The SM Lagrangian

: how to build a model?s.park@skku.edu

Open KIAS winter school on collider physics, 2013 Feb 17-23

prelims

•can there be a particle with a spin other than 0,1/2,1,3/2,2,... (i.e. Z+/2 or Z+)?

•How many traceless, Hermitian (2X2)matrices exist? (3X3)? (NXN)?

•Can you write the QED Lagrangian for a Dirac spinor?

test your understanding of previous lectures

M.C.Escher [Relativity,1953]

Andrew Lipson’s [Relativity]http://www.andrewlipson.com/escher/ascending.html

Lesson?LEGO® world is large

You may find some place where even the law of physics is not

applicable!Well, Lipson was actually cheating. But worth trying making models

and he could even sell the pictures.

Some thoughts

• “Theory space” is large

• ...probably the existing theory space is not large enough to describe the nature...

• we may lose our ways without having guiding principles.

• Particle physicist like to follow fundamental principles -not only like to think of the constituents of matter-

• Uncertainty principle (QM) and the principle of relativity (SR) provide some clues to approach the right answer.

• Logically Quantum Field Theory should be the right language [S.Weinberg QFT vol.1]

• So, now we regard QFT as the tool to attack the problem (...like LEGO® is the right toy for kids.)

• and try to build the right ‘model’ for Nature

• The SM is the best answer we have found so far.

• In a sense, we may call the SM the greatest intellectual achievement.

• Be proud of what you will understand!

• Choose spacetime

• Introduce matters

• Choose symmetry (Reps) & its breaking

• Construct Lagrangian density

Modern way of building a particle

physics model

The SM

• The spacetime: M4=R1,3

• Matters: 3 copies of {Q,u,d,L,e}

• Symmetry: Local SU(3)cXSU(2)wXU(1)y +SSB due to ⟨H⟩ ➔ SU(3)cXU(1)em

• LSM : the goal of this class

The Space time

• M4 has L4+ symmetry (Proper Lorentz Symmetry) + (inversions)

• L4+ = SU(2)XSU(2) the covering group

• A particle state is in (n, m)-representation where n’s and m’s are 0,1/2,1,3/2,...as angular momentum operator allows

all possibilities

Only a right ‘block’ fits in

• All possible particles

✴ scalar (s=0), observed 2012

✴ Weyl & Dirac spinors (s=1/2), observed 1899

✴ 4-vector (s=1), (2-form fields), observed 1905

✴ Rarita-Schwinger(s=3/2), never observed

✴ graviton (s=2), indirectly 1995

can play as a fundamental state of particle

[Weinberg-Witten theorem ,1980] prevent s>2.

• Their Lorentz invariant combinations can be a part of Lagrangian, which is again Lorentz invariant or follows the principle of Relativity

Lorentz invariant combinations

*There are infinite number of Lorentz invariant combinations. *but further constrained by symmetries, requiring better UV-behavior(a.k.a renormalizability)*..almost uniquely defined up to [some finite number of inputs] [See Prof. Ko’s lecture]

(ex) Kinetic term

*quadratic term with a field twice*often describes a free field (i.e. a field in no other interaction)

Only 3 terms are allowed up to Mass dimension 4. [why 4?]

[why not the other term?]

[check Dirac, Maxwell, YM actions]

with gravity ?

L4 is now defined only local inertial frame.(in freely falling frame, no gravity is felt by an observer)

negligible when M~1019 GeV is much bigger thanthe energy scale we are interested in E<10 TeV.

other possibilities?

fermionic directions for supersymmetry

extended spacetime

These two extensions of spacetime have been dominantly studied over decades.

VOCABULARYfor SM Matters

• Quarks and Leptons : strongly interacting or not

• Doublets and Singlets : weakly interacting or not

짧은 글짓기

• 다음 낱말을 넣어서 짧은 글을 지으시오.

Lepton, Singlet, Gluon

The Particles in the SM

Q. what is Y?

+two more copies

3 generations

Q. why 3? more?

what they are about?

