Introduction to gauge theory

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1 Introduction to gauge theory 2008 High energy lecture 1 장상현 연세대학교 September 24, 2008 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

2 Table of Contents 1 Introduction 2 Dirac equation 3 Quantization of Fields 4 Gauge Symmetry 5 Spontaneous Gauge Symmetry Breaking 6 Standard Model 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

3 References for quantum field theory Quark and Leptons Halzen and Martin Quantum Field Theory Ryder Quantum Field Theory Mandl and Show Gauge Theory of Elementary Particle Physics Cheng and Li Quantum Field Theory in a Nutshell Zee An Introduction to Quantum Field Theory Peskin and Schroeder and many more... 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

4 Introduction The Standard Model (SM) is The basis of High Energy Physics. SM is a local quantum gauge field theory with spontaneous gauge symmetry breaking mechanism a.k.a. Higgs Mechanism. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

5 Introduction The Standard Model (SM) is The basis of High Energy Physics. SM is a local quantum gauge field theory with spontaneous gauge symmetry breaking mechanism a.k.a. Higgs Mechanism. Object of this lecture is to learn the basic concept of the gauge symmetries and their breaking mechanism to understand SM. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

6 The lecture will be a short introduction course to Quantum field theory and gauge theory. A modern approach to this subject is to use path integral and propagator theory. However, we will follow traditional Lagrangian approach. For the path integral method, look for the references. In elementary particle physics, we use the unit where = c = k B = 1. Mas, length, time, energy, momentum, temperatures can be measured in ev or ev 1 in this unit. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

Paul Adrien Maurice Dirac, (1902 1984) was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics.

7 Paul Adrien Maurice Dirac, ( ) was a British theoretical physicist. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the so-called Dirac equation, which describes the behavior of fermions and which led to the prediction of the existence of antimatter. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

8 Dirac equation The classical field theory which describes EM field is consistent with Special theory of relativity 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

9 Dirac equation The classical field theory which describes EM field is consistent with Special theory of relativity but not with Quantum mechanics. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

10 Dirac equation The classical field theory which describes EM field is consistent with Special theory of relativity but not with Quantum mechanics. The Schrödinger equation describes low energy electrons in atom 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

11 Dirac equation The classical field theory which describes EM field is consistent with Special theory of relativity but not with Quantum mechanics. The Schrödinger equation describes low energy electrons in atom but it is not consistent with relativity. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

12 Dirac equation The classical field theory which describes EM field is consistent with Special theory of relativity but not with Quantum mechanics. The Schrödinger equation describes low energy electrons in atom but it is not consistent with relativity. Non-relativistic quantum mechanics cannot describe High energy particle interactions. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

13 Dirac equation The classical field theory which describes EM field is consistent with Special theory of relativity but not with Quantum mechanics. The Schrödinger equation describes low energy electrons in atom but it is not consistent with relativity. Non-relativistic quantum mechanics cannot describe High energy particle interactions. Need to combine quantum mechanics with special relativity. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

14 In 1928, Dirac realized that the wave equation can be linear to the space time derivative µ / x µ. (iγ µ µ m)ψ = 0 (1) Applying (iγ µ µ m) to (1) leads ( ) 1 2 {γµ, γ ν } µ ν + m 2 ψ = 0 (2) where {A, B} = AB + BA is anticommutator. If {γ µ, γ ν } = 2η µν (3) η µν is the Minkowski metric η 00 = 1, η jj = 1 and otherwise zero. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

15 Then the Dirac equation becomes ( 2 + m 2 )ψ = 0 This is the same form as Klein-Gordon equation for the scalar fields. ( 2 + m 2 )φ = 0 γ µ satisfies Clifford algebra (3) can be written as 4 4 matrices. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

16 One representation of γ µ satisfies (3) is ( ) ( γ 0 I 0 = γ i = 0 I 0 σ i σ i 0 ) (4) I is 2 2 identity matrix and σ i (i = 1, 2, 3) are Pauli matrices. It is called Dirac basis. Some useful notations: γ µ η µν γ µ p γ µ p µ e.g. Dirac equation (i m)ψ = 0 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

