What is electron? In Gee Kim GIFT, POSTECH December 5, 2013 Contents History and Discovery Fundamental properties Free relativistic quantum fields Free field quantizations Electromagnetic interaction Renomalization Renormalization Group and Higgs mechanism Summary History 1600 De Magnete (William Gilbert): new L. electricus, L. < electrum, Gk. < η¯λεκτ ρoν, amber. 1838 Richard Laming: P Atom = core matter + surrounding (unit electric charge). P P 1846 William Weber: Electricity = fluid(+) + fluid(−) . 1881 Hermann von Helmholtz: “behaves like atoms of electricity.” 1891 George Johnstone Stoney: electron = electr(ic) + (i)on. Discovery Crookes tube: 1869 Johann Wilhelm Hittorf: A glow emitted from the cathode. 1876 Eugen Goldstein: Cathode rays. 1870 Sir William Crookes: The luminescence rays comes from the cathod rays which I I I carried energy, moved from cathod to anode, and bent in magnetic field as negative charged. 1890 Arthur Schuster: The charge-to-mass ratio, e/m Discovery 1892 Hendrik Antoon Lorentz: mass ⇐ electric charge. 1896 J. J. Thomson with John S. Townsend and H. A. Wilson: e/m was independent of cathode material. 1896 George F. Fitzgerald: The universality of e/m and again proposed the name electron. 1896 Henri Becquerel: Radioactivity. 1896 Ernest Rutherford designated the radioactive particles, alpha (α) and beta (β). 1900 Becquerel: The β-rays have the same e/m as electrons. 1909 Robert Millikan and Harvey Fletcher: The oil-drop experiments (published in 1911). 1913 Abram Ioffe confirmed the Millikan’s experiments. Fundamental properties Mass: m = 9.109 × 10−31 kg = 0.511 MeV/c2 , where c = 2.998 × 108 m/s. Charge: e = −1.602 × 10−19 C Spin: Intrinsic spin angular momentum with I I S 2 = s(s + 1)~2 , the square of the spin magnitude, h = 1.0546 × 10−34 Js. where s = ± 21 and ~ = 2π µ = −gµB s, the spin magnetic moment, e~ where µB = 2mc = 0.927 × 10−20 emu and g is the Lande´ g-factor, for free-electron g = 2.0023. Size: A point particle, no larger than 10−22 m, I I α~ re = mc = 2.818 × 10−15 m, the classical electron radius, e2 1 where α = 4π~c = 137.04 = 0.00730, the fine structure constant. ~ λC = mc = 3.862 × 10−13 m, the electron Compton wavelength. Free relativistic quantum fields A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data. – P. A. M. Dirac ¨ The Schrodinger equation The time development of a physical system is expressed by the ¨ Schrodinger equation i~ ∂ψ = Hψ ∂t where the Hamiltonian H is a linear Hermitian operator. For an isolated free particle, the Hamiltonian is H= p2 2m and the quantum mechanical transcriptions are H → i~ ∂ , ∂t p→ ~ ∇ i leads to a relativistically incorrect equation i~ ∂ψ ~2 2 =− ∇ ψ. ∂t 2m Systems of units It is convenient to introduce the natural unit system for describing relativistic theories. I The natural unit system is defined by the constants c = ~ = 1. I In this system, −1 [length] = [time] = [energy] I I −1 = [mass] . The mass of a particle is equal to the rest energy (mc2 ) and to its inverse Compton wavelength (mc/~). The thermal unit system is the same as the natural unit system with the additional Boltzmann constant