• SU(3) is for interactions among different colors {r, g, y}

• SU(2)XU(1): W+(-) is for up-down (i.e. charged) interaction, Z is for up-up, down-down (i.e. neutral) interactions

-triplet of SU(3): 3 colors-singlelet of SU(2): no SU(2) interaction -also have Hypercharge of U(1)

-triplet of SU(3): 3 colors ~ strongly int.-doublet of SU(2) : up and down-also have Hypercharge of U(1)

color index suppressed

등장인물(1)

-singlelet of SU(3): no strong interaction-doublet of SU(2) : up and down-also have Hypercharge of U(1)

-singlelet of SU(3): no strong interaction-singlet of SU(2) :no SU(2) interaction-also have Hypercharge of U(1)

등장인물(2)

With respect to SU(2) interaction(and also U(1) interaction)

i.e. Parity (a symmetry L-R conversion) is maximally broken!

Parity violation

질문

• 그래서 eL와 eR 가 같은 입자인가요?

[reminder]SU(N) Theory

SU(N) generators

Lagrangian:

[prove this is invariant under SU(N) transformation]

[why 1/2 ?]

Gauge covariant derivatives

for a particle in fundamental representation

[Gell-Mann-Nishijima]

[Pauli]

[Gell-Mann]

Pauli for SU(2)

[Ex] Show

Gell-Mann for SU(3)

[Ex] Show

[Q. what is the role of gluon of lambda’s?]

SU(5) for GUT

[note] SU(3)XSU(2) is in SU(5)

[see project]

SU(N)

structure constantnormalization

Hypercharge assignment(1)

as it should be!

Hypercharge assignment(2)

as expected!

Left-handed-lepton-doublet

Hypercharge assignment(3)

Right-handed-lepton-singlet

Hypercharge assignment(4)

지금껏 배운 것 요약

i=1,2,3 generations

This is the SM Lagrangianbefore the electroweak symmetry breaking

QED in the SM

1. for electron, current-photon interaction allowed.2. for neutrino, no QED interaction allowed.3. electron is massive, Dirac spinor4. photon is massless (U(1)em is a good symmetry)

should be included

QED in the SM

diagonal, neutral current

off-diagonal, charged current

Q. Why no self-interactions for photon?

charged current

Q. more complication involved in quark sector.

Why?

neutral current

(W3, B) not aligned to the mass eigenstates:

Should reproduce:

[1] [2]

Now we can read Z-couplings

for neutrinofor electron

The SM interactions

[because eR has the same coupling structure]

[Ex] Work out the same structure in quark sector.

The SM w/o EWSB

• The SM Lagrangian is invariant under SU(3)XSU(2)XU(1)

• Successfully reproduce QED :-)

• CC interaction is described :-)

but long range :-(

• NC interaction is described :-)

but long range :-(

• all gauge bosons massless! [why?]

• all fermions massless! [why?]

A man who wrote a 2 pages-long paper

“It may be expected that when a. further mechanism (presumably related to the weak interactions) is introduced in order to break Y conservation, one of these gauge fields will acquire mass, leaving the photon as the only massless vector particle. ”

Higgs mechanism

• Only SU(3)XU(1)em is respected by Vacuum!

• CC and NC interactions are short-ranged.

• Symmetry is “spontaneously broken” when the vacuum does not respect the symmetry of Lagrangian.

[This part will be covered in Prof.Song’s talk]

some examples of SSB

• (in Weinberg’s dream) a chair in space

• (in my daughter’s math book) x2 =1

• (in physics) ferromagnet, superconductor, the Higgs

How things are related

G: Global symmetry of Lagrangian (equation)H: Global symmetry of Vacuum (solution)

:Nambu-Goldstonebosons

[Prof. Chun’s lecture]

for gauge fields?

•M gauge fields (massless in the beginning)•Vacuum respects only H•(M-N) gauge bosons becomes massive by eating the would-be-NGBs.

Electroweak SB

Q. How do you break G to H?

Q. Why should be scalar for EWSB?

What happens if

we want [electroweak symmetry][electromagnetic symmetry]

Note)

Gauge bosons got masses!