17 and anticommute with γ µ {γ 5, γ µ } = 0 The matrix γ 5 iγ 0 γ 1 γ 2 γ 3 has the form in Dirac basis γ 5 = ( 0 I I 0 ) (5) (γ 5 ) = γ 5, (γ 5 ) 2 = 1 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

18 With the 6 matrices σ µν i 2 [γµ, γ ν ] {1, γ µ, σ µν, γ µ γ 5, γ 5 } form a complete basis of 16 elements. All 4 4 matrices can be written as a linear combination of above 16 matrices. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

19 With the 6 matrices σ µν i 2 [γµ, γ ν ] {1, γ µ, σ µν, γ µ γ 5, γ 5 } form a complete basis of 16 elements. All 4 4 matrices can be written as a linear combination of above 16 matrices. γ µ can have different basis with the same physics. e.g. Weyl basis, ( ) ( ) ( γ 0 0 I =, γ i 0 σ i = I 0 σ i, γ 5 = 0 I 0 0 I ) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

20 If we transform the spinors to momentum space d 4 p ψ(x) = (2π) 4 e ipx ψ(p) The Dirac equation becomes (γ µ p µ m)ψ(p) = 0 (6) Dirac spinor ψ can be divided into two 2-component spinors, ( ) φ ψ = χ 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

21 In Dirac basis, (γ 0 1)ψ(p) = 0 in the rest frame p µ = (m, 0). Only φ describes electron, which has two component. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

22 In Dirac basis, (γ 0 1)ψ(p) = 0 in the rest frame p µ = (m, 0). Only φ describes electron, which has two component. For slowly moving electron, χ(p) is very small. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

23 Lorentz transformation is defined as Λ = e 1 ωµνj µν 2 anti-symmetric ω µν = ω νµ are 3 rotation and 3 boost parameters. J ij are rotation generators and J 0i are boost generators. The coordinate x α transforms x α = Λ α β xβ 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

24 Spinors transforms under Lorentz transformation is ψ (x ) = S(Λ)ψ(x) where Also and S(Λ) = e i 4 ωµνσµν Sγ ν S 1 = Λ ν µγ µ S(Λ) = γ 0 e i 4 ωµνσµν γ 0 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

25 Define ψ = ψ γ 0 then ψ(x)ψ(x) is invariant under Lorentz transformation (scalar). ψ(x)γ µ ψ(x) transform as Lorentz vector. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

26 Define ψ = ψ γ 0 then ψ(x)ψ(x) is invariant under Lorentz transformation (scalar). ψ(x)γ µ ψ(x) transform as Lorentz vector. ψ(x)γ 5 ψ(x) transform as a pseudoscalar. ψ(x)γ 5 γ µ ψ(x) transform as Lorentz pseudovector. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

27 Lagrangian L and Lagrangian density L is defined from the action S = dtl = d 4 xl In High energy physics(hep) Lagrangian means Lagrangian density L. If the L is a function of field φ(x), the Euler-Lagrange eq. of motion should satisfied. ( ) L µ L ( µ φ) φ = 0 (7) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

28 Lagrangian of free Dirac field L = ψ(iγ µ µ m)ψ (8) From the eq. of motion ( ) L µ L ( µ ψ) ψ = 0 Dirac equation can be obtained (iγ µ µ m)ψ = 0 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

29 Define chiral projection, The projection operators ψ L (x) = P L ψ(x), ψ R (x) = P R ψ(x). P L = 1 γ5 2, P R = 1 + γ5 2 Then the Dirac spinor is sum of two chiral components ψ(x) = ψ L (x) + ψ R (x). 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

30 In Weyl basis Thus Some properties to notice ( ) γ 5 I 0 =. 0 I ( ) ψ L ψ(x) =. ψ R P 2 L = P L, P 2 R = P R, P L P R = 0 γ 5 ψ L = ψ L, γ 5 ψ R = +ψ R. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

31 Dirac Lagrangian can be written in chiral components L = ψ(iγ µ µ m)ψ = ψ L iγ µ µ ψ L + ψ R iγ µ µ ψ R m( ψ L ψ R + ψ R ψ L ) (9) If m = 0, ψ L and ψ R are independent and have additional symmetry ψ L e iθ L ψ L, ψ R e iθ R ψ R, 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

