}$ reads

$\displaystyle
h_{D}(\bm{r})=\bm{\alpha}\cdot\bm{p}+V(\bm{r})+\beta\left[M+S(\bm{r})\right],$
(13)

where $V(\bm{r})/S(\bm{r})$ is the vector/scalar potential. The pairing
potential $\Delta$ reads

$\displaystyle\Delta(\bm{r_{1}},\bm{r_{2}})=V^{pp}(\bm{r_{1}},\bm{r_{2}})\kappa(\bm{r_{1}},\bm{r_{2}}),$
(14)

with $\kappa$ the pairing tensor. The pairing interaction in the particle-
particle channel $V^{pp}$ takes the form of a density-dependent $\delta$
function

$\displaystyle
V^{pp}(\bm{r_{1}},\bm{r_{2}})=V_{0}\frac{1-P^{\sigma}}{2}\delta(\bm{r_{1}}-\bm{r_{2}})\left(1-\frac{\rho(\bm{r_{1}})}{\rho_{\text{sat}}}\right),$
(15)

where $V_{0}$ refers to the pairing strength, $\rho_{\text{sat}}$ denotes the
saturation density of nuclear matter, and $\frac{1-P^{\sigma}}{2}$ is the
projector into the spin $S=0$ component.

For axially deformed nuclei, the scalar potential, vector potential, and
densities are expanded in terms of the Legendre polynomials Zhang  _et al._
(2022b)

$\displaystyle
f(\bm{r})=\sum_{\lambda}f_{\lambda}(r)P_{\lambda}(\cos\theta),~{}~{}~{}\lambda=\text{even
number}.$ (16)

The single proton potential equals to the summation of vector potential
$V(\bm{r})$ and scalar potentials $S(\bm{r})$, which can be obtained in a
self-consistent manner. One can find the details in Ref. Zhang  _et al._
(2020) and references cited therein.

##  III Results and discussion

We calculate the half-life of proton emitter 149Lu in the WKB approximation
with the proton-nucleus potential provided by the DRHBc theory. In this
approach, three relativistic density functionals, namely PC-PK1 Zhao  _et al._
(2010), NL3* Lalazissis  _et al._ (2009), and TMA Sugahara and Toki (1994),
are selected for the study. Due to the similarity of the results, we take the
density-functional PC-PK1 as an example to show in Fig. 1 the single-proton
levels of 149Lu around the proton Fermi energy. A dominant $\pi h_{11/2}$
component for the ground-state wave-function of 149Lu is suggested in Ref.
Auranen  _et al._ (2022), which splits into six orbitals from ${1}/{2}^{-}$ to
${11}/{2}^{-}$ once the nucleus becomes deformed. The proton orbital is
supposed to be blocked at orbitals $5/2^{-}$ and $7/2^{-}$ respectively, since
both are consistent with the emission of a proton with $l_{p}=5$ and their
level energies are close to the proton Fermi energy. The ground state of 149Lu
is identified by minimizing the energy of the system.

Figure 1: Ground-state (g.s.) single-proton levels nearby the Fermi surface
obtained in the DRHBc theory with the density functional PC-PK1. $\lambda_{p}$
denotes the Fermi energy. The length of the solid/dot lines represent the
occupation/non-occupation probabilities of the single-particle orbitals.
Quantum number $\Omega^{\pi}$ and the main Woods-Saxon components are given
for orbitals in the $sd$ shell and $h$ shell.

In Fig. 2, the potential energies of 149Lu as a function of the quadrupole
deformation are displayed, which are obtained in the DRHBc theory with density
functionals PC-PK1, NL3*, and TMA, respectively. One can see that the
predicted quadrupole deformation of the ground state of 149Lu is relatively
close, i.e., $-0.178$ for PC-PK1, $-0.185$ for NL3*, and $-0.197$ for TMA,
although the energies of the ground states obtained with the three density
functionals are different. Moreover, the minima of the potential energy curve
support our choice to block the valence proton at the orbital $5/2^{-}$ for
all the three density functionals. We note that the second minimum is slightly
higher than the first minimum by about 0.5 MeV.

