The document discusses a model of spin-1 vector dark matter interacting with a light vector mediator particle. It begins by reviewing previous related work on other models of dark matter with vector or spin-1 properties. It then outlines plans to build a new model with a massive spin-1 mediator between the standard model and vector dark matter, and hints at possible new phenomena that could arise from this setup. Figures and equations are included to illustrate concepts like the symmetry structure and interactions between the sectors.
Talk for the 26th Fr. Ciriaco Pedrosa, O.P. Memorial Lecture Series and 8th International Symposium on Mathematics and Physics at the University of Santo Tomas (Manila, Philippines). Presented remotely on Nov 26, 2021
Slides for my "lightning talk" at Science Hack Day: San Francisco (2015) on open data sets in particle physics. Small discussion of why some data sets are closed (e.g. LHC) while others are open (e.g. Fermi). Includes some suggested open science projects for the engaged public.
Talk for the 26th Fr. Ciriaco Pedrosa, O.P. Memorial Lecture Series and 8th International Symposium on Mathematics and Physics at the University of Santo Tomas (Manila, Philippines). Presented remotely on Nov 26, 2021
Slides for my "lightning talk" at Science Hack Day: San Francisco (2015) on open data sets in particle physics. Small discussion of why some data sets are closed (e.g. LHC) while others are open (e.g. Fermi). Includes some suggested open science projects for the engaged public.
Attending to Diversity in the Classroom: AnnotatedFlip Tanedo
Talk to high school physics teachers to initiate a discussion about inclusion and diversity in the classroom. Part of the UCR Summer Physics Teacher Academy program. Annotated version. (Slides with green bars represent content that was discussed verbally but did not appear on the slides explicitly.) Some teacher responses included at the end.
The overwhelming observational evidence for the existence of dark matter is only matched by the awkward scarcity of information about what it might actually be. Laboratory searches for dark matter now appear to exclude many of the "weakly interacting massive particle" models that were favored by particle physicists for decades. Where does that leave the hunt for dark matter? If we've left the WIMP behind, what are we looking for? We give a brief, biased, and largely fictional history of the WIMP in order to establish what has and has not been excluded, and why it matters.
This general-interest presentation grew out of discussions with astronomers who wanted to understand why some of their particle physics colleagues are "searching for WIMPs" while the others
have decided to live in a "post-WIMP world."
First identification of_direct_collapse_black_holes_candidates_in_the_early_u...Sérgio Sacani
The first black hole seeds, formed when the Universe was younger than ⇠ 500Myr, are recognized
to play an important role for the growth of early (z ⇠ 7) super-massive black holes.
While progresses have been made in understanding their formation and growth, their observational
signatures remain largely unexplored. As a result, no detection of such sources has been
confirmed so far. Supported by numerical simulations, we present a novel photometric method
to identify black hole seed candidates in deep multi-wavelength surveys.We predict that these
highly-obscured sources are characterized by a steep spectrum in the infrared (1.6−4.5μm),
i.e. by very red colors. The method selects the only 2 objects with a robust X-ray detection
found in the CANDELS/GOODS-S survey with a photometric redshift z & 6. Fitting their
infrared spectra only with a stellar component would require unrealistic star formation rates
(& 2000M# yr−1). To date, the selected objects represent the most promising black hole seed
candidates, possibly formed via the direct collapse black hole scenario, with predicted mass
> 105M#. While this result is based on the best photometric observations of high-z sources
available to date, additional progress is expected from spectroscopic and deeper X-ray data.
Upcoming observatories, like the JWST, will greatly expand the scope of this work.
USC Physics & Astronomy Colloquium, 22 Oct 2018. Laboratory searches for dark matter now exclude many of the “weakly interacting massive particle” models that were favored by particle physicists for decades. We discuss what this means for the theoretical and experimental frontier of particle physics and address what we really mean when we say “WIMP”.
Example Task1 for INT1 Please note that this is a task exa.docxelbanglis
Example Task1 for INT1
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
In the 19th century matter was thought to be made up of
tiny units, called atoms.
John Dalton applied ancient Greek concept from the philosopher,
Democritus (Van Helden, 1995).
Electron is discovered by J. J. Thompson (Stern and
Peredo, 2004).
He discovered negatively charged particles, electrons.
Electrons were smaller that atoms, changing previous ideas.
Subsequent discoveries in quantum mechanics radically
changed our understanding of electron behavior
(Krumeich, n. d.).
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
Circa 1803 John Dalton
introduced Atomic
Theory (Frostburg,
2005).
Matter was composed
of tiny, indivisible units
called atoms.
Atoms were the
smallest unit of matter.
John Dalton
(Worthington, 1895)
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
J. J. Thompson in 1897
discovered electrons while
working with cathode ray
tubes. (Stern and Peredo,
2004).
Since no atoms of gas
were present in the
cathode ray tubes, smaller
particles must have been
present.
This meant that particles
smaller than atoms exist, a
big change from Dalton’s
theory.
J. J. Thompson
(Stotesbury, 1900)
A cathode ray tube
(Crookes, 1879)
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
In 1924 Louis de Broglie
successfully theorized that
electrons act like waves
and particles at the same
time. (Krumeich, n. d.).
This discovery meant that
electrons exhibit wave-
particle duality.
This was a change from
Thompson’s view as
electrons being only
particles.
Louis de Broglie,
(Author unknown)
Artist
conception
of electron
as a standing
wave,
(Kuiper, n.d.)
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
Published May 23, 1833 in the United Kingdom
Michael Faraday continued his work in
discovering the connection between electricity
and magnetism (Faraday, 1833).
First recorded evidence of semiconductors
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
William IV was the ruling monarch
of the United Kingdom (Johnson,
2013).
1833 the UK abolished all slavery
in its colonies (The National
Archives, n. d.).
...
The Phase Field Method: Mesoscale Simulation Aiding Materials DiscoveryPFHub PFHub
Two types of computational materials science, model development and materials discovery. PF is used less than atomic scale methods. PF focused on model development not discovery. How to use PF for materials discovery?
Attending to Diversity in the Classroom: AnnotatedFlip Tanedo
Talk to high school physics teachers to initiate a discussion about inclusion and diversity in the classroom. Part of the UCR Summer Physics Teacher Academy program. Annotated version. (Slides with green bars represent content that was discussed verbally but did not appear on the slides explicitly.) Some teacher responses included at the end.
The overwhelming observational evidence for the existence of dark matter is only matched by the awkward scarcity of information about what it might actually be. Laboratory searches for dark matter now appear to exclude many of the "weakly interacting massive particle" models that were favored by particle physicists for decades. Where does that leave the hunt for dark matter? If we've left the WIMP behind, what are we looking for? We give a brief, biased, and largely fictional history of the WIMP in order to establish what has and has not been excluded, and why it matters.