32 Dirac Lagrangian can be written in chiral components L = ψ(iγ µ µ m)ψ = ψ L iγ µ µ ψ L + ψ R iγ µ µ ψ R m( ψ L ψ R + ψ R ψ L ) (9) If m = 0, ψ L and ψ R are independent and have additional symmetry ψ L e iθ L ψ L, ψ R e iθ R ψ R, Weak interaction is called chiral because it only interacts with left-handed leptons. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

33 Further readings Check the references for Charge conjugation, Parity transformation, CP and CP T. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

Amalie Emmy Noether, (1882 1935) was a German mathematician described by Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of

34 Amalie Emmy Noether, ( ) was a German mathematician described by Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. She also is known for her contributions to modern theoretical physics, especially for the first Noether s theorem which explains the connection between symmetry in physics and conservation laws. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

35 Noether theorem relates a continous symmetry to a conservation law. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

36 Noether s theorem If a Lagrangian L with a field φ a is invariant of a continuous transformation φ a φ a + δφ a 0 = δl = δl δφ a + δl δφ a δ( µ φ a ) δ( µφ a ) (10) use the eq. of motion (10) becomes ( ) δl δl = µ δφ a δ( µ φ a ) ( ) δl 0 = µ δ( µ φ a ) δφ a 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

37 We define a current Then J µ δl δ( µ φ a ) δφ a µ J µ = 0 We have a conserved current J µ. Noether s Theorem A conserved current is associated with a continuous symmetry of the Lagrangian. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

38 Charge is defined as Q = d 3 xj 0 = d 3 δl x δ( 0 φ a ) δφ a. (11) Since dq/dt = 0, the charge is conserved. π(x) δl δ( 0 φ a ) is canonical momentum (density) corresponding to φ a. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

39 Free scalar field Lagrangian L = µ φ 2 m 2 φ 2 (12) of Klein-Gordon equation of motion ( 2 + m 2 )φ = 0. The Noether current J µ = i[( µ φ )φ φ ( µ φ)], (13) corresponds the symmetry φ e iθ φ. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

40 For free fermion L = ψ(iγ µ µ m)ψ (14) The Noether current J µ = ψγ µ ψ (15) corresponds the symmetry ψ e iθ ψ. µ J µ = ( µ ψ)γ µ ψ + ψγ µ µ ψ = (im ψ)γ µ ψ + ψγ µ ( imψ) = 0 This symmetry is called U(1) global symmetry, since θ is the same for any space-time x. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

41 Quantization of scalar field For free scalar field, the canonical momentum is π = φ. Being a quantum field theory requires: 1. φ(x) and π(x) becomes operator 2. and they satisfy canonical commutator relation. [φ( x, t), π( y, t)] = iδ (3) ( x y) [φ(x), φ(y)] = [π(x), π(y)] = 0 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

42 If E p = p 2 + m 2, φ(x) = π(x) = 0 φ(x) d 3 p 1 ( (2π) 3 a( p)e ip x + a( p) e ip x) p 2E p 0 =E p (16) [a( p), a( p ) ] = (2π) 3 δ (3) ( p p ), (17) a( p) creates one particle state from vacuum 0 p = 2E p a( p) 0 a( p) destroys vacuum a( p) 0 = 0. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

43 Quantum field is a harmonic oscillator with continuous degree of freedom. φ(x) acting on vacuum, create a particle at x. φ(x) 0 = d 3 p 1 (2π) 3 e ip x p 2E p 0 φ(x) p = e ip x is free particle wave function. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

44 Dirac field quantization For free Dirac field, canonical momentum is π = δl δ( 0 ψ) = iψ. Not like the scalar case, fermion field should satisfy anticommutation relation a, b are spinor components {ψ a ( x, t), ψ b ( y, t)} = δ(3) ( x y)δ ab {ψ a ( x, t), ψ b ( y, t)} = {ψ a( x, t), ψ b ( y, t)} = 0 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

45 We can write the field operators ψ(x) = ψ(x) = d 3 p 1 (2π) 3 d 3 p (2π) 3 1 s = 1, 2 is spin index. (b(p, s)u(p, s)e ip x + d (p, s)v(p, s)e ip x) 2E p s (d(p, s) v(p, s)e ip x + b (p, s)ū(p, s)e ip x) 2E p s {b(p, s), b (p, s )} = {d(p, s), d (p, s )} = (2π) 3 δ (3) ( p p )δ ss 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