Figure 2: Potential energy of 149Lu as a function of quadrupole deformation
calculated in the DRHBc theory with PC-PK1, NL3*, and TMA density functional,
respectively. The experimental energy range in gray is obtained from the
binding energy of 148Yb from the NNDC database cit (2023) and the experimental
decay energy of 149Lu Auranen  _et al._ (2022) taking into account
experimental uncertainties. The red solid/ black dotdashed/ blue dashed lines
represent the potential energies obtained with PC-PK1/ NL3*/ TMA density
functionals. The band in the light red area covers the deformation of the
ground states self-consistently obtained in the DRHBc theory.

To appreciate the fact that the DRHBc theory can successfully describe the
ground-state properties of 149Lu, we list the nuclear structure inputs in
Table 1. With the WKB approximation, the half-life of this proton emitter is
computed and compared with the experimental measurement as well. It can be
seen that 149Lu is found to be oblately deformed in the DRHBc theory for all
three density functionals, which agrees with one of the predictions of the
nonadiabatic quasiparticle model, i.e., $\beta_{2}\lesssim-0.17$. The
calculated decay energy is of the same order of the magnitude as the
experimental one. Since the half-life of a proton emiter is very sensitive to
the $Q$ value Blank and Borge (2008); Delion  _et al._ (2006a); Qi  _et al._
(2019), we utilize the experimental decay energy $Q$ in the calculations. In
this way, the half-life evaluated in the DRHBc + WKB approach is in good
agreement with the experimental value at $1\sigma$ significance level, without
any adjustable parameter. This indicates that the DRHBc theory correctly
predicts the dominant contribution of the mean-field potential for 149Lu and
the (non)occupation probability for 148Yb.

Table 1: Half-life of proton emitter 149Lu. Note that the experimental quadrupole deformation here refers to that of the nonadiabatic quasiparticle model instead of the experimental measurement as explained in Ref. Auranen  _et al._ (2022). 149Lu |  Exp. Auranen  _et al._ (2022) | DRHBc (+ WKB)
---|---|---
|  |  PC-PK1 |  NL3* |  TMA
Orbital |  $1h_{{11/2}^{-}}$ |  $5/2^{-}$ |  $5/2^{-}$ |  $5/2^{-}$
$Q$ (MeV) |  1.92(2) |  1.42 |  1.16 |  1.22
$S_{F}$ |  / |  0.77 |  0.76 |  0.68
$\beta_{2}$ |  $\lesssim$ $-$ 0.17 or $\approx$ 0.15 |  $-$ 0.178 |  $-$ 0.185 |  $-$ 0.197
$T_{1/2}$ (ns) |  $450_{-100}^{+170}$ |  $538_{-131}^{+159}$ |  $469_{-110}^{+146}$ |  $544_{-128}^{+169}$

In the nonadiabatic quasiparticle model, a prolate deformation of
$\beta_{2}\approx 0.15$ Auranen  _et al._ (2022) was also recommended. In our
model, the minimum at $\beta_{2}\approx 0.15$ is the second minimum, about 0.5
MeV above the ground state as shown in Fig. 2. Therefore, we further study the
half-life dependence on the deformation of 149Lu so that we can constrain the
deformation by the measured half-life. Our predictions in the DRHBc + WKB
approach with the PC-PK1, NL3* and TMA density functionals are plotted in Fig.
3, respectively. The calculated half-lives agree with the experimental data
denoted by the grayed shadow region for the deformation range of
$-0.25\lesssim\beta_{2}\lesssim 0.06$, and are independent of the density
functionals. With the increase of the deformation $\beta_{2}$, our predictions
exceed the gray region. We studied the rapid enhancement of half-life at
$\beta_{2}\approx 0.1$ and found that it is caused by a switch of the orbit of
emitted proton from ${5/2^{-}}$ to ${7/2^{-}}$. As a result, the spectroscopic
factors shift from $0.77$, $0.76$, $0.68$ to $0.44$, $0.38$, $0.28$ for the
PC-PK1, NL3* and TMA density functionals, respectively. Hence, the calculated
half-lives suddenly increase according to Eq. (1).