This general-interest presentation grew out of discussions with astronomers who wanted to understand why some of their particle physics colleagues are "searching for WIMPs" while the others
have decided to live in a "post-WIMP world."
First identification of_direct_collapse_black_holes_candidates_in_the_early_u...Sérgio Sacani
The first black hole seeds, formed when the Universe was younger than ⇠ 500Myr, are recognized
to play an important role for the growth of early (z ⇠ 7) super-massive black holes.
While progresses have been made in understanding their formation and growth, their observational
signatures remain largely unexplored. As a result, no detection of such sources has been
confirmed so far. Supported by numerical simulations, we present a novel photometric method
to identify black hole seed candidates in deep multi-wavelength surveys.We predict that these
highly-obscured sources are characterized by a steep spectrum in the infrared (1.6−4.5μm),
i.e. by very red colors. The method selects the only 2 objects with a robust X-ray detection
found in the CANDELS/GOODS-S survey with a photometric redshift z & 6. Fitting their
infrared spectra only with a stellar component would require unrealistic star formation rates
(& 2000M# yr−1). To date, the selected objects represent the most promising black hole seed
candidates, possibly formed via the direct collapse black hole scenario, with predicted mass
> 105M#. While this result is based on the best photometric observations of high-z sources
available to date, additional progress is expected from spectroscopic and deeper X-ray data.
Upcoming observatories, like the JWST, will greatly expand the scope of this work.
USC Physics & Astronomy Colloquium, 22 Oct 2018. Laboratory searches for dark matter now exclude many of the “weakly interacting massive particle” models that were favored by particle physicists for decades. We discuss what this means for the theoretical and experimental frontier of particle physics and address what we really mean when we say “WIMP”.
Example Task1 for INT1 Please note that this is a task exa.docxelbanglis
Example Task1 for INT1
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
In the 19th century matter was thought to be made up of
tiny units, called atoms.
John Dalton applied ancient Greek concept from the philosopher,
Democritus (Van Helden, 1995).
Electron is discovered by J. J. Thompson (Stern and
Peredo, 2004).
He discovered negatively charged particles, electrons.
Electrons were smaller that atoms, changing previous ideas.
Subsequent discoveries in quantum mechanics radically
changed our understanding of electron behavior
(Krumeich, n. d.).
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
Circa 1803 John Dalton
introduced Atomic
Theory (Frostburg,
2005).
Matter was composed
of tiny, indivisible units
called atoms.
Atoms were the
smallest unit of matter.
John Dalton
(Worthington, 1895)
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
J. J. Thompson in 1897
discovered electrons while
working with cathode ray
tubes. (Stern and Peredo,
2004).
Since no atoms of gas
were present in the
cathode ray tubes, smaller
particles must have been
present.
This meant that particles
smaller than atoms exist, a
big change from Dalton’s
theory.
J. J. Thompson
(Stotesbury, 1900)
A cathode ray tube
(Crookes, 1879)
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
In 1924 Louis de Broglie
successfully theorized that
electrons act like waves
and particles at the same
time. (Krumeich, n. d.).
This discovery meant that
electrons exhibit wave-
particle duality.
This was a change from
Thompson’s view as
electrons being only
particles.
Louis de Broglie,
(Author unknown)
Artist
conception
of electron
as a standing
wave,
(Kuiper, n.d.)
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
Published May 23, 1833 in the United Kingdom
Michael Faraday continued his work in
discovering the connection between electricity
and magnetism (Faraday, 1833).
First recorded evidence of semiconductors
Please note that this is a task example. No portion of this example should be used
in your submission for this assessment. This example cannot be cited as a source.
William IV was the ruling monarch
of the United Kingdom (Johnson,
2013).
1833 the UK abolished all slavery
in its colonies (The National
Archives, n. d.).
...
The Phase Field Method: Mesoscale Simulation Aiding Materials DiscoveryPFHub PFHub
Two types of computational materials science, model development and materials discovery. PF is used less than atomic scale methods. PF focused on model development not discovery. How to use PF for materials discovery?
Seminar slides presenting work on dark matter annihilation into light mediators which subsequently decay in to Standard Model particles. This is motivated by indirect detection signals in gamma rays, such as the recent excess seen in the Fermi Large Area Telescope.
Maxwell Boltzmann Summary Answer the following questions anAbramMartino96
Maxwell Boltzmann: Summary
Answer the following questions and submit your responses as a PDF.
1. Write down one major conclusion you can draw from this week’s laboratory.
Please explain.
2. Describe the experimental evidence that supports your conclusion. Please
explain.
3. Give one example of applications/situations for the finding(s) you described
above in your everyday life outside of physics lab.
4.What
did
you
like
and
dislike
about
this
week;s
lab
Lab: The Maxwell-Boltzmann Distribution* Phys 242
*Some components of this lab are based on the activity developed by Julia Chamberlain & Ingrid Ulbrich
(PhET, UC Boulder; https://phet.colorado.edu/en/contributions/view/3687)
In this lab we will study several macroscopic quantities that can be used to describe a gas and explore the
relationships among these quantities. using a simulation from the PhET team:
https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_en.html
This is a variant of the simulation you used for the Gas Properties lab. The simulation can be run in a
browser. If you have issues with the simulation, try using another browser. If you are unable to run the
simulation, your TA will provide you with remote assistance. When you run the simulation, choose the
“Energy” option. At the very bottom of the screen you will see the other options for the simulation,
including a home button, “Ideal,” “Explore,” “Energy,” and “Diffusion.” If you accidentally navigate to
another area, you can return to the Energy option by clicking the button.
The simulation shows a preset volume. In its initial configuration the box is empty. On the right side of
the screen there is a menu labelled “Particles.” By expanding this menu, you can choose to add so many
heavy or light particles. These particles will enter the volume at a temperature of 300 K in the initial
setup.
Once there are particles in the box, the temperature and pressure in the box can be read off the scales on
the right corner of the box. The units can be changed for these values. To adjust the temperature of the
particles in the box, move and hold the slider bar below the box.
To the left of the box is a graph showing the speed of the particles. This is a histogram. By clicking the
blue and red boxes below the graph, you can see the distributions of the heavy and light particles,
respectively. The box above this shows the average speed of the heavy and light particles.
Below the speed distribution graph is a menu that can be expanded to show the kinetic energy distribution
of the particles. Again, by clicking the blue and red boxes below the graph, you can see the distributions
of the heavy and light particles, respectively.
On the left there is a handle to change the size of the box. There is also a lever at the top of the box that
can be lifted to open the box, allowing particles to escape. Particles ...