46 Both b(p, s) and d(p, s) annihilate vacuum b(p, s) 0 = d(p, s) 0 = 0 b (p, s) and d (p, s) creates particle with energy momentum p but they are charge conjugated state with each other. We define b (p, s) creates a fermion and d (p, s) creates an anti-fermion. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

47 From the Noether current J µ = ψγ µ ψ, there is a conserved charge d Q = d 3 xψ 3 p ( ) (x)ψ(x) = (2π) 3 b (p, s)b(p, s) d (p, s)d(p, s) b (p, s) creates a fermion with +1 charge and d (p, s) creates a fermion with 1 charge. For instance Qe is the electric charge of electrons. s 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

48 From the Noether current J µ = ψγ µ ψ, there is a conserved charge d Q = d 3 xψ 3 p ( ) (x)ψ(x) = (2π) 3 b (p, s)b(p, s) d (p, s)d(p, s) b (p, s) creates a fermion with +1 charge and d (p, s) creates a fermion with 1 charge. For instance Qe is the electric charge of electrons. s U(1) symmetry must be related with electric charge conservation! 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

49 Maxwell equation The Maxwell equation is eq. of motion for photon field A µ (x) µ F µν = 0 or 2 A ν ν µ A µ = 0 (18) where F µν = µ A ν ν A µ The Lagrangian for photon is L Max = 1 4 F µνf µν (19) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

50 L int = ea µ ψγ µ ψ is a covariant interaction term between Dirac and Maxwell field where e is a coupling constant. Then the combination of electro and fermion Lagrangian is L = L Dirac + L Max + L int D µ = µ iea µ is a covariant derivative. = ψ(iγ µ D µ m)ψ 1 4 F µνf µν (20) Then the Dirac equation with electromagnetic interaction is [iγ µ ( µ iea µ ) m]ψ = 0 (21) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

51 Gauge invariance The most significant property of the Lagrangian (20) is that it is invariant under gauge transformation. L Max = 1 4 F µνf µν is invariant under the transformation for any scalar function Λ(x) A µ (x) A µ (x) + 1 e µλ(x) While L Dirac is invariant under only if Λ(x) = constant. ψ(x) e iλ(x) ψ(x) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

52 However the total Lagrangian with interaction term (20) L = ψ(iγ µ D µ m)ψ 1 4 F µνf µν and covariant Dirac equation (21) [iγ µ ( µ iea µ ) m]ψ = 0 are invariant under local U(1) gauge symmetry. ψ(x) e iλ(x) ψ(x) A µ (x) A µ (x) + 1 e µλ(x) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

53 The gauge boson mass term MA 2 Aµ A µ is not invariant under the gauge transformation A µ (x) A µ (x) + 1 e µλ(x). Thus, the gauge invariant field A µ should be massless. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

54 Complex Scalar Field The Lagrangian of a free complex scalar field φ = (φ 1 + iφ 2 ) 2, φ = (φ 1 iφ 2 ) 2 L = ( µ φ)( µ φ ) m 2 φ φ (22) is invariant under global gauge transformation φ e iλ φ, φ e iλ φ, where Λ is a real constant. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

55 Complex Scalar Field The Lagrangian of a free complex scalar field φ = (φ 1 + iφ 2 ) 2, φ = (φ 1 iφ 2 ) 2 L = ( µ φ)( µ φ ) m 2 φ φ (22) is invariant under global gauge transformation φ e iλ φ, φ e iλ φ, where Λ is a real constant. However, it is not invariant for local gauge Λ(x) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

56 For small Λ(x), the local gauge transformation can be written as φ φ + iλ(x)φ, µ φ µ φ + iλ(x)( µ φ) + i( µ Λ(x))φ. Then Euler-Lagrange equation leads (10) δl = J µ µ Λ(x) (23) where the conserved current is J µ = i[( µ φ )φ φ ( µ φ)]. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