We note that the calculated half-lives fall again within the experimental
measurements as the quadrupole deformation $\beta_{2}$ increases to a
relatively large value of $\sim 0.3$. However, such a large prolate
deformation of 149Lu is not supported by the DRHBc theory because the ground
state energy is much larger than the experimental value. Therefore, the
prolate deformation of $\beta_{2}>0.06$, including $\beta_{2}\approx 0.15$
predicted by the nonadiabatic quasiparticle model and $\beta_{2}<-0.25$ are
out of the deformation range predicted by the DRHBc theory, which are denoted
in green and distinguished from the strongly recommended region in light red
in Fig. 3. The deformation dependence of the half-life of 149Lu is still in
debate and needs to be studied in other microscopic approaches and checked by
future experiments.

Figure 3: Calculated and experimental half-life of 149Lu as a function of
quadrupole deformation. The red solid/ black dot-dashed / blue dashed line
represents the calculated half-life in the DRHBc theory with the PC-PK1/ NL3*/
TMA density functional. The grey line denotes the central value of the
experimental half-life and the shadow region represents the experimental error
bar. The solid dots in the light red region correspond to the quadrupole
deformations and half-lives of the ground-states obtained with three density
functionals. The light green regions cover the values of the quadrupole
deformations of 149Lu excluded in our calculations.

As detailed above, the DRHBc + WKB approach is applicable to evaluate the
half-life of 149Lu, which agrees well with the latest experimental
measurement. We naturally expect that it also works for other lutetium proton-
emitters, such as 150,151Lu and their isomers 150m,151mLu. If so, we will have
the confidence to study other candidates, such as 148Lu. To study isomeric
states, we take the same nuclear ground-state potentials Lim  _et al._ (2016).
In Table 2 we show the calculated half-lives of 150,151Lu and their isomers
150m,151mLu, and predict the half-life of 148Lu in the DRHBc + WKB approach.
The results obtained in the spherical relativistic continuum Hartree-
Bogoliubov (RCHB) theory + WKB approach Lim  _et al._ (2016) are also listed
for comparison. It can be seen that the results from the DRHBc + WKB approach
based on the PC-PK1 density functional are in general consistent with those
from the RCHB + WKB calculations. Define the deviation of theoretical
predictions from the experimental data $\chi$ by
$\chi=\left|\frac{T_{1/2}^{\text{theor.}}-T_{1/2}^{\text{exp.}}}{T_{1/2}^{\text{exp.}}}\right|$,
we can see from Table 2 that the $\chi$ obtained in the DRHBc + WKB approach
is overall half of that obtained in the RCHB + WKB approach except for 150mLu.
This highlights the importance of deformation effects to accurately describe
the half-lives of the deformed proton emitters. The deviation between the
DRHBc + WKB result and the experimental value for 150mLu may be due to the
overestimated spectroscopic factor, which is reduced in ab initio calculations
Srivastava and Kumar (2016). For 148Lu, our study suggests that it is a more
oblately deformed proton emitter with a half-life longer than that of 149Lu,
which can be checked by future experiments. Although the NL3* density
functional can better reproduce the half-life for the case of 149Lu which can
be seen from Fig. 3, it is not as good as the PC-PK1 density functional in the
overall description of the half-lives of lutetium proton emitters, such as
150,150m,151,151mLu.