A preponderance of scientific evidence over the last hundred years tells us that our galaxy is filled with an unknown substance called dark matter. In fact, there is five times as much dark matter in the universe than there is ordinary matter: we are swimming in an ocean of dark matter and we have no firm idea what it is. We suspect that dark matter is composed of undiscovered elementary particles whose properties may, in turn, unlock some of the most pressing open questions in fundamental physics. So why haven't we figured out how to study dark matter in the lab, and why should we be optimistic that we may make progress in the coming decades?
Presentation about ParticleBites.com efforts in the context of sustainability as part of the Sustainable HEP 2nd ed. workshop. https://indico.cern.ch/event/1160140/timetable/
Presented at the 2022 APS April Meeting, session Z05.00009
Abstract: We present a novel approach for student assessment in large physics lecture courses on student-recorded videos. Students record 5-minute videos teaching how to solve a problem to other students and are partially graded based on peer reviews from other students. After piloting this method during COVID-19 remote teaching over the last year and a half, we have found encouraging indications that it (1) promotes student self-efficacy and metacognition, (2) builds in a deeper engagement with the material, (3) encourages student creativity, (4) develops technical and critical communication ability, and (5) avoids long-standing issues with digital plagiarism. Though the method was developed during pandemic teaching, we propose that aspects can be readily applied to in-person teaching and scales with class size. We comment on the potential to support diverse student retention in physics and outline potential pedagogical trade-offs of this method.
Invited talk at the American Physical Society April Meeting, 9 April 2022.
Like many physical systems, the challenge to make physics more equitable is multiscale. The way in which one perceives and is able to change inequities changes over the early phase of an academic career. These changes reflect the scope of one's academic community, the evolving set of career incentives, and a growing ability to directly influence institutional norms. In this talk we provide a framework for how we engage with equity as early career academics. From this framework, we highlight the ways in which early career academics are uniquely qualified to affect change, and the ways institutions can ensure that these academics continue to be agents for positive change as mid-career scientists.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
Travis Hills' Endeavors in Minnesota: Fostering Environmental and Economic Pr...Travis Hills MN
Travis Hills of Minnesota developed a method to convert waste into high-value dry fertilizer, significantly enriching soil quality. By providing farmers with a valuable resource derived from waste, Travis Hills helps enhance farm profitability while promoting environmental stewardship. Travis Hills' sustainable practices lead to cost savings and increased revenue for farmers by improving resource efficiency and reducing waste.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...
Vector Self-Interacting Dark Matter
1. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
Flip Tanedo
1 MAY 2019
UC Riverside Particle Theory
SPIN-1 DARK MATTER
& W H Y W E M I G H T C A R E
Work in progress with
Ian Chaffey
f l i p . t a n e d o @ u c r . e d u TEXAS A&M / MITCHELL INSTITUTE
&
2. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
2
Team Flip
March 2019
Ian Chaffey
Ian Chaffey
work with
3. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
3
the plan
via pexels.com
Figure 8: Cartoon of the Goldstone excitation for a ‘Mexican hat’ potential. Image from [148].
4.3.1 Framework
We begin with the concrete example of low-energy qcd that we described above. Given that the
chiral condensate hq̄qi breaks SU(3)A, we proceed to write down the e↵ective theory describing
the interaction of the resulting Goldstone bosons. Let us write U0 to refer to the direction in field
SM
med
DM
4. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
4
WIMP Complementarity
Dark matter searches related by crossing symmetry:
How Dark Matter talks to the Standard Model
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R E L I C A B U N DA N C E YO U ’ R E K I L L I N G M E N OT G R E AT, E I T H E R
5. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
5
Light Mediators
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Dark Matter
can keep thermal relic!
6. f l i p . t a n e d o @ u c r . e d u 56
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6
New Searches with Light Mediators
e
e
e
e
e
e
e
e
e e
capture
a
n
n
i
h
i
l
a
t
i
o
n
A0
A0
INDIRECT DIRECT MEDIATOR PRODUCTION
N N
A
N
N
I
H
I
L
ATI
ON
COLLI
DER
D I R E C T
A0
A0
Halo Morpholo
• SIDM particles follow the
0 2 4 6 8
0
2
4
6
8
R HkpcL
z
HkpcL
constant density contours
Kaplinghat, Linden, Keeley, HBY (2013) (P
C
d
SELF
Standard Model
Mediator
Dark Matter
SM
SM
SM
SM
accelerators astro
R E L I C
A B U N DA N C E
7. f l i p . t a n e d o @ u c r . e d u 56
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7
This talk
Build a dark sector with
vector dark matter,
light vector mediator.
MASS
2
SU(2) GAUGE PION HIGGS
New model
New phenomena?
(not this study)
Cool plots
New directions
Technical naturalness
8. f l i p . t a n e d o @ u c r . e d u 56
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Is this actually new?
KK DM: Servant & Tait (hep-ph/0206071)
SU(2) VDM: Hambye (0811.0172), Gross et al. (1505.07480)
Topology: Murayama & Shu (0905.1720 ), Baek et al. (1311.1035),
Ko & Tang (1609.02307), Khoze & Ro (1406.2291)
Simplified Model: Dent et al. (1505.03117)
Confined: Boddy et al. 1408.6532 & 1402.3629
Recent: Elahi & Khatibi 1902.04384, Choi et al. 1904.04109
Apologies for papers that I’ve missed
This work: massive spin-1 mediator, no fermions.
9. f l i p . t a n e d o @ u c r . e d u 56
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the plan
via pexels.com
Figure 8: Cartoon of the Goldstone excitation for a ‘Mexican hat’ potential. Image from [148].
4.3.1 Framework
We begin with the concrete example of low-energy qcd that we described above. Given that the
chiral condensate hq̄qi breaks SU(3)A, we proceed to write down the e↵ective theory describing
the interaction of the resulting Goldstone bosons. Let us write U0 to refer to the direction in field
SM
med
DM
10. f l i p . t a n e d o @ u c r . e d u 56
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10
the plan
via pexels.com
1. symmetry structure
2. model building
3. hint of pheno
11. f l i p . t a n e d o @ u c r . e d u 56
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11
Vector Dark Matter
SU(2)V
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g
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Wa
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COUPLING TRIPLET
We use common Standard Model particle names to
emphasize analogy to SM symmetries.
Most of talk: completely in the dark sector.
12. f l i p . t a n e d o @ u c r . e d u 56
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Baek, Ko, Park (1311.1035 )
Vector Dark Matter
SU(2) GAUGE TRIPLET
These vevs break the global symmetrie
3.1 Would-be Goldstones
We parameterize the Goldstone fields
the broken generators [31]:
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generators
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
h i =
1
2
✓
f
f
◆
=
?
