Circumscribed circle: Difference between revisions

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{{expert-subject|Physics|date=July 2009}}
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{{Refimprove|date=January 2008}}
 
[[File:SO10.svg|300px|right|thumb|The pattern of [[weak isospin]], W, weaker isospin, W', strong g3 and g8, and baryon minus lepton, B, charges for particles in the SO(10) model, rotated to show the embedding of the [[Georgi-Glashow model]] and [[Standard Model]], with electric charge roughly along the vertical. In addition to Standard Model particles, the theory includes thirty colored X bosons, responsible for [[proton decay]], and two W' bosons.]]
 
[[File:E6GUT.svg|300px|right|thumb|The pattern of charges for particles in the SO(10) model, rotated to show the embedding in [[E6 (mathematics)|E6]].]]
 
In [[particle physics]], one of the [[grand unified theory|grand unified theories]] (GUT) is based on the '''[[special orthogonal group|SO(10)]]''' [[Lie group]]. (The [[Lie group]] involved is not really the [[special orthogonal group]] SO(10), but rather its [[Double covering group|double cover]] [[spin group|Spin(10)]]; but calling it SO(10) is the standard convention.)
 
Before [[Georgi-Glashow model|SU(5)]], [[Harald Fritzsch]] and [[Peter Minkowski]]  and independently [[Howard Georgi]] found that all the matter contents are incorporated into a single representation, [[spinorial]] 16 of SO(10). (Historical note: the ''before'' in the previous sentence is misleading: Georgi found the SO(10) theory a few hours before finding SU(5) at the end of 1973.<ref>This story is told in various places; see for example, [http://ptp.ipap.jp.ezp-prod1.hul.harvard.edu/link?PTPS/170/119/ Yukawa-Tomonaga 100th Birthday Celebration]; Fritzsch and Minkowski analyzed SO(10) in 1974.</ref>)
 
==Important subgroups==
 
It has the [[branching rule]]s to [SU(5)&times;U(1)<sub>χ</sub>]/'''Z'''<sub>5</sub>.
:<math> 45 \rightarrow 24_0 \oplus 10_{-4} \oplus \overline{10}_4 \oplus 1_0</math>
:<math> 16 \rightarrow 10_1 \oplus \bar{5}_{-3} \oplus 1_5.</math>
:<math> 10 \rightarrow 5_{-2} \oplus \bar{5}_2.</math>
If the [[hypercharge]] is contained within SU(5), this is the conventional [[Georgi-Glashow model]], with the 16 as the matter fields, the 10 as the electroweak Higgs field and the 24 within the 45 as the GUT Higgs field. The [[superpotential]] may then include [[renormalizable]] terms of the form ''Tr''(45 ⋅ 45); ''Tr''(45 ⋅ 45 ⋅ 45); 10 ⋅ 45 ⋅ 10, 10 ⋅ 16* ⋅ 16 and 16* ⋅ 16. The first three are responsible to the [[gauge symmetry]] breaking at low energies and give the [[Higgs]] mass, and the latter two give the matter particles masses and their [[Yukawa coupling]]s to the Higgs.
 
There is another possible branching, under which the hypercharge is a linear combination of an SU(5) generator and χ. This is known as [[flipped SU(5)]].  
 
Another important subgroup is either [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/'''Z'''<sub>2</sub> or '''Z'''<sub>2</sub> ⋊ [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/'''Z'''<sub>2</sub> depending upon whether or not the [[left-right symmetry]] is broken, yielding the [[Pati-Salam model]], whose branching rule is
:<math> 16 \rightarrow (4,2,1)\oplus (\bar 4,1,2). </math>
 
==Spontaneous symmetry breaking==
The symmetry breaking of SO(10) is usually done with a combination of (( a 45<sub>H</sub> OR a 54<sub>H</sub>) AND ((a 16<sub>H</sub> AND a <math>\overline{16}_H</math>) OR (a 126<sub>H</sub> AND a <math>\overline{126}_H</math>)) ).
 
Let's say we choose a 54<sub>H</sub>. When this Higgs field acquires a GUT scale [[VEV]], we have a symmetry breaking to '''Z'''<sub>2</sub> ⋊ [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/'''Z'''<sub>2</sub>, i.e. the [[Pati-Salam model]] with a '''Z'''<sub>2</sub> [[left-right symmetry]].
 
If we have a 45<sub>H</sub> instead, this Higgs field can acquire any VEV in a two dimensional subspace without breaking the standard model. Depending on the direction of this linear combination, we can break the symmetry to SU(5)&times;U(1), the [[Georgi-Glashow model]] with a U(1) (diag(1,1,1,1,1,-1,-1,-1,-1,-1)), [[flipped SU(5)]] (diag(1,1,1,-1,-1,-1,-1,-1,1,1)), SU(4)&times;SU(2)&times;U(1) (diag(0,0,0,1,1,0,0,0,-1,-1)), the minimal [[left-right model]] (diag(1,1,1,0,0,-1,-1,-1,0,0)) or SU(3)&times;SU(2)&times;U(1)&times;U(1) for any other nonzero VEV.
 
The choice diag(1,1,1,0,0,-1,-1,-1,0,0) is called the [[Dimopoulos-Wilczek mechanism]] aka the [[missing VEV mechanism]] and it is proportional to [[B−L]].
 