Table 2: Calculated and measured half-lives of lutetium proton emitters. The $\beta_{2}$ values are provided by the DRHBc theory. The experimental data are taken from Refs. Blank and Borge (2008); Auranen  _et al._ (2022). The RCHB results from Ref. Lim  _et al._ (2016) are shown for comparison. Both DRHBc and RCHB results are based on the PC-PK1 density functional. The $5/2^{-}$ orbital is a component of $1h_{11/2}$, while the $3/2^{+}$ orbital is a mixture of the configuration $2d_{3/2^{+}}$ and $2d_{5/2^{+}}$. The deviation $\chi$ is defined as $\chi=\left|\frac{T_{1/2}^{\text{theor.}}-T_{1/2}^{\text{exp.}}}{T_{1/2}^{\text{exp.}}}\right|$. Nucleus |  $\beta_{2}$ | Orbital |  $Q$ (MeV) | $S_{F}$ | $T_{1/2}$ |  $\chi$ ($\%$)
---|---|---|---|---|---|---
|  |  Exp. |  DRHBc |  RCHB |  Exp. |  DRHBc |  RCHB |  DRHBc |  RCHB |  Exp. |  DRHBc |  RCHB |  DRHBc |  RCHB
148Lu |  $-0.184$ |  / |  $5/2^{-}$ |  / |  / |  1.70 |  / |  0.788 |  / |  / |  13.92$\mu$s |  / |  / |  /
149Lu |  $-0.178$ |  $1h_{11/2^{-}}$ |  $5/2^{-}$ |  / |  1.920(20) |  1.42 |  / |  0.771 |  / |  $450^{+170}_{-100}$ns |  $538^{+159}_{-131}$ns |  / |  19.56 |  /
150Lu |  $-0.166$ |  $1h_{11/2^{-}}$ |  $5/2^{-}$ |  $1h_{11/2^{-}}$ |  1.283(3) |  1.08 |  1.31 |  0.753 |  0.698 |  64.0(56)ms |  $51.11^{+3.89}_{-7.58}$ms |  88.46ms |  20.14 |  38.22
150mLu |  / |  $2d_{3/2^{+}}$ |  $3/2^{+}$ |  $2d_{3/2^{+}}$ |  1.306(5) |  / |  / |  0.692 |  0.552 |  $43^{+7}_{-5}\mu$s |  $27.34^{+3.40}_{-3.00}\mu$s |  35.72$\mu$s |  36.42 |  16.93
151Lu |  $-0.157$ |  $1h_{11/2^{-}}$ |  $5/2^{-}$ |  $1h_{11/2^{-}}$ |  1.253(3) |  0.82 |  1.03 |  0.733 |  0.610 |  127.1(18)ms |  $109.37^{+8.67}_{-7.95}$ms |  176.38ms |  13.95 |  38.78
151mLu |  / |  $2d_{3/2^{+}}$ |  $3/2^{+}$ |  $2d_{3/2^{+}}$ |  1.332(10) |  / |  / |  0.686 |  0.579 |  16(1)$\mu$s |  $14.99^{+3.84}_{-3.03}\mu$s |  18.03$\mu$s |  6.31 |  12.69

##  IV Summary and prospects

In summary, we have studied the one-proton emission of the newly identified
proton emitter 149Lu in the DRHBc + WKB approach. Using the self-consistently
achieved proton-nucleus potential and spectroscopic factor obtained in the
DRHBc theory with three different density functionals, we evaluated the half-
life of 149Lu proton emission in the WKB approximation. Our theoretical
predictions are in good agreement with the experiment data within
uncertainties, i.e., $538^{+159}_{-131}$ns, $469^{+146}_{-110}$ns,
$544^{+169}_{-128}$ns for PC-PK1, NL3* and TMA density functionals
respectively, which imply the robutness of our approach. Using the half-life
as a constraint, we determine the quadrupole deformation of 149Lu,
$\beta_{2}\approx-0.18$. Our study rules out the possibility of a prolately
deformed 149Lu suggested in the nonadiabatic quasiparticle model. This can be
confirmed by future $\gamma$-ray spectroscopy experiments and checked by other
theoretical works.

In the same approach, we also studied one-proton emission of other lutetium
isotopes near 149Lu. It turns out that the results of the DRHBc+ WKB approach
are on average better than those of the RCHB + WKB approach, which shows that
the deformation effects play a relevant role in the study of proton-emission.
Furthermore, we predicted 148Lu as a promising one-proton emitter with a
oblate deformation $\beta_{2}=-0.184$, which can be measured in the future.