0 massless dark photon
SU(2) GAUGE
h i =
2 f
=
ries su(2)H ! ? and su(2) ! u(1), r
s
ds as spacetime-dependent transforma
'H · T =
p
2'+
HT+
+
p
2'H
p p
13. f l i p . t a n e d o @ u c r . e d u 56
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13
Baek, Ko, Park (1311.1035 )
Vector Dark Matter
SU(2) GAUGE TRIPLET
0
SU(2) GAUGE
let’s make this massive
need to break U(1)
h i =
2 f
=
ries su(2)H ! ? and su(2) ! u(1), r
s
ds as spacetime-dependent transforma
'H · T =
p
2'+
HT+
+
p
2'H
p p
14. f l i p . t a n e d o @ u c r . e d u 56
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Vector Dark Matter
Hi =
✓
0
v/
p
2
◆
h i
he global symmetries su(2)H ! ? and
e Goldstones
he Goldstone fields as spacetime-depe
ors [31]:
'H ·T
v/2 hHi ' · T =
p
h i =
2 f
=
ries su(2)H ! ? and su(2) ! u(1), r
s
ds as spacetime-dependent transforma
'H · T =
p
2'+
HT+
+
p
2'H
p p
W stability?
Extra particles?
15. f l i p . t a n e d o @ u c r . e d u 56
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Standard Model Interlude
SU(2)L ⇥ U(1)Y ! U(1)EM
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SU(2)L ⇥ SU(2)R ! SU(2)V
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GAUGE HIGGS
PIONS
W±
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A
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Z
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⇡±,0
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h
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'±
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'0
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GLOBAL SYMMETRY
SU(2)H ⇥ SU(2)0
L ⇥ U(1)H
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EW is a gauged subgroup
16. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
16
Global Symmetry
al representation.
no interactions, the particles re
su(2) ⇥ su(2)H ⇥ u(1)H =
calar fields transform as
! U U†
su(2)H : H !
gonal (vector) subgroup su(2)V
(x) ! U (x)U†
,
unitary matrix and Ta
= 1
2
a
are the generato
articles respect. a global “flavor” symmetry
u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
as
2)H : H ! UHH u(1)H : H ! ei✓H
H .
p su(2)V of su(2) ⇥ su(2)H composed of transf
exp(i✓a
Ta
) is a 2 ⇥ 2 special unitary matrix an
mental representation.
mit of no interactions, the particles respect. a g
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥
the scalar fields transform as
) : ! U U†
su(2)H : H ! UHH
he diagonal (vector) subgroup su(2)V of su(2)
UH. The orthogonal combination is the ‘axial’
Higgs number” symmetry is analogous to hype
! UH(x) (x) ! U (x)U
2 ⇥ 2 special unitary matrix and Ta
= 1
2
a
are
ntation.
ctions, the particles respect. a global “flavor” s
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H
s transform as
†
su(2)H : H ! UHH u(1)H : H
tor) subgroup su(2)V of su(2) ⇥ su(2)H compo
{
gauge diagonal subgroup
TRIPLET (REAL)
DOUBLET
17. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
Global Symmetry Breaking
I think the h is actually lifted to m2
h ⇠ µf.]
p
2 +
3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
of the fields by
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
! ? and su(2) ! u(1), respectively.
me-dependent transformations of the vacuum by
Figure 1: Spectrum. [Flip: Check this... I
3 Symmetry Breaking
A linear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
We parameterize the vacuum expectation values o
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H !
3.1 Would-be Goldstones
al representation.
no interactions, the particles re
su(2) ⇥ su(2)H ⇥ u(1)H =
calar fields transform as
! U U†
su(2)H : H !
gonal (vector) subgroup su(2)V
{
gauge diagonal subgroup
18. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
18
Leftover symmetry
) is a 2 ⇥ 2 special unitary matrix and T = 2
are t
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
lar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
nal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compos
orthogonal combination is the ‘axial’ symmetry su(2)A
umber” symmetry is analogous to hypercharge in the St
Renormalizable Lagrangian
alizable Lagrangian satisfying the global symmetries of
= exp(i✓ T ) is a 2 ⇥ 2 special unitary matrix and T =
damental representation.
limit of no interactions, the particles respect. a global “
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A
ch the scalar fields transform as
(2) : ! U U†
su(2)H : H ! UHH u
the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)
UH. The orthogonal combination is the ‘axial’ symme
“Higgs number” symmetry is analogous to hypercharge
eneral, Renormalizable Lagrangian
al, renormalizable Lagrangian satisfying the global symm
⇥ 2 special unitary matrix and T = 2
are the genera
ation.
ons, the particles respect. a global “flavor” symmetry
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
transform as
su(2)H : H ! UHH u(1)H : H ! ei✓H
H .
r) subgroup su(2)V of su(2) ⇥ su(2)H composed of tran
nal combination is the ‘axial’ symmetry su(2)A, for whic
ymmetry is analogous to hypercharge in the Standard M
malizable Lagrangian
agrangian satisfying the global symmetries of the partic
T ) is a 2 ⇥ 2 special unitary matrix and T = 2
are
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H
alar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
onal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compo
e orthogonal combination is the ‘axial’ symmetry su(2)
umber” symmetry is analogous to hypercharge in the S
Renormalizable Lagrangian
malizable Lagrangian satisfying the global symmetries of
do not include any such terms.
2 Spectrum, Symmetry, Stability
ualitative overview of the model is as follows. The vacuum o
aks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1)H0
u(1)H0 : T3
V +
1
2
TH ,
logous to electric charge in the electroweak sector. In wha
ark sector particle with respect to the u(1)V ⇢ su(2)V gaug
ge bosons eat three of the five Goldstone modes. We sug
ns,’ ⇡±
. We take the limit where the triplet vev is much la
hTr 2
i =
f2
2
h|H|2
i =
we do not include any such terms.