kB = 1. I I In this system, [energy] = [temperature]. Especially, 1 eV = 11605 K. I The atomic Hartree unit system is defined by the constants ~ = e2 = m = 1, but c = α−1 . I The atomic Rydberg unit system is the same as the atomic Hartree unit system, but 2e2 = 1. Special theory of relativity Einstein concluded that the Maxwell’s equations are correct. c, c0 u → S0 λ S So every physical law has to satisfy the condition c = c0 . The corresponding space-time transformation group is called I I Homogeneous Lorentz group if λ = 0, Poincare´ group or inhomogeneous Lorentz group if λ 6= 0. Relativistic notions I I I I I I I x is the four-vector of space and time. xµ (µ = 0, 1, 2, 3) are the contravariant components of this vector. xµ are the covariant components effected by the Minkowski metric tensor,   1 0 0 0  0 −1 0 0  . gµν = g µν =   0 0 −1 0  0 0 0 −1 P xµ = x0 , x and xµ = gµν xν = 3ν=0 gµν xν = x0 , −x . The scalar product is defined by x · x ≡ xµ xµ = t2 − x2 . The equation for the lightcone: x2 ≡ xµ xµ = 0. The displacement vector is naturally raised, xµ , while the derivative operator is naturally lowered ∂ ∂ = ,∇ . ∂µ = ∂xµ ∂x0 Lightcone x0 future timelike elsewhere xi elsewhere spacelike past Relativistic quantum mechanics I Momentum vectors are similarly defined pµ = (E, px , py , pz ) I and the scalar product is defined by p · p = pµ pµ = E 2 − p · p = m2 . I Likewise p · x = pµ xµ = Et − p · x. I The quantum mechanical transcriptions will be (~ = 1) ∂ , ∂x0 pµ = i∂ µ . E=i or I p = −i∇ For a relativistic free p particle, we may try a relativistic Hamiltonian H = p2 + m2 (c = 1) to obtain i ∂ψ p = −∇2 + m2 ψ ∂t (?) The causality violation The amplitude for a free particle to propagate from x0 to x: U (t) = hx| e−iHt |x0 i √ 2 2 = hx| e−it p +m |x0 i Z √ 1 3 it p2 +m2 ip·(x−x0 ) = e d p pe (2π)3 Z ∞ √ 1 −it p2 +m2 . = dp p sin (p |x − x |) e 0 2π 2 |x − x0 | 0 At the point x2 p t2 (well outside the lightcone), the phase function px − t p2 + m2 has a stationary point at p = i √xmx . 2 −t2 We will have the propagation amplitude as U (t) ∼ e−m √ x2 −t2 , which is small but nonzero outside the lightcone. Causality is violated! The action principle The action S in local field theory is defined by the time integral of the Lagrangian density L of the set of the components of the field φr (x) and their derivatives ∂µ φr (x): Z σ L (φr , ∂µ φr ) d4 x, S= σ0 where a general spacelike plane σ at an instance τ is characterized by an equation of plane σ: n · x + τ = 0, n2 = +1, where nµ is a unit timelike normal vector. The variation of the action is Z σ ∂L ∂L − ∂µ δS = δ0 φr d4 x + F (σ) − F (σ0 ), ∂φr ∂ (∂µ φr ) σ0 Z ∂L ∂L ν µν F (σ) = ∂ φr − g L δxν − δφr dσµ . ∂ (∂µ φr ) ∂ (∂µ φr ) σ The equation of motion If we choose a variation which vanishes at the boundary planes σ0 and σ, the observables at the boundary are unchanged for the total variation δφr = δ0 φr + ∂µ φr δxµ , F (σ) = F (σ0 ) = 0 Z σ ∂L ∂L − ∂µ δ0 φr d4 x = 0. δS = ∂φ ∂ (∂ φ ) r µ r σ0 This equation is satisfied if the integrand vanishes at every point: ∂L ∂L − ∂µ = 0. ∂φr ∂ (∂µ φr ) These are the field equations. Note that we may