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Special Topics in Particle Physics
Beyond the Standard Model
Jonghee Yoo
Korea Advanced Institute of Science and Technology 2017 Fall Physics Course Lecture Series
PH489 Note 04
PH489 Contact
2
Professor Yoo, Jonghee E-mail: [email protected]
- E-mail is the easiest way to reach me Classes: E11-208 (PM 2:30 - 4:00, Monday and Wednesday)Office hours: There will be no regular office hours, but if you e-mail me we
can schedule meetings (any subject, not necessarily physics topics) - Office#1: KAIST Main Campus, E6-2, room 2306 (2nd floor) - Office#2: KAIST Munji Campus, Creation Hall, room C307-A (3rd floor)Web-page: yoo.kaist.ac.kr/lectures/
- course materials, corrections, useful links etc.
Teaching Assistant: Kim, Jongkuk ([email protected])
KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
PH489 Schedule In Nov/Dec 2017
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29 30 31
PH489 Seminar Schedule & Location
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2017-11-20 2:30PM~4:00PM (Monday) E11-208Shiers, Elizabeth: SUSY Searches with ATLAS at The Large Hadron ColliderYi, Kunwoo: Measuring the Density Parameters of the Universe
2017-11-24 7:00PM~8:30PM (Friday) Munji Creation Hall (문지캠퍼스 창조관 C306)Yildiz, Merve: A naive introduction to QCD, gluons & eight-gluons problemKim, Min-gi: EPR(Einstein Podolsky Rogen) Paradox and Bell’s theorem
2017-11-27 2:30PM~4:00PM (Monday) E11-208Park, Hyeonbin: Matter-antimatter asymmetryKim, Moonsik: Chameleon Field Theory
2017-12-01 7:00PM~8:30PM (Friday) Munji Creation Hall (문지캠퍼스 창조관 C306)Capurso, Filippo: The life of MuonsShim, Jaehyu: Particle accelerators
2017-12-04 2:30PM~4:00PM (Monday) E11-208Oh, Jaewhan: MWPC, Charged Particle Trajectory Tracking SystemLee, Dongjin: Effects of Gravitational Waves on Quantum Interference
2017-12-06 2:30PM~4:00PM (Wednesday) E11-208ChoeJo, YeolLin: Quarks, Colors, and Confinement
Note: Both speakers should arrive the class room 10min ahead of the class and check out the presentation system. The first talk will begin on time (2:30PM/7:00PM exact).
Standard Model of Particle Physics
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Standard Model of Particle Physics
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http://yoo.kaist.ac.kr/lectures/2017/1/files/YooKAISTPH450Lecture03.pdfIntroduction to the Standard Model can be found at:
Particle PhysicsWhat are the fundamental constituents of the UniverseHow do they interact each other?
Matter Particles: FermionsLeptons and quarks
Force Carrier Particles: BosonsElectromagnetic force (photon)Strong force (gluons)Weak force (W/Z bosons)
Standard Model of Particle Physics
Standard Model of Particle Physics
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Matter Particles
Leptons
Quarks
note:missing right-handedneutrinos
x 3 colors (r,g,b)
Gauge Principle
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The gauge principle is based on the fact that both classical physics and quantum theory involve quantities which, in principle, cannot be measured. ➔ It is possible to gauge a theory by a suitable choice of the non-measurable parameters in order to simplify the equation of motion.