2.2 Spectrum, Symmetry, Stability
A qualitative overview of the model is as follows. The vacuum
breaks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1
u(1)H0 : T3
V +
1
2
TH
analogous to electric charge in the electroweak sector. In w
a dark sector particle with respect to the u(1)V ⇢ su(2)V g
gauge bosons eat three of the five Goldstone modes. We
pions,’ ⇡±
. We take the limit where the triplet vev is much
hTr 2
i =
f2
2
h|H|
{
Figure 1: Spectrum. [Flip: Check this
3 Symmetry Breaking
A linear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
We parameterize the vacuum expectation value
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H
+1/2
-1/2
+1
+1
TV3 TH
Analog of electromagnetism
after electroweak breaking
gauged
19. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
19
Leftover symmetry
) is a 2 ⇥ 2 special unitary matrix and T = 2
are t
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
lar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
nal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compos
orthogonal combination is the ‘axial’ symmetry su(2)A
umber” symmetry is analogous to hypercharge in the St
Renormalizable Lagrangian
alizable Lagrangian satisfying the global symmetries of
= exp(i✓ T ) is a 2 ⇥ 2 special unitary matrix and T =
damental representation.
limit of no interactions, the particles respect. a global “
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A
ch the scalar fields transform as
(2) : ! U U†
su(2)H : H ! UHH u
the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)
UH. The orthogonal combination is the ‘axial’ symme
“Higgs number” symmetry is analogous to hypercharge
eneral, Renormalizable Lagrangian
al, renormalizable Lagrangian satisfying the global symm
⇥ 2 special unitary matrix and T = 2
are the genera
ation.
ons, the particles respect. a global “flavor” symmetry
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
transform as
su(2)H : H ! UHH u(1)H : H ! ei✓H
H .
r) subgroup su(2)V of su(2) ⇥ su(2)H composed of tran
nal combination is the ‘axial’ symmetry su(2)A, for whic
ymmetry is analogous to hypercharge in the Standard M
malizable Lagrangian
agrangian satisfying the global symmetries of the partic
T ) is a 2 ⇥ 2 special unitary matrix and T = 2
are
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H
alar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
onal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compo
e orthogonal combination is the ‘axial’ symmetry su(2)
umber” symmetry is analogous to hypercharge in the S
Renormalizable Lagrangian
malizable Lagrangian satisfying the global symmetries of
do not include any such terms.
2 Spectrum, Symmetry, Stability
ualitative overview of the model is as follows. The vacuum o
aks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1)H0
u(1)H0 : T3
V +
1
2
TH ,
logous to electric charge in the electroweak sector. In wha
ark sector particle with respect to the u(1)V ⇢ su(2)V gaug
ge bosons eat three of the five Goldstone modes. We sug
ns,’ ⇡±
. We take the limit where the triplet vev is much la
hTr 2
i =
f2
2
h|H|2
i =
we do not include any such terms.
2.2 Spectrum, Symmetry, Stability
A qualitative overview of the model is as follows. The vacuum
breaks the global symmetry su(2) ⇥ su(2)H ⇥ u(1)H ! u(1
u(1)H0 : T3
V +
1
2
TH
analogous to electric charge in the electroweak sector. In w
a dark sector particle with respect to the u(1)V ⇢ su(2)V g
gauge bosons eat three of the five Goldstone modes. We
pions,’ ⇡±
. We take the limit where the triplet vev is much
hTr 2
i =
f2
2
h|H|
{
Figure 1: Spectrum. [Flip: Check this
3 Symmetry Breaking
A linear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2
We parameterize the vacuum expectation value
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H
+1/2
-1/2
+1
+1
TV3 TH
gauged
order parameter of U(1) breaking
is charge 1/2
20. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
20
Goldstone smorgasborg
More Goldstones
than gauge bosons
Leftovers are pions.
+ radial modes
EAT
EAT
21. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
21
Leftover symmetry
) is a 2 ⇥ 2 special unitary matrix and T = 2
are t
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
lar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
nal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compos
orthogonal combination is the ‘axial’ symmetry su(2)A
umber” symmetry is analogous to hypercharge in the St
Renormalizable Lagrangian
alizable Lagrangian satisfying the global symmetries of
= exp(i✓ T ) is a 2 ⇥ 2 special unitary matrix and T =
damental representation.
limit of no interactions, the particles respect. a global “
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A
ch the scalar fields transform as
(2) : ! U U†
su(2)H : H ! UHH u
the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)
UH. The orthogonal combination is the ‘axial’ symme
“Higgs number” symmetry is analogous to hypercharge
eneral, Renormalizable Lagrangian
al, renormalizable Lagrangian satisfying the global symm
⇥ 2 special unitary matrix and T = 2
are the genera
ation.
ons, the particles respect. a global “flavor” symmetry
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
transform as
su(2)H : H ! UHH u(1)H : H ! ei✓H
H .
r) subgroup su(2)V of su(2) ⇥ su(2)H composed of tran
nal combination is the ‘axial’ symmetry su(2)A, for whic
ymmetry is analogous to hypercharge in the Standard M
malizable Lagrangian
agrangian satisfying the global symmetries of the partic
T ) is a 2 ⇥ 2 special unitary matrix and T = 2
are
representation.
o interactions, the particles respect. a global “flavor” sy
su(2) ⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H
alar fields transform as
! U U†
su(2)H : H ! UHH u(1)H : H !
onal (vector) subgroup su(2)V of su(2) ⇥ su(2)H compo
e orthogonal combination is the ‘axial’ symmetry su(2)
umber” symmetry is analogous to hypercharge in the S
Renormalizable Lagrangian
malizable Lagrangian satisfying the global symmetries of
2.2 Spectrum, Symmetry, Stability
A qualitative overview of the model is as follows. T
breaks the global symmetry su(2) ⇥ su(2)H ⇥ u(
u(1)H0 :
analogous to electric charge in the electroweak se
a dark sector particle with respect to the u(1)V ⇢
gauge bosons eat three of the five Goldstone mo
‘pions,’ ⇡±
. We take the limit where the triplet v
hTr 2
i =
f2
2
{
gauged
EATEN GOLDSTONES
UNEATEN PIONS
Explicitly break SU(2)A,
give mass to pions
PSEUDO-GOLDSTONES
22. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
22
Standard Model Interlude
SU(2)L ⇥ U(1)Y ! U(1)EM
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SU(2)L ⇥ SU(2)R ! SU(2)V
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GAUGE HIGGS
PIONS
W±
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A
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Z
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⇡±,0
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h
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'±
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'0
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GLOBAL SYMMETRY
SU(2)H ⇥ SU(2)0
L ⇥ U(1)H
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Mass from explicit
breaking of global sym.
FROM QUARK MASSES
23. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
23
the plan
via pexels.com
1. symmetry structure
2. model building
3. hint of pheno
24. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
24
Potential H =
✓
hu
hd
◆
=
We parameterize the vacuum expectati
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetrie
3.1 Would-be Goldstones
We parameterize the Goldstone fields
the broken generators [31]:
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generators
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
3 Symmetry Breaking
A linear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
We parameterize the vacuum expectation values of the fields
hHi =
✓
0
v/
p
2
◆
h i =
1
2
These vevs break the global symmetries su(2)H ! ? and su
3.1 Would-be Goldstones
We parameterize the Goldstone fields as spacetime-depende
the broken generators [31]:
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'
= ei
' ·T
f h i e i
' ·T
f ' · T =
p
2'
with respect to the su(2)H, generators T±
= T1
± iT2
, T3
. T
✓ ◆
DOUBLET
TRIPLET
Figure 2: Fields a and b acquire unequal vacuum expectation values fa > fb. The Goldstone excitations
with respect to a transformation by parameter ✓ have correspondingly di↵erent magnitudes, 'a > 'b.
