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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

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Page 1: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

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

Page 2: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

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

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PH489 Schedule In Nov/Dec 2017

3KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

29 30 31

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PH489 Seminar Schedule & Location

4KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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).

Page 5: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Standard Model of Particle Physics

5KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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Standard Model of Particle Physics

6KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 7: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Standard Model of Particle Physics

7KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Matter Particles

Leptons

Quarks

note:missing right-handedneutrinos

x 3 colors (r,g,b)

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Gauge Principle

8KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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:

Page 9: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Gauge Transformation

9KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 10: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Standard Model of Particle Physics

10KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 11: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Standard Model of Particle Physics

11KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

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Standard Model of Particle Physics

12KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

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Standard Model of Particle Physics

13KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Flavor mixings in

Neutrino Mixing

Quark Mixing

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Standard Model of Particle Physics

14KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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.

Page 15: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Standard Model of Particle Physics

15KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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)

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Standard Model of Particle Physics

16KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

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Standard Model of Particle Physics

17KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

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Hierarchy Problem in Standard Model

18KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

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Supersymmetry

19KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

● 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

Page 20: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Supersymmetry

20KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

● 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.

~ ~ ~~ ~ ~~ ~ ~

~~~

~

~~

~

~

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Supersymmetry

21KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

● 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

Page 22: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Supersymmetry

22KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 23: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

Supersymmetry

23KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

● 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....

Page 24: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

24KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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)

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MSSM

25KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

● 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)

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MSSM Particles

26KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Neutralinos and charginos are often denoted as: X0, X±

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GUT: Running Coupling Constants

27KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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)

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Running Coupling Constant & Unification of Forces

28KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

↵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

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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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30KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Supersymmetry: Experimental Search

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31KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Supersymmetry: Experimental Search

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Models of Grand Unification

32KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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)

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SU(5) Model

33KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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)

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SU(5) Model: Interaction

34KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 35: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

SU(5) Model: Consequence

35KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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36KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 37: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

37KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Proton Decay

https://www.youtube.com/watch?time_continue=1&v=7NMs0Vnwd1Q

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38KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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39KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

Page 40: Special Topics in Particle Physicsyoo.kaist.ac.kr/lectures/2017/2/files/YooKAISTPH489... · 2017-12-27 · Gauge Principle KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model 8

40KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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41KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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42KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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43KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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44KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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45KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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46KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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47KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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48KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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49KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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

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y

R~10-31 m

Compactification

50KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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51KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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52KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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53KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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54KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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55KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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56KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Kaluza-Klein Universal Extra Dimension

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57KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Kaluza-Klein Universal Extra Dimension

Interaction Lagrangian of Kaluza-Klein Dark Matter with quarks

Interaction cross section

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58KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Phys.Rev.D78:056002 (2008)

Kaluza-Klein Universal Extra Dimension

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59KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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60KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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61KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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62KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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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63KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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).

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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

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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:

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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)

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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

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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)

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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

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70KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

Unification of Forces

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The Origin of Everything

71KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model

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The History of Everything

72KAIST-PH489-Yoo-2017-Note04: Beyond the Standard Model