Maxwell Equations
Coulomb Gauge: Lorentz Gauge:
Gauge Transformation
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Global gauge transformation:
For example Dirac equation: ➔
Local gauge transformation: The requirement for invariance under a local transformation is much more stringent For example the Dirac equation is not invariant under the local transformationand requires modification of derivative ➔ covariant derivative
For gauge transformation:
the Dirac equation retains its original form if the gauge field is transformed as:
�i�
µ(@µ � ieA
0µ(x))�m
0(x) = 0
{i�µ(@µ � ieAµ � ie@µ⇠(x))�m} 0(x) = 0
(i�µD0µ �m) 0(x) = 0
Standard Model of Particle Physics
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Three gauge forces (based on local gauge invariance)
● 1 vector field (B) coupled to hypercharge
● 3 vector fields (W) coupled to weak isocharge
● 8 vector fields (G) coupled to color charge
Standard Model Yang-Mills Theory + Higgs Mechanism
g1 = e/cosθW
g2 = e/sinθW
g3 = gs
Standard Model of Particle Physics
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Electro-Weak mixing
Gauge fields mix and produce physical fields
Photon-field
Z-field
Photon field coupled to electric charge
Z-field coupled to weak charge
Standard Model of Particle Physics
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Higgs mechanism
Higgs gives masses to W/Z, higgs, and fermions
Higgs potential:
Spontaneous Symmetry Breaking & Vacuum Expectation Value
Mass of particles via Higgs mechanism
W/Z and Higgs
Fermions
Standard Model of Particle Physics
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Flavor mixings in
Neutrino Mixing
Quark Mixing
Standard Model of Particle Physics
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Free parameters in the Standard Model
● Coupling constants: e, g, sinθW
● Boson masses: mW, mZ, mH
● Fermion masses: (mνe, mνµ, mντ, me, mµ, mτ), (mu, md, ms,mc, mt, mb)
● Quark mixing parameters UCKM : (θ1,θ2,θ3,δCP)CKM
● Neutrino mixing parameters UMNS: (θ1,θ2,θ3,δCP)MNS
More than 20 free parameters!Depends on who you are asking to the number of free parameters and representation of the parameters in SM may vary. For example Majorana phases, number of fermion generations are not included in the above number counting. The higgs mass can be expressed with free parameters of VEV (v) and λ etc.
Standard Model of Particle Physics
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Standard Model is a beautiful theory based on a simple principle of local gauge invariance
It describes almost all particle physics observations up to 100 GeV (down to length scale of 10-18m)
It has been tested better than 0.1% of accuracy
Discoveries and achievements- W/Z bosons- top quark discovery - CP violation in B-meson system- higgs discovery- ……
Full Lagrangian: SU(3)C × SU(2)L × U(1)
Standard Model of Particle Physics
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Features of the Standard Model
● No transitions between leptons and quarksthe lepton number L and baryon number B are separately conserved
● The charge of the proton is exactly the same as that of the positron● Neutrinos are massless● A family contains only the left-handed neutrino and the associated right-handed anti-neutrino● The weak interaction has a pure V-A structure (maximal parity violation)
Predictions of the Standard Model
● The proton is stable ● The neutrinoless double beta decay is forbidden
Standard Model of Particle Physics
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Unanswered questions in Standard Model
- gravity?
- why 3 generations?
- why 3 forces?
- neutrino masses?
- hierarchy problem?
- dark matter and dark energy?
- ad hoc higgs mechanism (µ2 <0)?
- too many (>20) free parameters?
➔ There must be physics beyond the Standard Model
Hierarchy Problem in Standard Model
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Contribution of fermion loops to Higgs mass is quadratically divergent
● If there is no new physics at higher energy scale, the Λ is the Planck mass scale (MP = 1019 GeV). ● The mass of higgs is measured to be MH = 120 GeV ➔ a fine-tuning over the level of 10-17 (=102GeV/1019GeV)
This unnatural cut-off is called “hierarchy problem”
Renormalization is the procedure of eliminating divergences in calculations of higher order corrections
Λ : cut-off parameter on the magnitude of the 4-momentum in the loop
…
Supersymmetry
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● Supersymmetric GUT model was introduced by Akulov and Volkov (1972) and Wess and Zumino (1974) — renomalizable theory
● Supersymmetry introduce a symmetry between fermions and bosons; ➔ fermions and bosons are combined into supermultiplets
Supersymmetry
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● The symmetry between fermions and bosons is such that every fermion has a bosonic partner in the same multiplet, and vice versa.
● In case of an unbroken symmetry the two partners have the same mass.
