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Introduction to Spontaneous Symmetry Breaking Introduction to Spontaneous Symmetry Breaking Ling-Fong Li Carnegie Mellon University 2011 BCVSPIN, 25, July 2011 Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 1 / 30 Outline 1 Introduction 2 Symmetry and Conservation Law 3 Explicit vs Spontaneous Symmetry Breaking 4 Goldstone Theorem 5 Spontaneous symmetry breaking in global symmetry 6 Spontaneous symmetry breaking in Local symmetry-Higgs Phenomena 7 Standard Model of Electroweak Interaction Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 2 / 30 Introduction Fundamental Interactions in nature 1 Strong interactions–Quantum Chromodynamics(QCD) : gauge theory based on SU (3) symmetry Electromagnetic interaction 2 Electroweak interaction–gauge theory based on Weak interaction SU (2) U (1) symmetry with spontaneous symmetry breaking 3 Gravitational interaction: gauge theory of local Lorentz symmetry Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 3 / 30 Symmetry and Conservation Law Noether’s Theorem: Any continuous transformation which leaves the action S = d 4x Z L invariant, will give a conserved current µ ∂ Jµ = 0 which gives conserved charge dQ = 0, Q = d 3x J dt 0 Z Symmetry Transformation Conserved Charge time translation t t + a Energy ! space translation !x !x + !b Momentum rotation ! Angular momentum Other conserved quantities: Electric charge, Baryon number,... Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 4 / 30 Remarks Conservation laws = stability of particles e.g. Baryon number conservation) = proton is stable, Charge conservation = electron is stable, Dark Matter ? ) ) Conservation laws might change as we gain more knowledge e.g. muon number is violated only when ν oscillations were observed Parity violation was discovered only in late 50’s, CP violation ... In quantum system, symmetries = degenercies of energy levels e.g. rotational symmetry = (2l +)1) degeneracies ) Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 5 / 30 2) Symmetry Breaking Most of the symmetries in nature are approximate symmetries. (a) Explicit breaking– H = H0 + H1, H1 does not have the symmetry of H0 Example: Hydrogen atom in external magnetic …eld B! 2 ! 2 p Ze 1 ! H0 = , H1 = !µ B 2m 4πε0 r ! H0 is invariant under all rotations while H1 is invariant only for rotation along direction of B Add small non-symmetric terms to the Hamiltonian= degeneracies reduced or removed ) (b) Spontaneous breaking: Hamiltonian has the symmetry, [Q, H ] = 0 but ground state does not, Q 0 = 0 (Nambu 1960, Goldstone 1961) j i 6 Example: Ferromagnetism T > TC (Curie Temp) all magnetic dipoles are randomly oriented–rotational symm T < TC all magnetic dipoles are in the same direction-not invariant under rotation Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 6 / 30 Ginsburg-Landau theory: Write the free energy u as function of magnetization M! in the form, ! ! 2 ! ! ! ! 2 u(M ) = (∂t M ) + α1 (T ) (M M ) + α2 (M M ) , where α2 > 0, α1 = α (T TC ) α > 0 Here u has O (3) rotational symmetry. The ground state is at ! ! ! M (α1 + 2α2M M ) = 0 ! T > TC minimum at M = 0. ! α1 ! T < TC minimum at M = = 0 . If we choose M to be in some direction the 2α2 6 r rotational symmetry is broken. Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 7 / 30 Goldstone Theorem Noether theorem: continuous symmetry = conserved current, ) dQ ∂ J µ = 0, Q = d 3xJ 0 (x ) , = 0 µ dt Z Suppose A (x ) and B (x ) are some local operators and transform into each other under the symmetry charge Q, [Q, A (0)] = B (0) Suppose 0 [Q, A (0)] 0 = 0 B (0) 0 = v = 0, symmetry breaking condition h j j i h j j i 6 This implies Q 0 = 0, j i 6 and Q is a broken charge. Then En = 0, as !p n = 0, for some state n This means zero energy excitations. Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 8 / 30 To show this, write 0 [Q, A (0)] 0 = d 3x 0 J 0 (x ) , A (0) 0 h j j i Z Inserting a complete set of states and integating over x 0 [Q, A (0)] 0 h j j i 3 3 0 iEn t 0 iEn t = ∑ (2π) δ (!p n ) 0 J (0) n n A 0 e 0 A n n J (0) 0 e = υ = 0 n h j j i h j j i 6 n o iEn t RHS is time independent while LHS depends explicitly on time from e . This relation can be satis…ed only if there exists an intermediate state n for which j i En = 0, for !p n = 0 For relativistic system, energy momentum relation yields 2 2 En = !p + m mn = 0, Goldstone boson n n ) q Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 9 / 30 Remarks 1 State n with property j i 0 A (0) n = 0, h j j i 6 is the massless Goldstone boson while local …eld B corresponds to some massive particle. This state will also have the property n Q 0 = 0 h j j i 6 This means n can connect to vacuum through broken charge Q. j i 2 Spectrum Here Q is a broken symmetry charge. In general, each broken charge will have a Goldstone boson, # of Goldstone bosons= # of broken generators In many cases, there are symmetry charges which remain unbroken, Qi 0 = 0, i = 1, 2, , l j i These unbroken charges Q1, Q2, Ql form a group H, a subgroup of original symmetry group G . As a consequence, the particles will form multiplets of symm group H. For example, if symmetry SU (2) SU (2) is spontaneously broken to SU (2) , particles will form SU (2) multiplets Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 10 / 30 3 In relativistic …eld theory, massless particle = long range force. In real world , no massless particls except photon. ) In strong interaction, we have approximate spontaneous symmetry breaking. = almost Goldtone bosons ) 4 Goldstone bosons can either be elementary …elds or composite …elds a) Elementary …elds Here local operators A, B are …elds in the Lagrangian–this is the case in Standard electroweak theory b) Composite …elds In this case A, B correspond to some bound states of elementary …elds. For example in QCD the ‡avor chiral symmetry is broken by quark condensates. Here we have 5 _ _ Q , qγ5q = qq 5 Q some_ chiral charge, q quark …eld _ If 0 qq 0 = 0 ( quark condensate), then qγ5q will correspond to a Goldstone boson. This is usually6 referred to as dynamical symmetry breaking. Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 11 / 30 Global symmetry We now discuss the spontaneou symmetry in some simple relativistic …eld theory. As an example, consider 1 = [(∂ σ)2 + (∂ π)2 ] V (σ2 + π2 ) L 2 µ µ with µ2 λ V (σ2 + π2 ) = (σ2 + π2 ) + (σ2 + π2 )2 (1) 2 4 This potential has O (2) symmetry σ σ cos α sin α σ 0 = π ! π0 sin α cos α π where α is a constant, global symm Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 12 / 30 To get the minimum we solve the equations ∂V = µ2 + λ(σ2 + π2 ) σ = 0 ∂σ ∂V = µ2 + λ(σ2 + π2 ) π = 0 ∂π Solution µ2 σ2 + π2 = = ν2 circle in σ π plane λ Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 13 / 30 Choose σ = ν, π = 0 which are …elds con…gurations for classical vacuum. New quantum …elds σ0 = σ ν , π0 = π correspond to oscillations around the minimum. Note that σ = v = 0 h i 6 is the symmetry breaking condition in Goldstone’s theorem. The new Lagrangian is 1 2 2 2 2 2 2 λ 2 2 2 = [(∂ σ0 + (∂ π) ] µ σ0 λνσ0(σ0 + π0 ) (σ0 + π0 ) L 2 µ µ 4 There is no π 2 term, π massless Goldstone boson 0 ) 0 Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 14 / 30 Local symmetry: gauge symmetry Maxwell Equations: ρ ! !E = , ! B! = 0 r ε0 r ! ! ! ! ∂B 1 ! ! ∂E ! E + = 0, B = ε0 + J r ∂t µ0 r ∂t Source free equations can be solved by introducing scalar and vector potentials, φ, !A, ∂!A B! = ! !A, !E = !φ r r ∂t These solutions are not unique because of gauge invariance, ∂α φ φ , !A !A + !α ! ∂t ! r Or A A ∂ α (x ) µ ! µ µ Classically, it is not so clear what the physical role is played by gauge invariance. Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 15 / 30 In quantum mechanics, Schrodinger eq for charged particle is, 1 h 2 ∂ψ ! e !A eφ ψ = i h 2m i r ∂t " # This requires transformation of wave function, e ψ exp i α (x ) ψ ! h to get same physics. Thus gauge invariance is now connected to symmetry (local) transformation. Ling-Fong Li (Carnegie Mellon University) Introduction to Spontaneous Symmetry Breaking2011 BCVSPIN, 25, July 2011 16 / 30 Spontaneous symmetry breaking in local symmetry Local symmetry : symm parameter depends on space-time. Consider a …eld theory with U (1) local symmetry, 2 1 L = D φ † (D µφ) + µ2 φ†φ λ φ†φ F F µν µ 4 µν where D φ = ∂ igA φ, F = ∂ A ∂ A µ µ µ µν µ ν ν µ This local U (1) phase transformation is of the form, iα φ (x ) φ0 (x ) = e φ (x ) , α = α (x ) ! The derivative transforms as iα ∂ φ (x ) ∂ φ0 (x ) = e ∂ φ (x ) i∂ α µ ! µ µ µ which is not a phase U (1) transformation.
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