The choice of a 16<sub>H</sub> and a <math>\overline{16}_H</math> breaks the gauge group down to the Georgi-Glashow SU(5). The same comment applies to the choice of a 126<sub>H</sub> and a <math>\overline{126}_H</math>.
 
It is the combination of BOTH a 45/54 and a 16/<math>\overline{16}</math> or 126/<math>\overline{126}</math> which breaks SO(10) down to the [[Standard Model]].
 
==The electroweak Higgs and the doublet-triplet splitting problem==
The electroweak Higgs doublets come from an SO(10) 10<sub>H</sub>. Unfortunately, this same 10 also contains triplets. The masses of the doublets have to be stabilized at the electroweak scale, which is many orders of magnitude smaller than the GUT scale whereas the triplets have to be really heavy in order to prevent triplet-mediated [[proton decay]]s. See [[doublet-triplet splitting problem]].
 
Among the solutions for it is the Dimopoulos-Wilczek mechanism, or the choice of diag(0,0,0,1,1,0,0,0,-1,-1) of <45>. Unfortunately, this is not stable once the 16/<math>\overline{16}</math> or 126/<math>\overline{126}</math> sector interacts with the 45 sector.<ref>*{{cite arXiv |author=[[John Baez|J.C. Baez]], J. Huerta |eprint=0904.1556 |title=The Algebra of Grand Unified Theories |year=2009 |class=hep-th }}</ref>
 
==Matter==
 
[[File:SO(10) - 16 Weight Diagram.svg|right|400px]]
 
The matter representations come in three copies (generations) of the 16 representation. The [[Yukawa coupling]] is 10<sub>H</sub> 16<sub>f</sub> 16<sub>f</sub>. This includes a right-handed neutrino. We can either include three copies of [[singlet]] representations φ and a Yukawa coupling <math><\overline{16}_H>16_f \phi</math> (see [[double seesaw mechanism]]) or add the Yukawa interaction <math><\overline{126}_H> 16_f 16_f</math> or add the [[nonrenormalizable]] coupling <math><\overline{16}_H><\overline{16}_H>16_f 16_f</math>. See [[seesaw mechanism]].
 
==Proton decay==
<gallery caption="These graphics refer to the [[X and Y bosons|X bosons]] and [[Higgs boson]]s." widths="125px" heights="150px" perrow="3">
 
Image:Proton_decay2.svg|Dimension 6 proton decay mediated by the ''X'' boson <math>(3,2)_{-\frac{5}{6}}</math> in SU(5) GUT
 
Image:proton decay3.svg|Dimension 6 proton decay mediated by the ''X'' boson <math>(3,2)_{\frac{1}{6}}</math> in flipped SU(5) GUT
 
</gallery>
 
Note that SO(10) contains both the Georgi-Glashow SU(5) and flipped SU(5).
 
== See also ==
* [[Flipped SO(10)]]
 
==Notes==
{{reflist}}
 
[[Category:Particle physics]]

Revision as of 22:57, 28 February 2014

When no positive effects are gained from this treatment either, the option is to keep on trying to see if it takes more time (and money) or to try something different. We'll be talking a little about where cellulite comes from and how you can get rid of it without dieting or exhausting yourself at the gym. Stress produces a hormone called cortisol which can make your body store energy and fat. There are different solutions for the problem, and, medical therapy is one, which helps the problem of cellulite that women experience. Make-up and cheap cosmetic products will only work at a glance; meaning if anyone gets a closer look, the jig will be up.

Take glucosamine supplements to improve skin resilience. If you are looking to get serious about your career path and have visible tattoos, you either have to find an industry that will not be put off by your tattoos or get. Also, getting Smart - Lipo is not only safer, but is cheaper than traditional liposuction. Sharon Bell is an avid health and fitness enthusiast and published author. Merely placed, bipolar RF gadgets have two electrodes.

A few women find symptoms on the stomach, what you find is the vast majority of women have it on their thighs, hips and butt. Basically they are used to keep cellulite from showing through your clothing, and they really do work for this sole purpose. Cellulite treatment with the help of the latest and innovative Cellulaze workstation ensures greater safety and convenience. So try these lower body exercises to get rid of cellulite, and make sure to start each exercise with a 10-min warm-up. If you haven't had fortune with scammy cellulite lotions, tablets, and other gadgets, give this a try.

Muscles in our body are covered with a padding of fatty connective tissue. Your doctor will want a specific antibiotic depending on the place of your cellulites and the possible cause of your infection. At this stage, the fibrous septae, which is caused by constricted circulation and drainage problems, will start to thicken and form hardened fibers. As your fat content is also one culprit that pushes the tissue up against the skin to form cellulite, you must get rid of the fat by burning it off. Massage and even invasive treatment methods will only provide temporary relief.

Identify a snug, quiet identify to take a chair for ten to 15 moments on a regular groundwork, quit all of your hustling and efficient. To start off, you need to make a daily dietary menu to follow so you will consistently eat the right foods that will help you remove fat from your skin tissue. So basically, if you can avoid caffeine, you will be able to avoid cellulite build up to some degree. For this reason you should limit the amount of fat in your diet by reducing the amount of processed foods you ingest while increasing the amount of natural food you consume such as fruits, vegetables, and grains. During this treatment you will have to wear a suit specially designed for the procedure, while walking on a treadmill for 30 minutes.

If you have any sort of inquiries pertaining to where and ways to utilize how to treat cellulite, you could contact us at our own web site.

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