57 As the case of fermions, we can add the interaction therm L int = ej µ A µ (24) between scalar field and gauge field. Then δl int = e(δj µ )A µ J µ µ Λ, (25) where for δj µ = 2 φ 2 µ Λ A µ (x) A µ (x) + 1 e µλ(x). 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

58 To cancel the extra term e(δj µ )A µ, we must add L ext = e 2 A µ A µ φ 2 The total Lagrangian L scalar = (D µ φ)(d µ φ) m 2 φ φ 1 4 F µν F µν (26) is local U(1) gauge symmetric, where D µ φ = ( µ iea µ )φ is a covariant derivative. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

59 With the gauge invariant Lagrangian (20), the total Lagrangian L = ψ(iγ µ D µ m f )ψ + (D µ φ)(d µ φ) m 2 sφ φ 1 4 F µν F µν (27) gives a complete description of the world with 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

60 With the gauge invariant Lagrangian (20), the total Lagrangian L = ψ(iγ µ D µ m f )ψ + (D µ φ)(d µ φ) m 2 sφ φ 1 4 F µν F µν (27) gives a complete description of the world with 1 a local U(1) symmetric charged scalar with mass m s, 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

61 With the gauge invariant Lagrangian (20), the total Lagrangian L = ψ(iγ µ D µ m f )ψ + (D µ φ)(d µ φ) m 2 sφ φ 1 4 F µν F µν (27) gives a complete description of the world with 1 a local U(1) symmetric charged scalar with mass m s, 2 a local U(1) symmetric charged fermion with mass m f, 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

62 With the gauge invariant Lagrangian (20), the total Lagrangian L = ψ(iγ µ D µ m f )ψ + (D µ φ)(d µ φ) m 2 sφ φ 1 4 F µν F µν (27) gives a complete description of the world with 1 a local U(1) symmetric charged scalar with mass m s, 2 a local U(1) symmetric charged fermion with mass m f, 3 a local U(1) symmetric neutral massless gauge boson(photon). 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

63 With the gauge invariant Lagrangian (20), the total Lagrangian L = ψ(iγ µ D µ m f )ψ + (D µ φ)(d µ φ) m 2 sφ φ 1 4 F µν F µν (27) gives a complete description of the world with 1 a local U(1) symmetric charged scalar with mass m s, 2 a local U(1) symmetric charged fermion with mass m f, 3 a local U(1) symmetric neutral massless gauge boson(photon). Or, in one sentence, Quantum electrodynamics (QED). 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

64 For a free (non-interacting) gauge field ( E p = p 2 + m 2 ) A µ (x)= d 3 p 1 (2π) 3 2E p 3 r=0 ɛ µ : polarization vector, r: indices of polarization. ( a r ( p)ɛ µ r e ip x + a r ( p) ɛ µ r e ip x) (28) [a r ( p), a s ( p ) ] = (2π) 3 δ rs δ (3) ( p p ), (29) is a quantization condition for a photon field, where one photon state is p a r ( p) 0 and a r ( p) 0 = 0. Photon is neutral since it is a real field 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

65 Renormalization Only a rough description of renormalization will be presented. Read the Field Theory references for the details of this subject. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

66 If a system of particles {k i } in scatters to {p f } out ({k i }, {p f } are set of initial and final momenta of particles) The matrix element M is defined as out {p f } {k i } in = (2π) (4) δ( k i p f ) im. Without going through the details, the scattering cross section is dσ M 2. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

67 M can be expanded with time-evolution k t = e iht k 0 (the Hamiltonian H). Thus in the infinite time limit, out {p f } {k i } in = lim T {p f } e ih(2t ) {k i } (30) Amplitude of scattering can be expanded with interaction terms. Interaction term of fields are proportional to coupling constant G. e.g. (e, e 2,... ) from ea µ ψγ µ ψ, e 2 A µ A µ φ 2 The leading order term of (30) is proportional to G and higher order term will be G 2, G 3, etc. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

68 M can be expanded with time-evolution k t = e iht k 0 (the Hamiltonian H). Thus in the infinite time limit, out {p f } {k i } in = lim T {p f } e ih(2t ) {k i } (30) Amplitude of scattering can be expanded with interaction terms. Interaction term of fields are proportional to coupling constant G. e.g. (e, e 2,... ) from ea µ ψγ µ ψ, e 2 A µ A µ φ 2 The leading order term of (30) is proportional to G and higher order term will be G 2, G 3, etc. This is a basic idea of perturbation expansion. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