In summary, we showed that the DRHBc+WKB approach is an effective and
efficient method to evaluate the half-lives of the newly discovered 149Lu and
its neighbours. The sensitivity of the half-live to the deformation effects
allows one to use the half-life to determine the deformation. Similar studies
of two-proton emission are in progress.

##  V Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  VI Data availability

Data will be made available on request.

##  VII Acknowledgement

We would like to thank Dr. Yi-Bin Qian and Dr. Kai-Yuan Zhang for valuable
discussion. We also thank the DRHBc Mass Table collaboration for providing the
DRHBc code and for useful suggestions. This work was partially supported by
the National Natural Science Foundation of China (Grant Nos. 12175010,
11975041, 11961141004, 1220530), the China Postdoctoral Science Foundation
under Grant No. 2022M720360, the Deutsche Forschungsgemeinschaft (DFG) and the
National Natural Science Foundation of China (NSFC) through the funds provided
to the Sino-German Collaborative Research Center TRR110 “Symmetries and the
Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG Project-ID
196253076).

## References

  * Delion  _et al._ (2006a) D. Delion, R. Liotta, and R. Wyss, Physics Reports 424, 113 (2006a).
  * Marsden and Lantsberry (1915) E. Marsden and W. Lantsberry, Phil. Mag. 30, 240 (1915).
  * Cerny  _et al._ (1970) J. Cerny, J. E. Esterl, R. A. Gough, and R. G. Sextro, Phys. Lett. B 33, 284 (1970).
  * Jackson  _et al._ (1970) K. P. Jackson, C. U. Cardinal, H. C. Evans, N. A. Jelley, and J. Cerny, Phys. Lett. B 33, 281 (1970).
  * Cerny  _et al._ (1972) J. Cerny, R. A. Gough, R. G. Sextro, and J. E. Esterl, Nucl. Phys. A 188, 666 (1972).
  * Woods and Davids (1997) P. J. Woods and C. N. Davids, Ann. Rev. Nucl. Part. Sci. 47, 541 (1997).
  * Blank and Borge (2008) B. Blank and M. Borge, Prog. Part. Nucl. Phys. 60, 403 (2008).
  * Pfutzner  _et al._ (2012) M. Pfutzner, M. Karny, L. V. Grigorenko, and K. Riisager, Rev. Mod. Phys. 84, 567 (2012).
  * Auranen  _et al._ (2022) K. Auranen _et al._ , Phys. Rev. Lett. 128, 112501 (2022).
  * Qi  _et al._ (2019) C. Qi, R. Liotta, and R. Wyss, Prog. Part. Nucl. Phys. 105, 214 (2019).
  * Davids  _et al._ (1997) C. N. Davids _et al._ , Phys. Rev. C 55, 2255 (1997).
  * Aberg  _et al._ (1997) S. Aberg, P. B. Semmes, and W. Nazarewicz, Phys. Rev. C 56, 1762 (1997), [Erratum: Phys.Rev.C 58, 3011–3011 (1998)].
  * Maglione  _et al._ (1998) E. Maglione, L. S. Ferreira, and R. J. Liotta, Phys. Rev. Lett. 81, 538 (1998).
  * Guzmán  _et al._ (1999) F. Guzmán, M. Gonçalves, O. A. P. Tavares, S. B. Duarte, F. García, and O. Rodríguez, Phys. Rev. C 59, R2339 (1999).
