1 of 38

QUANTUM FIELD THEORY AND STANDARD MODEL

양자장론과 표준모형

2 of 38

LECTURE 4

The Standard Model

3 of 38

1. While QFT is a plausible framework describing relativistic QM, it does not tell us how the nature should behave, e.g., which gauge group? Which representations for matter?

2. Due to our limit of accessibility, we just have a phenomenological effective model

(energy below TeV, feeble interaction with dark sector, gravity…)

3. Nevertheless, quite often, the low energy behavior can be a smoking gun (떡밥) of the physics beyond our probe when combined with the fundamental principles of e.g., QM

4. Standard Model (Weinberg-Salam-Glashow model : unification of electrodynamics and weak interaction)

Recommended reading : R. Kleiss, ‘Derivation of the minimal standard model Lagrangian’ in ‘physics up to 200TeV’ (Erice school 1990)

A. Agrawal, S. M. Barr, J. F. Donoghue, D. Seckel, hep-ph/9707380 Phys. Rev. D 57 (1998) 5480

See also Particle Data Group https://pdg.lbl.gov

4 of 38

Parity Violation : discovery of chiral nature of fermions

Parity : or

just like , but : under the parity,

That is, , in particular,

: the parity invariance of spin-1/2 particle means that the left-handed spinor (1/2, 0) and the right handed spinor (0, 1/2) must behave in the same way (always vector-like)

However, investigation of β decay shows that the weak interaction does not respect parity

(T. D. Lee and C. N. Yang, 1956 ; C. S. Wu, E. Amber, R. W. Hayward, D. D. Hoppes, R. P. Hudson, 1957)

5 of 38

6 of 38

mirror image

Before decay

After decay

parity

mirror

(1) What actually happens

(2) Expected one in the presence of parity invariance

7 of 38

If parity is a symmetry, detectors A and B should detect the same amount of electrons.

However, C. S. Wu et al’s experiment shows that detector A finds much more electrons

Electron at detector B can be thought of as a result of the transition of the right handed to the left handed through the left-right mixing in the mass term (suppressed by electron mass)

(1) What actually happens

(2) Expected one in the presence of parity invariance

Detector A

Detector B

8 of 38

This shows that only the left handed electron / neutrino are charged under the weak interaction

: fermions are chiral

Let’s ignore the masses for a moment (assume that all particles are massless)

(1/2, 0) : SU(2) doublet

(0, 1/2) : SU(2) singlet

Not discovered (not charged in any known interaction)

Can be ignored if neutrino is massless

Lepton

Quark

9 of 38

10 of 38

SU(2) doublet :

SU(2) singlet :

11 of 38

Higgs mechanism

12 of 38

13 of 38

‘Gauge singlet consists of the Higgs and bi-spinor’ : fermion mass term (Yukawa coupling)

1. Charged lepton mass (for a moment we assume neutrino to be massless)

: obviously SU(2)W singlet

: U(1)Y singlet

2. d-quark mass

Higgs VEV

14 of 38

Higgs VEV

15 of 38

Charge = +1

Determined by the coupling to Higgs

16 of 38

2. Neutral gauge bosons

Since Higgs has Y=1/2,

The massive, neutral combination

Then remaining combination is neutral, massless : photon

17 of 38

18 of 38

Then

19 of 38

Summary

or

20 of 38

Summary

Lepton

Quark

Higgs

21 of 38

-_-

22 of 38

1. three generations (families)

Three copies of have been found.

Gauge charges (representations) are the same, but Yukawa couplings are different : masses are different

23 of 38

diagonal

quarks in mass eigenbasis

Cabibbo-Kobayashi-Maskawa (CKM) matrix

24 of 38

25 of 38

26 of 38

27 of 38

One more reason that the operator is interesting: it breaks a global symmetry of the Standard Model (renormalizable) Lagrangian.

Two global symmetries of the Standard Model :

1. Baryon number

leptons are neutral

(point : all the quarks have the same charge; 1/3 is assigned to make baryons (qqq bound state) have the baryon number +1)

2. Lepton number

quarks are neutral

The operator is not a singlet under the lepton number! Indeed, Majorana mass term breaks the lepton number. (indeed anomalous)

28 of 38

In general, global symmetry is feasible :

If we put global charges into the black hole, we do not have any way to measure it.

(for U(1) gauge symmetry, charge inside the black hole changes the geometry and give the nontrivial electric field, the flux of which coincides with the amount of charge.)

If the global charge is preserved even inside the black hole, the entropy bound is violated

Global symmetry as an accidental symmetry

Global symmetry ‘emerges’ by some accidental structure of the model:

e.g., considering renormalizable operators only.

separation of quarks and leptons by the strong interaction

negligible Majorana mass term ….

29 of 38

30 of 38

s ?

31 of 38

32 of 38

33 of 38

SU(2)L

SU(2)R

SU(2)V

SU(2)A

‘vector’

:parity even

‘axial’

: parity odd

34 of 38

That is, in the massless limit, while the interaction (Lagrangian) respects the global (chiral in our case) symmetry, vacuum configuration breaks it : spontaneous symmetry breaking (‘hidden symmetry’)

The similar situation can be found in the magnetization of ferromagnet

In this case, quantum fluctuation in the direction of (spontaneously broken) symmetry breaking is massless : (Nambu-)Goldstone boson

In magnetization, this Goldstone boson of the spin wave mediates the long-range force which leads to infinitely long correlation length

: explains how the spin direction here is the same as the spin direction there.

Vacuum configuration fixes the direction of spin

35 of 38

SU(2)A generators

Would-be vacuum configuration

36 of 38

SU(2)A transformation of the vacuum configuration:

Invariant