69 Higher order term in M contains momentum integrals. And the integral diverges with p. It could be fatal problem of the field theory itself. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

70 Higher order term in M contains momentum integrals. And the integral diverges with p. It could be fatal problem of the field theory itself. The solution that the theorist found(?) is very simple. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

71 Cut it off! 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

72 Integrate momentum p to finite cut-off Λ to make the theory finite. By summing up the perturbation series and re-normalizing it, we can obtain the physical values. The final result can depend on at most log(λ). The physical parameters (couplings, masses) varies with the energy at the log scale. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

73 Renormalizability The leading order term is proportional to G. If dimension [G] = a, roughly the perturbation give G 2 Λ 2a contribution. If a < 0, the sum depends strongly on Λ and renormalization fails. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

74 Renormalizability The leading order term is proportional to G. If dimension [G] = a, roughly the perturbation give G 2 Λ 2a contribution. If a < 0, the sum depends strongly on Λ and renormalization fails. [G] 0 is a condition for renormalizable interaction. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

75 From [L] = 4 and [x] = 1, [ µ ] = [m] = 1, We obtain for scalar fields [φ] = [A µ ] = 1, and for fermion fields [ψ] = 3 2. Therefore, [e] = 0. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

76 Any term with dimension more than 4 need a coupling [G] < 0 and is not renormalizable. Only gauge invariant dimension 4 term is φ 4. The renormalizable Lagrangian with U(1) gauge symmetry is L = ψ(iγ µ D µ m f )ψ + D µ φ 2 V ( φ 2 ) F µν F µν, (31) where V ( φ 2 ) = µ 2 φ 2 + λ φ 4. (32) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

77 The minimum of potential (32) is at φ = 0 and φ 2 = µ2 2λ For λ > 0 and real mass µ, φ = φ = 0 is the absolute minimum. For µ 2 < 0, the potential has more than one minimum. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

78 If q = φ, there is two minimum in the potential. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

79 Since φ = (φ 1 + iφ 2 ) 2, V (φ 1, φ 2 ) has a shape of Mexican hat 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

80 Spontaneous Symmetry Breaking What will happen, if there are more than one ground state? 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

81 Spontaneous Symmetry Breaking What will happen, if there are more than one ground state? Like coin flipping, system can choose each ground state with equal probability. Even after the system select a specific ground state, the Lagrangian has the symmetry. However, the solution itself does not have a symmetry, anymore. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

82 Spontaneous Symmetry Breaking What will happen, if there are more than one ground state? Like coin flipping, system can choose each ground state with equal probability. Even after the system select a specific ground state, the Lagrangian has the symmetry. However, the solution itself does not have a symmetry, anymore. The symmetry is broken spontaneously. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

83 The spontaneous symmetry breaking of U(1) If the scalar field has imaginary mass, it can have continuous (Mexican hat shape) ground state. Choose a vacuum, µ 2 φ = v = 2λ and parametrize it as φ(x) = ρ(x) exp[iθ(x)] then ρ = v, θ = 0. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

84 Insert φ(x) = (v + χ(x))e iθ(x) to Lagrangian L = φ 2 µ 2 φ 2 λ φ 4, (33) = ( χ) 2 λv 4 4λv 2 χ 2 4λvχ 3 λχ 4 + (v + χ) 2 ( θ) 2. χ has a real mass and θ is massless. θ is called Nambu-Goldstone boson. (33) does not have global U(1) symmetry. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

85 Insert φ(x) = (v + χ(x))e iθ(x) to Lagrangian L = φ 2 µ 2 φ 2 λ φ 4, (33) = ( χ) 2 λv 4 4λv 2 χ 2 4λvχ 3 λχ 4 + (v + χ) 2 ( θ) 2. χ has a real mass and θ is massless. θ is called Nambu-Goldstone boson. (33) does not have global U(1) symmetry. It is broken spontaneously. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

86 Goldstone theorem Whenever a continuous symmetry is spontaneously broken, a massless (Nambu-Goldstone) boson emerge. Any degree of freedom moves along with the flat direction does not have a mass. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