write for the general variation of S simply δS = F (σ) − F (σ0 ). Lorentz transformations The components of a four-vector referred to two different inertial systems with the same origin are related by a homogeneous proper Lorentz transformation, which is defined as the real linear transformation which leaves x2 = x0 2 = 0 invariant, µ x0 = Λµ ν xν ; ν Λµ ν Λλ = g µλ , and which, in addition, satisfies Λµ ν real, det (Λµ ν ) > 0, Λ0 0 > 0. The inhomogeneous Lorentz transformation involves displacements, such that x0 = Lx: L: µ x0 = Λµ ν xν + λµ , where λµ is a four-vector independent of x. The field component φr (x) transforms according to the proper Lorentz transformation: φr 0 (Lx) = Sr s φs (x). Lorentz group For the Lorentz transformation of the field φr 0 (x) = U (L)φr (x)U −1 (L), we observe that the operators U (L) form a representation of the Lorentz group: U (L2 L1 ) = U (L2 )U (L1 ). The infinitesimal Lorentz transformations are defined by Λµ ν = gµ ν + αµ ν , λµ = αµ , where αµ ν and αµ are infinitesimals of first order. The relation ν Λµ ν Λλ = g µλ then leads to αµν + ανµ = 0. Poincare´ group The infinitesimal part of the transformation U may be written explicitly U = 1 + iK, where the generator K is written as a linear function of the α’s: 1 K = M µν αµν + P µ αµ , 2 M µν = −M νµ . Lie’s theorem asserts that such operators Xr satisfy X [Xr , Xs ] = crs t Xt , t where the coefficients crs t are called the structure constants of the group. This relation takes the form [P µ , P ν ] = 0, i −i M µν , P λ = g µν P ν − g νλ P µ , h −i [M µν , M ρσ ] = g µρ M νσ − g µσ M νρ + g νσ M µρ − g νρ M µσ . Poincare´ group The transformed field components φr 0 under the opertator U may be written φr 0 = U φr U −1 = (1 + iK) φr (1 − iK) ' φr + i [K, φr ] . So we have the increment of the field compoents after the transformation δφr = i [K, φr ] . The infinitesimal part of the transformation matrix Sr s : Sr s = δr s + Σr s , 1 Σr s = Σr sµν aµν , 2 where the coefficients Σr sµν = −Σr sνµ . The increment of φr becomes δφr (x) = 1 [Σr sµν φs (x) + (xµ ∂ ν − xν ∂ µ ) φr (x)] αµν −∂ µ φr (x)αµ . 2 Momentum operators We obtain the defining relations for the momentum operators: i [M µν , φr (x)] = Σr sµν φs (x) + (xµ ∂ ν − xν ∂ µ ) φr (x), i [P µ , φr (x)] = −∂ µ φr (x). Under an infinitesimal Lorentz transfromation the plane σ suffers a displacement: δxµ = αµ ν xν + αν , and the field at the displaced point x + δx is φr + δφr , with 1 δφr (x) = Σr sµν φs (x)αµν . 2 The generating operator F (σ) yields Z 1 rµ sνρ µν ρ F (σ) = T (ανρ x + αν ) − π Σr φs ανρ dσµ , 2 σ where π rµ = ∂L , ∂ (∂µ φr ) T µν = π rµ ∂ ν φr − g µν L. Momentum operators We write F (σ) in the form F (σ) = M µν 1 µν M αµν + P µ αµ , 2 Z (T ρµ xν − T ρν xµ − π rρ Σr sµν φs ) dσρ = Pµ = Zσ T ρµ dσρ . σ ¶The operator F (σ) is the generating operator for the variation of the field at a point on the boundary σ: This variation is 1 [Σr sµν φs + (xµ ∂ ν − xν ∂ µ ) φr ] αµν − ∂ µ φr αµ 2 = i [F (σ), φr ] , δ0 φr = which must hold for arbitrary values of the ten parameters αµν and αµ . Momentum operators So we obtain the set of equations i [M µν , φr ] = Σr sµν φs + (xµ ∂ ν − xν ∂ µ ) φr , i [P µ , φr ] = −∂ µ φr . The tensor T µν is called the canonical momentum tensor, while the angular momentum tensor M µν may be split into two parts, defined by M µν Lµν N µν = Lµν + N µν (total angular momentum), Z = (T ρµ xν − T ρν xµ ) dσρ (orbital angular momentum), σZ = − π rρ Σr sµν φs dσρ (spin angular momentum). σ Conservation laws For any set of functions f µ (x) which vanish sufficiently fast in spacelike directions Z σ Z Z ∂µ f µ d4 x = − f µ dσµ + f µ dσµ = 0, σ0 σ σ0 if the conservation law holds, so that we have ∂µ f µ = 0. Applying this results to the integrands of M µν and P µ , we obtain T µν − T νµ + ∂ρ H ρµν = 0, µν = 0, ∂µ T with H ρµν = π rρ Σr sµν φs = −H ρνµ . Conservation laws Defining the symmetrical momentum tensor Θµν = T µν + ∂ρ Gρµν , where 1 (Hρµν + Hµνρ + Hνµρ ) , 2 we obtain the following tensor properties Gρµν = Θµν Pν = Θνµ , Z = Θµν dσµ , σ ∂ µ Θµν = 0. Commutation rules The generating operator is given by Z F (σ) = − π rµ δφr dσµ , σ and the arbitrary variation of the field components are Z sµ δφr (x) = i φr (x), π (y)δφs (y)dσµ , for x ∈ σ. σ For any three operators A, B, and C, the Jacobi identity is [A, BC] = [A, B] C + B [A, C] = {A, B} C − B {A, C} . So that for the three operators φr (x), π sµ (y), and δφs (y), we have two possibilities (a) [φr (x), δφs (y)] = 0, (b) {φr (x), δφs (y)} = 0, [φr (x), π sµ (y)] = −δr s δ µ (x, y), {φr (x), π sµ (y)} = −δr s δ µ (x, y). Pauli’s principle Wolfgang Pauli (1936, 1940) suggested the new principles that 1. The total energy of the system must be a positive definite operator such that the vacuum state is the state of the lowest energy. 2. Observerbles at two points with space-like separation must commute with each other. The quantization of the fields I with half-integer spin according to case (a) would violate principle 1, → known as the exclusion principle, I while with integer spin according to case (b) would violate principle 2. Free field quantizations We have to remember that what we oberve is not nature herself, but nature exposed to our method of questioning. – Werner Heisenberg. The Klein-Gordon attempts Considering the Lagrangian of a scalar field L = = 1 1 ˙2 1 φ − (∇φ)2 − m2 φ2 2 2 2 1 2 2 1 2 (∂µ φ) − m φ , 2 2 we obtain the Klein-Gordon equation 2 ∂ 2 2 − ∇ + m φ = 0, ∂ µ ∂µ + m2 φ = 0, or + m2 φ = 0. 2 ∂t ˙ Noting that the conjugate to φ(x) is π(x) = φ(x), we can construct the Hamiltonian: Z Z 1 2 1 1 2 2 2 3 3 H = d xH = d x π + (∇φ) + m φ . 2 2 2 The Klein-Gordon attempts In the momentum space representation, the Klein-Gordon field is expanded as Z d3 p ip·x φ (x, t) = e φ (p, t) (2π)3 so that the Klein-Gordon equation will be 2 2 ∂ ∂ 2 2 2 + |p| + m + ωp φ (p, t) = 0, φ (p, t) = 0 or ∂t2 ∂t2 which is a simple harmonic oscillator (SHO) equation which can be easily solved by introducing the annihilation and creation operators such that h i ap , ap0 † = (2π)3 δ (3) p − p0 . The Klein-Gordon attempts We will expand the field φ (x) and π (x) in terms of the annihilation and creation operators as Z d3 p 1 † p φ (x) = a + a eip·x ; p −p (2π)3 2ωp r Z ωp d3 p † (i) π (x) = a − a eip·x . p −p 3 2 (2π) These expansions yield the field commutator relation Z 3 3 0 r d pd p −i ωp0 0 φ (x) , π x = ωp (2π)6 2 h i h i 0 0 × a−p † , ap0 − ap , ap0 † ei(p·x+p ·x ) = iδ (3) x − x0 . The Klein-Gordon attempts Then the Hamiltonian will be   Z h i  † d3 p ap ap + 1 ap , ap †  . H= ω p  (2π)3  } |2 {z =0 The total momentum operator is written as Z Z d3 p 3 P = − d xπ (x) ∇φ (x) = pap † ap . (2π)3 ⇒ The operator ap † creates momenum p and energy p ωp = |p| + m2 . ⇒ The state ap † aq † · · · |0i has momentum p + q + · · · . ⇒ We call these excitations particles. q ⇒ We will refer to ωp as Ep = + |p|2 + m2 , since it is the positive energy of the particle. The Klein-Gordon attempts The one-particle state |pi ∝ ap † |0i is normalized with the Lorentz invariance with a boost p0 i = γ (pi + pβE) and 0 E = γ (E + βpi ), where β = vi /c and γ = 1 − β 2 : dp0 i dpi dE 0 0 p −q γ 1+β dpi γ p0 − q 0 (E + βpi ) E E0 p0 − q 0 . E δ (3) (p − q) = δ (3) p0 − q0 = δ (3) = δ (3) = δ (3) We define |pi ≡ p 2Ep ap † |0i → hp|qi = 2Ep (2π)3 δ (3) (p − q) . Time-evolution of the Klein-Gordon fields The Heisenberg picture of the fields φ(x) = φ (x, t) = eiHt φ (x) e−iHt π(x) = π (x, t) = eiHt π (x) e−iHt exhibit the time-evolution by the Heisenberg equation of motion ∂ i ∂t O = [O, H]. The time dependences of the annihilation and creation operators are iHt aH ap e−iHt = ap e−iEp t , p ≡e † iHt aH ap † e−iHt = ap † eiEp t . p ≡e † H † Omitting the superscript H, aH p → ap and ap → ap , the fields are expanded by the operators: Z 1 d3 p −ip·x † ip·x p a e ; + a e φ (x, t) = 0 p p p =Ep (2π)3 2Ep ∂ π (x, t) = φ (x, t) : ∂t the explicit description of the particle-wave duality. The Dirac field I The Klein-Gordon Lagrangian 1 1 1 L = φ˙ 2 − (∇φ)2 − m2 φ2 2 2 2 I I I I has resolved the relativistic inconsistency of the ¨ Schrodinger equation. However, the quantization a, a† = 1 ; electron. Dirac (1928) suggested another Lagrangian: L = ψ¯ i∂/ − m ψ = ψ¯ (iγ µ ∂µ − m) ψ. (ψ¯ ≡ ψ † γ 0 ) The canonical momentum conjugate to ψ is iψ † , and thus the Hamiltonian is Z Z 3 ¯ H = d xψ (−iγ · ∇ + m) ψ = d3 xψ † −iγ 0 γ · ∇ + mγ 0 ψ Z α ≡ γ 0 γ, β ≡ γ 0 = d3 xψ¯ [−iα · ∇ + mβ] ψ. Dirac matrices The Dirac matrices follows the algebra {γ µ , γ ν } ≡ γ µ γ ν + γ ν γ µ = 2g µν × 14×4 → S µν = i µ ν [γ , γ ] . 4 Define i γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 = − µνρσ γµ γν γρ γσ . 4 There are 5 standard classes of the γ-matrices σ µν 1 γµ = 2i [γ µ , γ ν ] γµγ5 γ5 Explicitly, we have 0 1 0 ; γ = 1 0 i γ = scalar vector tensor pseudo-vector pseudo-scalar 0 σi −σ i 0 ; 5 1 4 6 4 1 16 γ = −1 0 0 1 . Dirac spinor Pauli spin matrices are defined by the Dirac algebra i j γ j ≡ iσ j ⇒ γ , γ = −2δ ij . The Lorentz algebra are then 1 S ij = ijk σ k , 2 the two-dimensional representation of the rotation group. The boost and rotation generators are i i 0 i i σ 0 S 0i = γ ,γ = − , 0 −σ i 4 2 1 ijk σ k 0 1 i i j ij γ ,γ = − ≡ ijk Σk , S = k 0 σ 4 2 2 which transform the four-component field ψ, a Dirac spinor. Dirac equation The action principle yields the Dirac equation (iγ µ ∂µ − m) ψ(x) = 0. This implies