~ ~ ~~ ~ ~~ ~ ~
~~~
~
~~
~
~
Supersymmetry
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● In Supersymmetry this is essentially the s-top (squark) loop cancelling the effect of the top quark loop
● The correction reduces to logarithmic:
● If every fermion is accompanied by two scalars with couplings λs=λf2
the quadratic divergences cancel
● Impose a symmetry between fermions and bosons ➔ Supersymmetry
Supersymmetry
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Supersymmetric Operator ➔ transforms fermion to boson and vice versa
SUSY operator Q: remove boson and put fermion:
�a†c+ c†a
|bi = |fi
Q̂ =�a†c+ c†a
annihilation operator of boson
creation operatorof fermion
Q̂|0i = 0
* We will skip SUSY algebra which is quite complicated to introduce in PH489 class. However, advanced students are encouraged to refer supersymmetry text books.
�a†c+ c†a
|fi = |bi
Supersymmetry
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● Particle spin changes under the supersymmetric transformations: Q● Supersymmetry introduces a new quantum number: R-parity:
Rp = (-1)3(B-L)+2S
Rp = 1 (Standard Model particles) Rp = -1 (Supersymmetry particles)
● Rp is conserved ➔ SUSY particles can only be produced in pairs of a SUSY particle and its antiparticle
● SUSY particles cannot decay directly to SM particles so the lightest SUSY particle has nothing to decay to.
for example: stable, weakly interacting Dark Matter candidate
lightest neutralino, sneutrino, Gravitino....
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MSSM Lagrangian
Superpotential
Chiral superfields
Vector superfields
Soft SUSY breaking term
➔ 124 free parameters in this minimal SUSY (MSSM) model!
Minimal Supersymmetric Standard Model (MSSM)
MSSM
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● The least number of particles added to the Standard Model to make a viable SUSY model (N=1 supersymmetry) ● Assume R-Parity is conserved (stable proton)
● Each SM particle has a SUSY partner➔ Supersymmetry requires two Higgs doublets to cancel gauge anomalies and provide mass to both up and down-type particles
SUSY is a broken symmetry Many different theories for SUSY breaking Generally spontaneous symmetry breaking in a hidden sector is communicated to the visible sector through corrections to the masses
Constrained MSSM (CMSSM sometimes called mSUGRA)Impose GUT scale (Mpl) relations on the MSSM
Set all scalar masses to one value m0 Set all gaugino masses to one value m1/2 Set trilinear couplings to one value A0 Set ratio of Higgs doublet VeVs to tanβTotal 5 free parameters (including sign of the higgsino mass term µ)
MSSM Particles
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Neutralinos and charginos are often denoted as: X0, X±
GUT: Running Coupling Constants
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At low energies where we are living in the GUT symmetry is broken.➔ The observed three forces might be a different aspect of a single fundamental force.
In addition to the known gauge bosons (𝛾, W±, Z0, g), we expect that there exist as yet undiscovered bosons (say X and Y).
How the interaction constants gs, g, g’ may be derived from gGUT?➔ running coupling constants:
The coupling constants indeed a variable of distance and energy (due to vacuum polarization and other higher order effects)
For example: due to the vacuum polarization the strength of electric charge increases at a very small distance (Lamb shift) while the bare charge screened due to the e+e- polarization in vacuum at relatively long distance.
One finds that gs, g, g’ approach one another in the energy region 1015~1016 GeV (GUT scale)
Running Coupling Constant & Unification of Forces
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↵i(Q2) =
↵i(M2)
1 +bi↵i(M2)
⇡ln(M2/Q2)
Standard Model Supersymmetry
With Supersymmetry the gauge coupling constants are unified at MX = 1016 GeV
b2 = �6 + 2Ngen +NH
2
29
Vacuum Polarization
Polarization of molecules around the electric charge (q) in a dielectric medium. The effective charge is given by qeff = q0/ϵ where ϵ is the dielectric constant
(a) A photon propagating through empty space undergoes a virtual transition into an electron-positron pair. (b) and (c) show such diagrams for the scattering of an electron image from: http://cerncourier.com/cws/article/cern/28487
In the QED, the vacuum itself behaves like dielectric, resulting vacuum polarization
KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
eR = e0
✓1� ↵
3⇡ln
⇤2
m2R
◆1/2
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Supersymmetry: Experimental Search
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Supersymmetry: Experimental Search
Models of Grand Unification
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Motivation for Grand Unified Theory (GUT)
The objective of a GUT is to explain the phenomenologically very different forces using a single elementary basic principle — a single fundamental coupling constant.