87 Goldstone theorem Whenever a continuous symmetry is spontaneously broken, a massless (Nambu-Goldstone) boson emerge. Any degree of freedom moves along with the flat direction does not have a mass. No mass term m 2 φ 2 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

88 Anderson-Higgs Mechanism If we add gauge field L = 1 4 F µν F µν + D µ φ 2 µ 2 φ 2 λ φ 4, (34) = 1 4 F µν F µν + e 2 ρ 2 (B µ ) 2 + ( ρ) 2 µ 2 ρ 2 λρ 4. where B µ = A µ (1/e) µ θ and F µν = µ A ν ν A µ = µ B ν ν B µ are invariant under U(1) transformation, φ e iλ φ (θ θ + Λ), A µ A µ + 1 e µλ 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

89 If the symmetry is broken spontaneously, µ 2 φ = v = 2λ and insert ρ = v + χ to (35), L = 1 4 F µν F µν + (ev) 2 (B µ ) 2 + e 2 (2vχ + χ 2 )(B µ ) 2 +( χ) 2 4λv 2 χ 2 4λvχ 3 λχ 4 λv 4 (35) 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

90 If the symmetry is broken spontaneously, µ 2 φ = v = 2λ and insert ρ = v + χ to (35), L = 1 4 F µν F µν + (ev) 2 (B µ ) 2 + e 2 (2vχ + χ 2 )(B µ ) 2 +( χ) 2 4λv 2 χ 2 4λvχ 3 λχ 4 λv 4 (35) There is no Goldstone boson θ, while B µ gains a mass M = 2ev. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

91 In the Anderson-Higgs (also, Landau-Ginzburg-Kibble) mechanism, the massless degree of freedom is eaten by gauge field. As a consequence, the gauge field becomes massive. Since the massive photon has an extra degree of freedom in addition to two polarizations, the total degree of freedom does not change. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

92 Ferromagnetism The Hamiltonian of Ferromagnet has rotational symmetry of spin, A spin can point any direction which is global SO(3) symmetry. If spin aligns one direction, SO(3) is spontaneously broken to SO(2): a symmetry of rotation around spin direction. Since a continuous symmetry is spontaneously broken, there exists Goldstone mode called spin wave. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

93 Superconductivity When temperature went down SO(2) which is equivalent to U(1) is also broken spontaneously. At low temperature a pair of electron in superconducting material act like a boson (Higgs scalar). This is the same case as we discussed, local U(1) gauge symmetry. There is no Goldstone mode, but the photon becomes massive. Which explains why electric force becomes short-ranged and magnetic field cannot penetrate in superconducting material. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

94 The Standard Model I will close this lecture with a comment about the Standard Model. The Standard Model(SM) is a SU(2) L U(1) Y local gauge theory with 6 quarks and leptons as a basis. SU(2) L is non-abelian gauge symmetry, which is rather complicated than Abelian gauge group U(1) But the basic concept is the same. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

95 In SM, instead of U(1), SU(2) is broken by Higgs mechanism, As a result, there exist three massive gauge bosons W ±, Z. Also the quarks and leptons which are chiral field to SU(2) L and originally massless, obtain the masses. SM is extremely successful, both in theory and experiments. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

96 In SM, instead of U(1), SU(2) is broken by Higgs mechanism, As a result, there exist three massive gauge bosons W ±, Z. Also the quarks and leptons which are chiral field to SU(2) L and originally massless, obtain the masses. SM is extremely successful, both in theory and experiments. Except, we have not seen Higgs scalar, yet. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

97 God Particle? There were rumors that LHC, which started this month, can create mini-black hole which eventually destroy the earth. 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

98 God Particle? Instead of the end of the world, LHC experiment will (probably) discover Higgs scalar (or so-called god-particle). 장상현 ( 연세대학교 ) Introduction to gauge theory September 24, / 72

Physics 582, Problem Set 1 Solutions

Physics 582, Problem Set 1 Solutions TAs: Hart Goldman and Ramanjit Sohal Fall 2018 1. THE DIRAC EQUATION [20 PTS] Consider a four-component fermion Ψ(x) in 3+1D, L[ Ψ, Ψ] = Ψ(i/ m)ψ, (1.1) where we use