  * Balasubramaniam and Arunachalam (2005) M. Balasubramaniam and N. Arunachalam, Phys. Rev. C 71, 014603 (2005).
  * Basu  _et al._ (2005) D. N. Basu, P. R. Chowdhury, and C. Samanta, Phys. Rev. C 72, 051601 (2005).
  * Delion  _et al._ (2006b) D. S. Delion, R. J. Liotta, and R. Wyss, Phys. Rev. Lett. 96, 072501 (2006b).
  * Bhattacharya and Gangopadhyay (2007) M. Bhattacharya and G. Gangopadhyay, Phys. Lett. B 651, 263 (2007).
  * Dong  _et al._ (2009) J. M. Dong, H. F. Zhang, and G. Royer, Phys. Rev. C 79, 054330 (2009).
  * Zhang  _et al._ (2010) H. F. Zhang, Y. J. Wang, J. M. Dong, J. Q. Li, and W. Scheid, J. Phys. G: Nucl. Part. Phys. 37, 085107 (2010).
  * Routray  _et al._ (2011) T. R. Routray, S. K. Tripathy, B. B. Dash, B. Behera, and D. N. Basu, Eur. Phys. J. A 47, 92 (2011).
  * and and and (2010) and and and, Chinese Physics Letters 27, 112301 (2010).
  * Ferreira  _et al._ (2011) L. S. Ferreira, E. Maglione, and P. Ring, Phys. Lett. B 701, 508 (2011).
  * Zhao  _et al._ (2014) Q. Zhao, J. M. Dong, J. L. Song, and W. H. Long, Phys. Rev. C 90, 054326 (2014).
  * Qian and Ren (2016) Y. Qian and Z. Ren, Eur. Phys. J. A 52, 68 (2016).
  * Lim  _et al._ (2016) Y. Lim, X. Xia, and Y. Kim, Phys. Rev. C 93, 014314 (2016).
  * Zdeb  _et al._ (2016) A. Zdeb, M. Warda, C. M. Petrache, and K. Pomorski, Eur. Phys. J. A 52, 323 (2016).
  * Santhosh and Sukumaran (2017) K. P. Santhosh and I. Sukumaran, Phys. Rev. C 96, 034619 (2017).
  * Chen  _et al._ (2019a) J.-L. Chen, X.-H. Li, J.-H. Cheng, J.-G. Deng, and X.-J. Wu, J. Phys. G 46, 065107 (2019a).
  * Chen  _et al._ (2019b) J.-L. Chen, J.-Y. Xu, J.-G. Deng, X.-H. Li, B. He, and P.-C. Chu, Eur. Phys. J. A 55, 214 (2019b).
  * Cheng  _et al._ (2020a) J.-H. Cheng, X. Pan, Y.-T. Zou, X.-H. Li, Z. Zhang, and P.-C. Chu, Eur. Phys. J. A 56, 273 (2020a).
  * Cheng  _et al._ (2020b) J.-H. Cheng, J.-L. Chen, J.-G. Deng, X.-H. Li, Z. Zhang, and P.-C. Chu, Nucl. Phys. A 997, 121717 (2020b).
  * Oishi (2023) T. Oishi, Phys. Rev. C 107, 034301 (2023).
  * Xu (2022) S.-Z. Xu, Master’s thesis, Beihang University (2022).
  * Xu and Zhang (2022) S.-Z. Xu and S.-S. Zhang, in  _European Physical Journal Web of Conferences_, European Physical Journal Web of Conferences, Vol. 260 (2022) p. 11037.
  * Lalazissis  _et al._ (1999) G. A. Lalazissis, D. Vretenar, and P. Ring, Phys. Rev. C 60, 051302 (1999).
  * Möller  _et al._ (2016) P. Möller, A. J. Sierk, T. Ichikawa, and H. Sagawa, Atom. Data Nucl. Data Tabl. 109-110, 1 (2016).
  * Möller  _et al._ (2019) P. Möller, M. R. Mumpower, T. Kawano, and W. D. Myers, Atom. Data Nucl. Data Tabl. 125, 1 (2019).
  * Geng  _et al._ (2004) L. S. Geng, H. Toki, and J. Meng, Prog. Theor. Phys. 112, 603 (2004).
  * Zhou  _et al._ (2010) S.-G. Zhou, J. Meng, P. Ring, and E.-G. Zhao, Phys. Rev. C 82, 011301 (2010).