Customary attempt is to achieve the GUT is to assign a simple group G which contain the Standard Model:
G � SU(3)C ⇥ SU(2)L ⇥ U(1)
In order to make the group to be simple it must not have a decomposition form. This ensures that the theory contains only one coupling constant.
Smallest group satisfying these conditions: G = SU(5) ⊃ SU(3)C × SU(2)L × U(1)Next simple group satisfying these conditions: G = SU(10) ⊃ SU(5) × SU(3)C × SU(2)L × U(1)
In GUT models, it is assumed that the symmetry group S of the SMis part of a larger simple group G, which is visible only at high energies (~1016 GeV)
SU(5) Model
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The simplest realization of a GUT model by Georgi and Glashow:
The SU(5) theory contains following 15 left-handed fermions as matter particles:
and arranged in two multiplets
The covariant matrix of the model look like:
➔ SU(5) has 24 generators (52 - 1 = 24)
SU(5) Model: Interaction
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The kinetic term of the Lagrangian is then:
➔ implicit in this form, there are interaction terms between gauge bosons and matter fields:
Inserting content of fermonic and bosonic fields we get:
Exchange of X bosons can turn leptons into quarks and vice versa.➔ violate lepton number conservation and leads to process like a proton decay
SU(5) Model: Consequence
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SU(5) model involves only the known fermions — no room for right-handed neutrinos nor left-handed anti-neutrinos
● The breaking of the SU(5) symmetry occurs spontaneously by coupling to Higgs ➔ SSB at GUT scale (~1015GeV) by a 24-dimensional Higgs fields ➔ Only X and Y bosons acquire masses during the GUT scale SSB Total 24 gauge bosons including 12 known (𝛾, W±, Z, 8-gluons) gauge bosons
● Neutrinos are massless (0νββ-decay is NOT allowed)
● Baryon number (B) and the lepton number (L) are not separately conserved but (B-L) is conserved
● Magnetic monopoles with masses ranging from 1015 ~ 1017 GeV are predicted
● At the unification point the Weinberg angle is expected to be: sinθW = 3/8
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Proton Decay
Proton decay is one of the important predictions (requirement) of the Grand Unified Theories.
Theory Proton lifetime
Minimal SU(5) 1030~1031 years
Minimal Supersymmetric SU(5) 1028~1032 years
SUGRA SU(5) 1032~1034 years
Supersymmetric SU(5) ~1034 years
Minimal SO(10) <~1035 years
Supersymmetric SO(10) 1032~1035 years
not a full list
Note the age of the universe is 1010 years
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Proton Decay
https://www.youtube.com/watch?time_continue=1&v=7NMs0Vnwd1Q
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Proton Decay
Simulation
All attempts to observe the proton decay have failed — the best upper bound of the proton’s lifetime is 1.67 × 1034 years (from the SuperKamiokande experiment)
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Korea Neutrino Observatory (Plan)Water Cherenkov detector (250 kton)
— 1000m underground — Location to be determined — 30~40 years of operation
Physics goal ● Proton decay ● Definite measurements of neutrino mass ordering and δCP phase ● Observation of supernova neutrinos and Neutrino astrophysics ● Solar neutrinos and geoneturinos ● Non-standard neutrino interactions
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Extra Dimension
In ancient times, it must be very hard to believe that the earth is actually a globe. To make things even worse it’s rotating on its axis and revolving around the Sun.