The Goldstone, 'V , for a vectorial transformation where ✓a = ✓b is thus not orthogonal to the corre-
sponding Goldstone, 'A for an axial transformation where ✓a = ✓b.
u(1)V and u(1)A by the same e↵ective order parameter, f2
V = f2
a + f2
b . Neither u(1)V nor u(1)A is
preferred over the other. Why, then, is it the case in (A.7) that the 'V eats more 'a while 'A
eats more of 'b? The root of this confusion is illustrated in Fig. 2: in the absence of gauging, the
naı̈ve description of the vector and axial Goldstones are not orthogonal to one another. The choice
of gauging a particular combination of the full global symmetry breaks the symmetry and gives
‘priority’ to the eaten Goldstone boson to have a larger admixture of the field that does most of
the symmetry breaking.
A.4 Gauging an Axial Combination
Figure 2: Fields a and b acquire unequal vacuum expectation values fa > fb. The Goldstone excitations
with respect to a transformation by parameter ✓ have correspondingly di↵erent magnitudes, 'a > 'b.
The Goldstone, 'V , for a vectorial transformation where ✓a = ✓b is thus not orthogonal to the corre-
sponding Goldstone, 'A for an axial transformation where ✓a = ✓b.
u(1)V and u(1)A by the same e↵ective order parameter, f2
V = f2
a + f2
b . Neither u(1)V nor u(1)A is
preferred over the other. Why, then, is it the case in (A.7) that the 'V eats more 'a while 'A
eats more of 'b? The root of this confusion is illustrated in Fig. 2: in the absence of gauging, the
naı̈ve description of the vector and axial Goldstones are not orthogonal to one another. The choice
of gauging a particular combination of the full global symmetry breaks the symmetry and gives
‘priority’ to the eaten Goldstone boson to have a larger admixture of the field that does most of
the symmetry breaking.
A.4 Gauging an Axial Combination
su(2) : ! U U su(2)H : H ! UHH u(1)H : H ! e H .
ge the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)H composed of trans
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respec
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The trilin
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
{
{
Figure 1: Spectrum. [Flip: Check this... I think the h is actually lifted to m2
h ⇠ µf.]
Symmetry Breaking
ear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
±
⌘
1
+ i 2
p
2
.
arameterize the vacuum expectation values of the fields by
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
e vevs break the global symmetries su(2)H ! ? and su(2) ! u(1), respectively.
Would-be Goldstones
arameterize the Goldstone fields as spacetime-dependent transformations of the
roken generators [31]:
'H ·T p p
his... I think the h is actually lifted to m2
h ⇠ µf.]
is
3
p
2 +
2 3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
ues of the fields by
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
)H ! ? and su(2) ! u(1), respectively.
cetime-dependent transformations of the vacuum by
SPONTANEOUS BREAKING
25. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
25
The Model
ge the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)H composed of trans
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respec
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The trilin
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
0 1
{
{
Figure 1: Spectrum. [Flip: Check this... I think the h is actually lifted to m2
h ⇠ µf.]
Symmetry Breaking
ear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
±
⌘
1
+ i 2
p
2
.
arameterize the vacuum expectation values of the fields by
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
e vevs break the global symmetries su(2)H ! ? and su(2) ! u(1), respectively.
Would-be Goldstones
arameterize the Goldstone fields as spacetime-dependent transformations of the
roken generators [31]:
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'+
HT+
+
p
2'HT + '0
HT3
his... I think the h is actually lifted to m2
h ⇠ µf.]
is
3
p
2 +
2 3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
ues of the fields by
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
)H ! ? and su(2) ! u(1), respectively.
cetime-dependent transformations of the vacuum by
' · T =
p
2'+
T+
+
p
2' T + '0
T3
(3.3)
EAT GOLDSTONES
(MASS TO VECTORS)
EXPLICIT SU(2)A
BREAKING
(MASS TO SCALARS)
SPONTANEOUS BREAKING
H =
✓
hu
hd
◆
=
We parameterize the vacuum expectati
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetrie
3.1 Would-be Goldstones
We parameterize the Goldstone fields
the broken generators [31]:
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generators
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
3 Symmetry Breaking
A linear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
We parameterize the vacuum expectation values of the fields
hHi =
✓
0
v/
p
2
◆
h i =
1
2
These vevs break the global symmetries su(2)H ! ? and su
3.1 Would-be Goldstones
We parameterize the Goldstone fields as spacetime-depende
the broken generators [31]:
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'
= ei
' ·T
f h i e i
' ·T
f ' · T =
p
2'
with respect to the su(2)H, generators T±
= T1
± iT2
, T3
. T
✓ ◆
DOUBLET
TRIPLET
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respec
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The trilin
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
0
term.1
✓ ◆
26. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
26
Gauge Boson Mass
EAT GOLDSTONES
(MASS TO VECTORS)
DOUBLET
TRIPLET
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respec
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The trilin
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
0
term.1
✓ ◆
hHi =
✓
0
v/
p
2
◆
These vevs break the global symmetries su(2)H
3.1 Would-be Goldstones
We parameterize the Goldstone fields as space
the broken generators [31]:
H = ei
'H ·T
v/2 hHi '
= ei
' ·T
f h i e i
' ·T
f '
with respect to the su(2)H, generators T±
= T
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the diagonal com
the covariant derivatives on the scalar fields ar
DµH = @µH igWa
µ Ta
H
H =
hd
=
2 2 3 ⌘ p
We parameterize the vacuum expectation values of the fields by
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
These vevs break the global symmetries su(2)H ! ? and su(2) ! u(1), respecti
3.1 Would-be Goldstones
We parameterize the Goldstone fields as spacetime-dependent transformations o
the broken generators [31]:
H = ei
'H ·T
v/2 hHi 'H · T =
p
2'+
HT+
+
p
2'HT + '
= ei
' ·T
f h i e i
' ·T
f ' · T =
p
2'+
T+
+
p
2' T ,
with respect to the su(2)H, generators T±
= T1
± iT2
, T3
. The radial modes are
H|radial =
1
p
2
✓
0
h
◆
|radial =
1
2
✓ ◆
.