the Klein-Gordon equation shown by acting (−iγ µ ∂µ − m) on the left (−iγ µ ∂µ − m) (iγ ν ∂ν − m) ψ = γ µ γ ν ∂µ ∂ν + m2 ψ 1 µ ν 2 {γ , γ } ∂µ ∂ν + m ψ = 2 = ∂ 2 + m2 ψ = 0. Since the canonical momentum conjugate to ψ is iψ † , the Hermitian conjugate form of the Dirac equation is ¯ µ − mψ¯ = 0. −i∂µ ψγ Solutions of the Dirac equation Since a Dirac field ψ obeys the Klein-Gordon equation, we can expand it as linear combinations of plane waves: ψ(x) = u(p)e−ip·x , ψ(x) = v(p)e+ip·x . Plugging them into the Dirac equation, we obtain (γ µ pµ − m) u(p) = p p2 = m2 , / − m u(p) = 0, (γ µ pµ + m) v(p) = p p2 = m2 , / + m v(p) = 0, p0 > 0, p0 > 0. In the rest frame, with σ µ ≡ (1, σ) and σ ¯ µ = (1, −σ), the column vectors u(p) and v(p) are in the form √ √ p · σξ s p · ση s s s √ , v (p) = , s = 1, 2, u (p) = √ p·σ ¯ξs − p·σ ¯ ηs where ξ s and η s are the bases of the two-component spinors. Spin sums The solutions are normalized accoring to u ¯r (p)us (p) = +2mδ rs , us† (p)us (p) = +2Ep δ rs , v¯r (p)v s (p) = −2mδ rs , v s† (p)v s (p) = +2Ep δ rs . The u’s and v’s are orthogonal to each other: u ¯r (p)v s (p) = v¯r (p)us (p) = 0, but ur† (p) v s (p) = v r† (−p) us (p) = 0. Then the completeness relations are X us (p)¯ us (p) = γ · p + m = γ µ pµ + m = p / + m, s X s v s (p)¯ v s (p) = γ · p − m = γ µ pµ − m = p / − m. The quantized Dirac field The Dirac field operators are expanded by plane waves Z d3 p 1 X s s −ip·x s† s ip·x p ψ(x) = a u (p)e + b v (p)e ; p p (2π)3 2Ep s Z d3 p 1 X s s −ip·x s† s ip·x ¯ p b v ¯ (p)e + a u ¯ (p)e ψ(x) = , p p (2π)3 2Ep s where the creation and annihilation operators obey the anticommutation relations n o n o arp , asq † = brp , bsq † = (2π)3 δ (3) (p − q) δ rs . The equal-time anticommutation relations for ψ and ψ † are then n o ψa (x) , ψb † (y) = δ (3) (x − y) δab ; n o {ψa (x) , ψb (y)} = ψa † (x) , ψb † (y) = 0. Physical meaning of the Dirac field The vacuum |0i is defined to be the state such that asp |0i = bsp |0i = 0. The Hamiltonian, with dropping the infinities, are written Z d3 p X s† s s† s E a a + b b H= p p p p p . (2π)3 s The momentum operator is Z Z 3 † P = d xψ (−i∇) ψ = d3 p X s † s s† s p a a + b b p p p p . (2π 3 ) s Thus both asp † and bsp † create particles with energy +Ep and p momentum p. The one-particle states |p, si ≡ 2Ep asp † |0i is defined so that hp, r|q, si = 2Ep (2π)3 δ (3) (p − q) δ rs is Lorentz invariant. Conservations of the Dirac field The Dirac field transforms according to ψ(x) → ψ 0 (x) = Λ 1 ψ Λ−1 x . 2 The change in the field at a fixed point is (Λ 1 ' i − 2i ωµν S µν = 1 − 2i θΣ3 , i.e. infinitesimal rotation angle θ 2 about z-axis) δψ = ψ 0 (x) − ψ(x) = Λ 1 ψ Λ−1 x − ψ(x) 2 i 3 = 1 − θΣ ψ (t, x + θy, y − θx, z) − ψ(x) 2 i 3 = −θ x∂y + y∂x + Σ ψ(x) ≡ θ∆ψ. 2 The conserved Noether currents are i ∂L 0 3 0 ¯ j = ∆ψ = −iψγ x∂y − y∂x + Σ ψ, ∂ (∂0 ψ) 2 Z 1 J = d3 xψ † x × (−i∇) + Σ ψ. 2 Spin- 12 Dirac field At t = 0, for simplicity, Z Z 3 3 0 d pd p 1 0 3 p Jz = d x e−ip ·x eip·x 6 2Ep 2Ep0 (2π) X 0† 0 0 r0 † 0 r0 r r −p × ap0 u p + b−p0 v r,r0 × Σ3 r r ap u (p) + br−p † v r (−p) . 