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Extra Dimension
There is no good reason why we live in 3-space and 1-time dimensions. Extra dimensions in physics are the proposals of additional space dimensions beyond the (3 + 1) observed space-time.
symmetrymegazine
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Recap: Special Relativity
● Laws of physics remain the same in all inertial frames● The speed of light in a vacuum is a universal constant
The differential of distance in Minkowski space:
where the Minkowski metric is
A generalized 4-vector Newtonian law in free space is
with 4-velocity of where τ is a proper time
These are the Lorentz invariant expression of Newtonian law
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Lorentz invariant form for Electrodynamics: introduce field strength tensor
The homogeneous Maxwell equations is :
The inhomogeneous Maxwell equations is :
with fully asymmetric Levi-Civita pseudo tensor 𝜖
The equation of motion of point like particle with charge q is given by
The energy momentum tensor can be written as
Recap: Special Relativity
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Recap: General Relativity
● Equivalence inertial mass and gravitational mass● Existence of a local inertial frame for every point in spacetime
In a local inertial frame (Minkowski space) the differential of distance is given by
with the Minkowski metric ηαβ and inertial coordinate 𝝃
transformation to Riemann space is accomplished by setting:
with the metric tensor
Introduce tensors which are invariant under general coordinate transformations.
Aα : Lorentz tensorAµ : Riemann tensor
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Recap: General Relativity
A useful expression of a generalized derivative is
where the Christoffel symbols used to the metric tensor:
A generalized equation of motion then
or rather
For generalization of electrodynamics, the Maxwell equation and equation of motion promoted as
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Recap: General Relativity
Just for the completeness, we list the Riemann curvature tensor, the Ricci tensor and the scalar curvature.
Einstein’s field equation cannot be derived. It can only be made plausible with assumptions. - Energy momentum conservation should hold ➔ the covariant derivative of the energy momentum tensor should vanish. - The field equation should yield the Newtonian limit in a weak gravitational field
Using these assumptions the Einstein’s field equation is written as:
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Extra Dimension: Kaluza-Klein Theory
Theodor Kaluza Oscar Klein
In 1920s only Gravitation and Electrodynamics are known. The Weak and Strong interactions was yet to be discovered.Adding an additional dimension in Einstein’s field equation is rather easy in Kaluza-Klein theory. Consider Einstein’s equations in a vacuum which means setting Tµν = 0. In this case contracting with gµν yields R = 0.
Kaluza’s approach (1919)
I = 0, 1, 2, 3, 4
(t, x, y, z) ⌘ (x0, x
1, x
2, x
3) ⌘ x
µ
(t, x, y, z, ⌘) ⌘ (x0, x
1, x
2, x
3, x
4) ⌘ x
I
µ = 0, 1, 2, 3
5D metric
gauge field from EM dilaton field
Cylinder condition
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Extra Dimension: Kaluza-Klein Theory
Klein’s approach (1926)
The equation of motion generalized in 5D as
a straight forward calculation using the cylinder condition yields
and for 5th dimension
note the similarity
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Extra Dimension: Kaluza-Klein Theory
Indeed the charge (q) is proportional to the canonical momentum in the fifth dimension (p5) and that this momentum is conserved due to the cylinder condition.
p5 = n2⇡
L
LP =pG ' 1.6⇥ 10�34m
L =2⇡
e
p16⇡G ' 0.8⇥ 10�31m
Since the charge q is always a multiple of electron charge: q = ne with
The boundary condition yields quantized momenta in 5th dimension:
Using the above two representations of p5 the length scale of the extra dimension estimated as:
note the Planck length scale
p5 =qp
16⇡G
y
xµ
R~10-31 m
Compactification
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For a compact space of size R, a momenta greater that 1/R can probe the extra dimensions. Exciting Kaluza-Klein modes of gravity (or other fields) will appear as discrete massive modes in units of 1/R.