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the diagonal combination of su(2)H ⇥su(2) . In ou
cquire vevs (3.2), then the kinetic terms yield the following mass terms for the
= m2
W W+
W +
1
2
m2
AA2
m2
W = g2
f2
+
g2
v2
4
m2
A =
g2
v2
4
.
he massive dark matter W±
= (W1
⌥iW2
)/
p
2 and mediator (dark photon) A
⌧ f2
yields a spectrum where the dark photon is much lighter than the dark m
derivatives with respect to the spin-1 mass eigenstates are
DµH = @µH i
g
p
2
W+
µ T+
+ Wµ T H igAµT3
H
Dµ = @µ i
g
p
2
W+
µ [T+
, ] + Wµ [T , ] igAµ[T3
, ] .
s Mechanism and Leftover Goldstones
linear combination of Goldstone bosons associated with su(2)V . Gauging the
su(2)V promotes this global symmetry to a local symmetry. In unitary gau
cal su(2)V transformation to remove 'V from the theory. It appears solely
polarization of the massive gauge bosons. We may express ' in terms of th
the fields acquire vevs (3.2), then the kinetic terms yield the following mass terms for the g
sons:
Lmass = m2
W W+
W +
1
2
m2
AA2
m2
W = g2
f2
+
g2
v2
4
m2
A =
g2
v2
4
.
e identify the massive dark matter W±
= (W1
⌥iW2
)/
p
2 and mediator (dark photon) A =
e limit v2
⌧ f2
yields a spectrum where the dark photon is much lighter than the dark ma
e covariant derivatives with respect to the spin-1 mass eigenstates are
27. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
27
Eaten Goldstones
CCWZ Phys. Rev. 177 (1969) 2247
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generato
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the di
the covariant derivatives on the scala
DµH = @µH igWa
µ Ta
s Mechanism and Leftover Goldstones
linear combination of Goldstone bosons associated with su(2)V . Gaugi
u(2)V promotes this global symmetry to a local symmetry. In unita
al su(2)V transformation to remove 'V from the theory. It appears
olarization of the massive gauge bosons. We may express 'V in term
this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
aks the u(1) symmetry so that the A eats the only neutral Goldston
e charged states for which there are two pairs of charged Goldstones
d gauge bosons; see Appendix A for an illustrative u(1) example. Fr
rmalized su(2)V Goldstone 'V and the orthogonal state 'A:
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
.
ge, '±
V only appears as the longitudinal mode of W±
. The ‘axial’ com
Goldstone boson that remains in the theory. We refer to these as pion
{
combination su(2)V promotes this global symmetry to a local
performs a local su(2)V transformation to remove 'V from the
longitudinal polarization of the massive gauge bosons. We may
by identifying this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W +
Only hHi breaks the u(1) symmetry so that the A eats the on
contrast to the charged states for which there are two pairs of
pair of charged gauge bosons; see Appendix A for an illustrativ
identify the normalized su(2)V Goldstone 'V and the orthogona
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f
In unitary gauge, '±
V only appears as the longitudinal mode of
is an uneaten Goldstone boson that remains in the theory. We r
them ⇡±
in anticipation of including explicit symmetry breaking
0
EATEN CHARGED GOLDSTONE
ONLY ONE NEUTRAL
GOLDSTONE
28. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
28
Eaten Goldstones
CCWZ Phys. Rev. 177 (1969) 2247
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generato
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the di
the covariant derivatives on the scala
DµH = @µH igWa
µ Ta
and Leftover Goldstones
on of Goldstone bosons associated with su(2)V . Gauging the vector
his global symmetry to a local symmetry. In unitary gauge one
rmation to remove 'V from the theory. It appears solely as the
massive gauge bosons. We may express 'V in terms of the 'H,
e kinetic terms:
|2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA . (3.10)
metry so that the A eats the only neutral Goldstone. This is in
for which there are two pairs of charged Goldstones and only one
see Appendix A for an illustrative u(1) example. From (3.10) we
Goldstone 'V and the orthogonal state 'A:
v/2)'±
H
(v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
. (3.11)
ears as the longitudinal mode of W±
. The ‘axial’ combination '±
A
that remains in the theory. We refer to these as pions and relabel
{
n su(2)V promotes this global symmetry to a local symmetry. In uni
local su(2)V transformation to remove 'V from the theory. It appea
polarization of the massive gauge bosons. We may express 'V in ter
ng this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
reaks the u(1) symmetry so that the A eats the only neutral Goldst
the charged states for which there are two pairs of charged Goldstone
ged gauge bosons; see Appendix A for an illustrative u(1) example.
normalized su(2)V Goldstone 'V and the orthogonal state 'A:
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
.
gauge, '±
V only appears as the longitudinal mode of W±
. The ‘axial’ c
en Goldstone boson that remains in the theory. We refer to these as pi
anticipation of including explicit symmetry breaking terms to make th
mmetry Breaking with , 0
EATEN CHARGED GOLDSTONE
u(2)V promotes this global symmetry to a local symmetry. In unitary
al su(2)V transformation to remove 'V from the theory. It appears so
olarization of the massive gauge bosons. We may express 'V in terms
this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
ks the u(1) symmetry so that the A eats the only neutral Goldstone.
charged states for which there are two pairs of charged Goldstones an
d gauge bosons; see Appendix A for an illustrative u(1) example. From
rmalized su(2)V Goldstone 'V and the orthogonal state 'A:
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
.
ge, '±
V only appears as the longitudinal mode of W±
. The ‘axial’ comb
Goldstone boson that remains in the theory. We refer to these as pions
ticipation of including explicit symmetry breaking terms to make them
metry Breaking with , 0
ORTHOGONAL GOLDSTONE
d Symmetries
ength g to a two scalar particles: a doublet Hi
and
tion, the su(2) transformation is
(x) ! U (x)U†
, (2.1)
ry matrix and Ta
= 1
2
a
are the generators of su(2)
s respect. a global “flavor” symmetry
= su(2)V ⇥ su(2)A ⇥ u(1)H , (2.2)
Particles and Symmetries
at couples with strength g to a two scalar particles: a d
In this representation, the su(2) transformation is
! UH(x) (x) ! U (x)U†
,
2 ⇥ 2 special unitary matrix and Ta
= 1
2
a
are the gene
ntation.
ctions, the particles respect. a global “flavor” symmetry
⇥ su(2)H ⇥ u(1)H = su(2)V ⇥ su(2)A ⇥ u(1)H ,
s transform as
NB: how does this know that
it should eat the “smaller” vev?
29. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
29
Goldstone smorgasborg
EAT
EAT
contrast to the charged states for which ther
pair of charged gauge bosons; see Appendix
identify the normalized su(2)V Goldstone 'V
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
In unitary gauge, '±
V only appears as the lon
is an uneaten Goldstone boson that remains
them ⇡±
in anticipation of including explicit
3.4 Symmetry Breaking with
The simplest form of this model takes only t
V | , 0 =
4!