2 The commutator rules for as0 † yields X σ3 1 X s† Σ3 r r† s† u (0) u (0) a0 |0i = ξ s† ξ r ar0 |0i ; Jz a0 |0i = 2m r 2 2 r the eigenvalues of Jz are ± 12 . ⇒ The Dirac field conveys spin- 21 . Conserved quantities of the Dirac field I µ ψ(x) is conserved by the Dirac ¯ A current j µ (x) = ψ(x)γ equation ¯ µ ∂µ ψ ∂µ j µ = ∂µ ψ¯ γ µ ψ + ψγ = imψ¯ ψ + ψ¯ (−imψ) = 0. I The charge associated with this current is Z d3 p X s † s s† s ap ap − bp bp Q= (2π)3 s I is conserved: there is a unit charge e. µ γ 5 ψ(x) is ¯ An axial vector current j µ5 (x) = ψ(x)γ conserved ¯ 5 ψ. ∂µ j µ5 = 2imψγ if m = 0. Discrete symmetries of the Dirac field Let C the charge conjugation, P the parity, T the time reversal operators. Use the shorthand (−1)µ ≡ 1 for µ = 0 and (−1)µ ≡ −1 for µ = 1, 2, 3. P T C CP T ¯ ψψ ¯ 5ψ iψγ ¯ µψ ψγ ¯ µγ5ψ ψγ ¯ µν ψ ψσ ∂µ +1 +1 +1 +1 −1 −1 +1 +1 (−1)µ (−1)µ −1 −1 −(−1)µ (−1)µ +1 −1 (−1)µ (−1)ν −(−1)µ (−1)ν −1 +1 (−1)µ −(−1)µ +1 −1 I The free Dirac Lagrangian L0 = ψ¯ (iγ µ ∂µ − m) ψ is invariant under C, P , and T separately. I The perturbation δL must be a Lorentz scalar. All Lorentz scalar combinations of ψ¯ and ψ are invariant I under the combined symmetry CP T . Propagators and causality The amplitude for a scalar Klein-Gordon particle to propagate from y to x is h0| φ(x)φ(y) |0i: Z d3 p 1 −ip·(x−y) ? = h0| φ(x)φ(y) |0i = D0 (x − y) = e . y x (2π)3 2Ep H I When x − y is purely in the time direction, (x − y)2 > 0; x0 − y 0 = t, x − y = 0: Z ∞ √ 4π p2 0 −i p2 +m2 t p D (x − y) = dp e (2π)3 0 2 p2 + m2 Z ∞ p 1 = dE E 2 − m2 e−iEt 2 4π m ∼ e−imt . t→∞ Propagators and causality I When x − y is purely spatial direction, (x − y)2 < 0; x0 − y 0 = 0, x − y = r: Z d3 p 1 ip·r 0 D (x − y) = e (2π)3 2Ep Z ∞ 2π p2 eipr − e−ipr = dp 2Ep ipr (2π)3 0 Z ∞ ipr −i pe = dp p 2 2 (2π) r −∞ p2 + m2 −mr ∼ e . r→∞ ⇒ Causality is still violated so we need to a correct form of the amplitude vanishing for (x − y)2 < 0. ⇒ Since φ(x) is a quantum field, let us consider a commutator [φ(x), φ(y)]. Propagators and causality I The amplitude for the commutator h0| [φ(x), φ(y)] |0i = h0| φ(x)φ(y) |0i − h0| φ(y)φ(x) |0i = D0 (x − y) − D0 (y − x) Z d3 p 1 −ip·(x−y) ip·(x−y) e − e = (2π)3 2Ep I preserves causality under the Lorentz transformation by taking (x − y) → −(x − y) on the second term to cancel each other. The amplitude integral can convey the frequency integral through the residue theorem: Z I 0 0 0 0 dp0 e−ip0 (x −y ) dp0 e−ip0 (x −y ) = ℘ 2π p2 − m2 2π p0 2 − Ep 2 = −2πi Res|p0 =+Ep + Res|p0 =−Ep 1 −iEp (x0 −y0 ) 0 0 = −i e − e+iEp (x −y ) . 2Ep Propagators and causality The amplitude for x0 > y 0 is then Z Z d3 p dp0 −1 h0| [φ(x), φ(y)] |0i = e−ip·(x−y) . 3 2 2 0 0 2πi p − m x >y (2π) Let us define a function DR (x − y) ≡ θ x0 − y 0 h0| [φ(x), φ(y)] |0i , which satisfies the Klein-Gordon Green’s function equation ∂ 2 + m2 DR (x − y) = −iδ (4) (x − y), ˜ R (p) = −i. or −p2 + m2 D which is known as the retarded Green’s function, explicitly Z i d4 p −ip·(x−y) . DR (x − y) = 4 p2 − m2 e (2π) For x0 < y 0 , we have the advanced Green’s function DA (x − y) = θ y 0 − x0 h0| [φ(x), φ(y)] |0i = −DR (x − y). Propagators and causality Let us define the Klein-Gordon Feynman propagator as DF0 (x − y) ≡ h0| Tˆ {φ(x)φ(y)} |0i = DR (x − y) + DA (y − x) = θ (t) h0| φ(x)φ(y) |0i + θ (−t) h0| φ(y)φ(x) |0i (t≡x0 −y 0 ) Z = i d4 p −ip·(x−y) . 