Extra Dimension: Kaluza-Klein Theory
The compactification can be implemented by imposing a periodic boundary condition
Examine a complex scalar field 𝛷 obeying the Klein-Gordon equation in 5D
with the Lagrangian
K = 0,1,2,3,4
The 5D action is given by
……… (*)
(xµ, y
a), µ = 0, 1, 2, 3, a = 1, ..., D
Suppose a coordinate system:
and D=1 for this example
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Extra Dimension: Kaluza-Klein Theory
➔ Incorporating the periodic boundary condition (*) yield the mode expansion
using the orthogonality relation
yields the effective 4D action:
The effective 4D theory describes an infinite number called Kaluza-Klein tower of Klein-Gordon fields (xµ) with the masses
The additional contribution to the mass of the states that is related to the momentum in 5D is evident:
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Kaluza-Klein Universal Extra Dimension
The Kaluza-Klein theory is impossible to gain a chiral gauge theory from a simple compactification on a topological smooth space. The theory contain undesirable fermionic degrees of freedom. In order to resolve this problem, Universal Extra Dimension theory introduces an additional discrete symmetry (Z2 symmetry).
The topological space defended above is denoted as:
This imposition of the boundary conditions switching the extra-dimension space from the former manifold to a orbifold.
which yields in 5-th D:
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Kaluza-Klein Universal Extra Dimension
Rewrite the complex field in terms of eigenfunctions of the parity operator acting on the extra dimension
where
If 𝛷 is taken to be even all 𝜙n(-) must vanish and if 𝛷 is taken to be odd all 𝜙n(+) including 𝜙0(+)
must vanish. Therefore orbifold compactification makes it possible to develop a chiral gauge theory by removing unwanted fermionic degrees of freedom.
All Standard Model particles have to be described by even wave functions, and 5th D particles in odd wave functions.
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Kaluza-Klein Universal Extra Dimension
The Fermions can be expressed as:
The masses of Kaluza-Klein modes at tree level:
Examining the matrix mixing the first level Kaluza-Klein modes and incorporating the first level radiative corrections the result
Hence the Kaluza-Klein photons is given by
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Kaluza-Klein Universal Extra Dimension
Kaluza-Klein number is conserved with respect to all interactions neglecting branes and orbifolds. A discrete subgroup called Kaluza-Klein parity PKK=(-1)n is conserved. Therefore at least lightest Kaluza-Klein mode is stable.
not allowed
B(1)
?(1)?(k)
Suppose B(1) is the lightest KK particle
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Kaluza-Klein Universal Extra Dimension
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Kaluza-Klein Universal Extra Dimension
Interaction Lagrangian of Kaluza-Klein Dark Matter with quarks
Interaction cross section
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Phys.Rev.D78:056002 (2008)
Kaluza-Klein Universal Extra Dimension
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Large Extra Dimension
The higher dimensional theory accompany a higher dimensional Planck scale which can be ~TeV. Gravity dilutes into the extra dimensions and the gravitational potential falls off faster at distances smaller than the radius of the extra dimensions. This explains why gravity is so much weaker than the other interactions. It means that at smaller distances, gravity is much stronger than what we expect in the our 3D space.
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Large Extra Dimension
n=1 R~1012m solar systemn=2 R~10-3m Pinheadn=3 R~10-9m Gold atom…
n=6 R~10fm Nucleus
Effective (higher dimensional) Planck scale can be at 1 TeV
The ADD-model (Arkani-Hamed, Dimopoulos and Dvali, 1998) adds extra space dimensions. In general, each of them is compactified to the same radius. All SM particles are confined to our brane, while gravitons are allowed to propagate freely in the bulk.
M2Planck = RnM2+n
Planck(4+n)
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Randal-Sundrum Extra Dimension
G
(L >> 1)
Mother Brane (y=1/k)Our World Brane(y=1/Wk)
Infinite 5-th dimension No compactification is required
Zero mode gravitation is trapped on the Mother Brane(Planck Brane)
Randal-Sundrum model (1999) is a 5-dimensional spacetime with a 'warped' geometry. The solution for the metric is found by analyzing the solution of Einstein's field equations with a constant energy density on our brane where the SM particles live. In the type I model the extra dimension is compactified, in the type II model it is infinite.