  * Li  _et al._ (2012a) L. Li, J. Meng, P. Ring, E.-G. Zhao, and S.-G. Zhou, Phys. Rev. C 85, 024312 (2012a).
  * Zhang  _et al._ (2022a) S. Zhang, S. Zhong, B. Shao, and M. S. Smith, J. Phys. G 49, 025102 (2022a).
  * Zhong  _et al._ (2022) S.-Y. Zhong, S.-S. Zhang, X.-X. Sun, and M. S. Smith, Sci. China Phys. Mech. Astron. 65, 262011 (2022).
  * Horiuchi  _et al._ (2012) W. Horiuchi, T. Inakura, T. Nakatsukasa, and Y. Suzuki, Phys. Rev. C 86, 024614 (2012).
  * Meng and Zhou (2015) J. Meng and S. G. Zhou, J. Phys. G 42, 093101 (2015).
  * Sun  _et al._ (2018) X.-X. Sun, J. Zhao, and S.-G. Zhou, Phys. Lett. B 785, 530 (2018).
  * Zhang  _et al._ (2019) K. Zhang, D. Wang, and S. Zhang, Phys. Rev. C 100, 034312 (2019).
  * Sun  _et al._ (2020) X.-X. Sun, J. Zhao, and S.-G. Zhou, Nucl. Phys. A 1003, 122011 (2020).
  * Yang  _et al._ (2021) Z. H. Yang _et al._ , Phys. Rev. Lett. 126, 082501 (2021).
  * Sun and Zhou (2021) X.-X. Sun and S.-G. Zhou, Sci. Bull. 66, 2072 (2021).
  * Sun (2021) X.-X. Sun, Phys. Rev. C 103, 054315 (2021).
  * Zhang  _et al._ (2023) K. Y. Zhang, P. Papakonstantinou, M. H. Mun, Y. Kim, H. Yan, and X. X. Sun, Phys. Rev. C 107, L041303 (2023).
  * In  _et al._ (2021) E. J. In, Y. Kim, P. Papakonstantinou, and S.-W. Hong, Int. J. Mod. Phys. E 30, 2150009 (2021).
  * He  _et al._ (2021) X.-T. He, C. Wang, K.-Y. Zhang, and C.-W. Shen, Chin. Phys. C 45, 101001 (2021).
  * Zhang  _et al._ (2022b) K. Zhang _et al._ (DRHBc Mass Table), Atom. Data Nucl. Data Tabl. 144, 101488 (2022b).
  * Pan  _et al._ (2022) C. Pan _et al._ (DRHBc Mass Table), Phys. Rev. C 106, 014316 (2022).
  * Sorensen  _et al._ (1966) R. A. Sorensen, E. K. Lin, and B. L. Cohen, Phys. Rev. 142, 729 (1966).
  * Buck  _et al._ (1992) B. Buck, A. C. Merchant, and S. M. Perez, Phys. Rev. C 45, 1688 (1992).
  * Zhou  _et al._ (2003) S.-G. Zhou, J. Meng, and P. Ring, Phys. Rev. C 68, 034323 (2003).
  * Li  _et al._ (2012b) L. Li, J. Meng, P. Ring, E.-G. Zhao, and S.-G. Zhou, Chin. Phys. Lett. 29, 042101 (2012b).
  * Zhang  _et al._ (2020) K. Zhang _et al._ (DRHBc Mass Table), Phys. Rev. C 102, 024314 (2020).
  * Kucharek and Ring (1991) H. Kucharek and P. Ring, Zeitschrift für Physik A Hadrons and Nuclei 339, 23 (1991).
  * Zhao  _et al._ (2010) P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C 82, 054319 (2010).
  * Lalazissis  _et al._ (2009) G. A. Lalazissis, S. Karatzikos, R. Fossion, D. Pena Arteaga, A. V. Afanasjev, and P. Ring, Phys. Lett. B 671, 36 (2009).
  * Sugahara and Toki (1994) Y. Sugahara and H. Toki, Nucl. Phys. A 579, 557 (1994).
  * cit (2023) https://www.nndc.bnl.gov/nudat3/ (2023).
  * Srivastava and Kumar (2016) P. C. Srivastava and V. Kumar, Phys. Rev. C 94, 064306 (2016).