2 Tr 2
These terms separately break the su(2) and
gauged vector combination of the two and gi
g this mixing in the kinetic terms:
|DH|2
+ Tr |D |2
g
⇣v
2
@'+
H + f@'+
⌘
W + h.c. g
v
2
@'0
HA .
eaks the u(1) symmetry so that the A eats the only neutral Goldston
he charged states for which there are two pairs of charged Goldstones
ed gauge bosons; see Appendix A for an illustrative u(1) example. Fr
normalized su(2)V Goldstone 'V and the orthogonal state 'A:
'±
V =
f'±
+ (v/2)'±
H
p
f2 + (v/2)2
'±
A =
f'±
H (v/2)'±
p
f2 + (v/2)2
.
auge, '±
V only appears as the longitudinal mode of W±
. The ‘axial’ com
n Goldstone boson that remains in the theory. We refer to these as pion
anticipation of including explicit symmetry breaking terms to make them
mmetry Breaking with , 0
form of this model takes only the first two terms in (2.5),
0
UNEATEN
massless modes?
this sucks.
30. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
30
The Model
su(2) : ! U U su(2)H : H ! UHH u(1)H : H ! e H .
ge the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)H composed of trans
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respec
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The trilin
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
{
{
Figure 1: Spectrum. [Flip: Check this... I think the h is actually lifted to m2
h ⇠ µf.]
Symmetry Breaking
ear parameterization of the scalar fields is
H =
✓
hu
hd
◆
=
1
2
✓ 3
p
2 +
p
2 3
◆
±
⌘
1
+ i 2
p
2
.
arameterize the vacuum expectation values of the fields by
hHi =
✓
0
v/
p
2
◆
h i =
1
2
✓
f
f
◆
= fT3
.
e vevs break the global symmetries su(2)H ! ? and su(2) ! u(1), respectively.
Would-be Goldstones
arameterize the Goldstone fields as spacetime-dependent transformations of the
roken generators [31]:
'H ·T p p
his... I think the h is actually lifted to m2
h ⇠ µf.]
is
3
p
2 +
2 3
◆
±
⌘
1
+ i 2
p
2
. (3.1)
ues of the fields by
h i =
1
2
✓
f
f
◆
= fT3
. (3.2)
)H ! ? and su(2) ! u(1), respectively.
cetime-dependent transformations of the vacuum by
EAT GOLDSTONES
(MASS TO VECTORS)
EXPLICIT SU(2)A
BREAKING
(MASS TO SCALARS)
SPONTANEOUS BREAKING
ge the diagonal (vector) subgroup su(2)V of su(2) ⇥ su(2)H composed of trans
= UH. The orthogonal combination is the ‘axial’ symmetry su(2)A, for which
H “Higgs number” symmetry is analogous to hypercharge in the Standard Mo
General, Renormalizable Lagrangian
eral, renormalizable Lagrangian satisfying the global symmetries of the particle
L =
1
4
Fa
µ⌫Faµ⌫
+ |DµH|2
+ Tr |Dµ |2
V
V =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
.
D are covariant derivatives for the fundamental and adjoint of su(2), respec
e potential V to imply that the scalars and H obtain vacuum expectation va
ntaneously break the symmetries of the theory. This breaking produces a sp
ne bosons, three of which are eaten by the massive gauge bosons. The trilin
y breaks the global axial su(2)A symmetry. This gives a mass to the remaining
stone modes. The 00
term mixes the radial modes of the H and . We syst
the theory starting from the symmetry breaking and 0
terms and subsequen
ts of the µ and 00
terms. Additional quartic terms obeying the global symmet
0
term.1
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generato
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the di
the covariant derivatives on the scala
DµH = @µH igWa
µ Ta
31. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
31
The Model
gian
e global symmetries of the particle content is
V (2.4)
v2
0
2
+ µH†
H + 00
|H|2
Tr 2
. (2.5)
ental and adjoint of su(2), respectively. We
d H obtain vacuum expectation values (vevs)
eory. This breaking produces a spectrum of
massive gauge bosons. The trilinear µ term
This gives a mass to the remaining the would-
modes of the H and . We systematically
king and 0
terms and subsequently include
terms obeying the global symmetries reduce
b
12⇥2
◆
H =
1
2
|H|2
Tr 2
.
EXPLICIT SU(2)A
BREAKING
H = ei
'H ·T
v/2 hHi
= ei
' ·T
f h i e i
' ·T
f
with respect to the su(2)H, generato
H|radial =
1
p
2
✓
0
h
◆
3.2 Gauge Boson Masses
The gauged su(2)V symmetry is the di
the covariant derivatives on the scala
DµH = @µH igWa
µ Ta
Invariant under SU(2)V,
but not SU(2)A
Inserting nonlinear fields gives explicit mass term
for the axial Goldstones (pions)
µv
+ µv2/4f 0v2/3
. (3.22)
erm induces mixes the charged Goldstones, '±
and
M2
G =
0
B
@
µv2
4f
µv
2
µv
2
µ
f
1
C
A . (3.23)
0. This corresponds to the massless Goldstone G±
dstone, ⇡±
, has a mass-squared given by the trace:
m2
⇡ = µf
✓
1 +
v2
4f2
◆
. (3.24)
y a rotation
32. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
32
Effect of mu
μ encodes post-NLΣM interactiosn of pions
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
-10 -5 0 5 10
-10
-5
0
5
10
f
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v
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µ
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metry Breaking with , 0
, µ
e µ term in the potential explicitly breaks su(2) ⇥ su(2)H ! su(2)V
oportional to µ:
V | , 0,µ =
4!
2 Tr 2
f2
0
2
+
0
4!
2|H|2
v2
0
2
+ µH†
H .
minimum of the potential from (3.13) to the following condition:
f2
= f2
0 +
3µv2
2 f
v2
= v2
0 +
3µf
0
.
33. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
33
34. f l i p . t a n e d o @ u c r . e d u 56
TEXAS A&M / MITCHELL INSTITUTE
34
f2
= f2
0 +
3µv2
2 f
v2
= v2
0
µ term causes the vev to shift the H vev, and vice versa.
ing for phenomenological hierarchy
nomenologically we require that the mediator is light and that
matter; this forces
g2
v2
⌧ g2
f2
. µf .
uming g . O(1), we see that the the vev f2
is perturbed by
0 value f2
0 . On the other hand, the hierarchy f, µ v and pe
v2
is shifted by a large amount relative to v2
0. Without loss o
then require that v2
0 is negative and tuned to give a small v2
⌧