ds
2 =1
k
2y
2(dy2 + ⌘µ⌫dx
µdx
⌫)
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Eot-Wash Group (University of Washington)
Experiment Searching for Extra Dimensions
Motivated by higher-dimensional theories that predict new effects. Testing gravitational 1/r2 law at separations ranging down to 218 μm using a 10-fold symmetric torsion pendulum and a rotating 10-fold symmetric attractor. (arxiv:hep-ph/0011014v1)
The simplest scenario with 2 large extra dimensions predicts λ=R∗ and α=3 or α=4 for compactification on an 2-sphere or 2-torus, respectively R⇤ =
1
⇤
✓MP
⇤
◆2/n
The extra dimension radii:
α
λ [m]
No deviation fromNewtonian law > 218µm
Modified Gravitational Potential:
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Magnetic Monopole
N
S N
N
S
S
NS
NS
NS
N
S
orbital motion of electrons
spin of electrons
The source of magnetic field is the motion of the electric charge. Understanding source of field generated by bar magnet essentially lies in understanding currents at atomic level within bulk matter.
If this was the full story, particle physicists may not be too much interested in the source of the magnetic field (potentially magnetic monopoles).
64KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Magnetic Monopole
electricmonopole
magneticmonopole
● We have electric monopole - electric dipole is generated by electrically charged particles with opposite polarity- magnetic dipole is generated by circular electric currents ➔ no way to separate S & N
● If magnetic monopoles exist, we can achieve ultimate symmetries in electromagnetism
65KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Magnetic Monopole
Dirac (1931) showed that if magnetic monopoles exist the charge quantization can be understood (from angular momentum quantization) ●
B
βt
r
●
x
z
be
g
Suppose a electric charge moves with a velocity of β in the z-direction. The charge is subject to a Lorentz force in the y-direction
The component of Bx is therefore
The momentum transmitted to the particle is given by
Hence the change in angular momentum is
implying a quantization of the electric charge:
66KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Magnetic Monopole
Dirac’s idea shows the properties of magnetic monopole.
for example: classical electron radius:
Assuming the magnetic monopole radius is similar to electron radius
There was a serious objection to the possible existence of magnetic monopole, since it violates the time reversal invariance. However, after the discovery of CP-violation in K0 system (which is equivalent of T-violation in CPT theorem) the objection to the possible existence of monopoles can no longer be justified.
(mM ~ 2.4 GeV)
67KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Magnetic Monopole
Symmetry of Maxwell equations with magnetic monopole (source term)
Define the angle 𝜙0 such that ρm and jm vanish
The electric charge of electron and magnetic monopole are chosen:
⇢m = ⇢0e
✓� sin�0 +
⇢0m⇢0e
cos�0
◆= 0
68KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Magnetic Monopole in GUT
● t’Hooft and Polyakov (1974) showed that the magnetic monopoles appear to be natural consequence in the framework of GUT — especially the SU(5) group
● The U(1) gauge theory (Abelian) extended by introduction of magnetic monopole — but it’s not required.
● t’Hooft extended it to Non-Abelian group SU(5). The spontaneous symmetry breaking leads to a stable solution with a property of magnetic monopole. — which is required!
Using a simplest Georgi and Glashow model, t’Hooft showed magnetic monopole field obtained after proper choice of gauge invariant Fµν in the model
Mass of GUT monopoles:
➔ this heavy mass explains why the monopoles escaped from detection so far
[t’Hooft : Nuclear Physics B79 (1974) 276-284]
choice of invariant field Fµν
vector and scalar field in GG model
radial B-field component (monopole)
(D⌫Qe)
69KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Magnetic Monopole Detector
Science Photo Library
February 14th, 1982
PRL V48, 20 (1982)
Superconducting ring magnetic monopole detector
➔ No further observation of the events nor supporting evidence afterwards.
As magnetic monopole passes the superconducting loop, magnetic field in the loop changes ➔ The induced electromagnetic force in the loop drive current
70KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
Unification of Forces
The Origin of Everything
71KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model
The History of Everything
72KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model