4 p2 − m2 + i e (2π) where Tˆ {· · · } is the time-ordering operator, and the integrand of the last line has the poles p0 = ± (Ep − i), for → 0+ . Similiarly, we can also define the Dirac Feynman propagator as ¯ SF0 (x − y) ≡ h0| Tˆ ψ(x)ψ(y) |0i Z 4 d p i p /+m e−ip·(x−y) . = (2π)4 p2 − m2 + i Electromagnetic interaction To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature . . . –Richard P. Feynman. Electromagnetic interaction I I I I We have understood the spin and dynamics of electron as a free Dirac field. However, a free particle is not measurable so we need interaction to really observe it. An electron is subjected to the electromagnetic interaction with the Lagrangian such that LQED = LDirac + LMaxwell + Lint 1 ¯ µ ψAµ , = ψ¯ i∂/ − m0 ψ + F µν Fµν − e0 ψγ 4 where Aµ is the electromagnetic vector potential, Fµν = ∂µ Aν − ∂ν Aµ is the skew-symmetrical electromagnetic field tensor, and e0 < 0 is the electron charge. Introducing Dµ ≡ ∂µ + ie0 Aµ we have a simpler form 1 / − m0 ψ − F µν Fµν . LQED = ψ¯ iD 4 Electromagnetic interaction I The QED Lagrangian is invariant under the gauge transformations ψ(x) → eiα(x) ψ(x), I I Aµ (x) → Aµ (x) + ∂µ α(x). The equation of motion for ψ is / − m0 ψ(x) = 0, iD which is jus the Dirac equation coupled to the electromagnetic field. The equation of motion for Aν is ¯ ν ψ = e0 j ν , ∂µ F µν = e0 ψγ I which is the inhomogeneous Maxwell equations, with the ¯ ν ψ. current density j ν = ψγ The quantization of Aµ fields are depending on the choice of gauges, such as the Coulomb gauge ∇ · A = 0 or the Lorentz gauge ∂µ Aµ = 0. Maxwell field In the relativistic notations, the maxwell field is defined by F µν ⇒ = ∂ µ Aν − ∂ ν Aν , j µ = (ρ, j) ∂A , B=∇×A E = −∇A0 − ∂t and written as  0 −E 1 −E 2 −E 3  E1 0 −B 3 B2  . = 3  E2 B 0 −B 1  E 3 −B 2 B1 0  F µν = −F νµ The four-vector potential Aµ does not determined uniquely for a gauge transformation Aµ (x) → Aµ (x) + ∂ µ α(x), but it yields the Lorentz invariant Maxwell equation Aµ − ∂ µ (∂ · A) = j µ . Radiation field I We modify the Lagrangian LMaxwell to 1 1 LMaxwell = − F µν Fµν − ξ (∂ · A)2 4 2 so that the Maxwell’s equation are replace by Aµ − (1 − ξ)∂µ (∂ · A) = 0 and the conjugate momenta π µ to Aµ are πρ = I ∂LMaxwell = F µ0 − ξg µ0 (∂ · A) , ∂ (∂0 Aµ ) where (∂ · A) is a scalar field such that (∂ · A) = 0. For radiation field, we conveniently choose ξ = 1 (Feynman gauge) to yield the Maxwell equation Aµ = 0. Dixitque Deus, et facta est lux. – Genesis 1:3 Radiation field The solutions of Aµ = 0 are the plane waves: Z d3 k 1 √ Aµ (x) = 3 (2π) 2Ek 3 h i X −ik·x (λ) † (λ) ∗ +ik·x × a(λ) (k)ε(λ) (k)e + a (k)ε (k)e , µ µ λ=0 where ε(λ) are the bases of polarization vectors, which satisfies ∗ (λ) X ε(λ) (k) µ (k)εν = gµν , ∗ (λ) (λ) ε (k) · ε (k) λ 0 ∗ 0 ε(λ) (k) · ε(λ ) (k) = g λλ . Real photons convey only the transverse polarizations ˆ, the right- and left-handed εµ = (0, ε), where k · ε = 0. For k k z polarization vectors are 1 εµ = √ (0, 1, ±i, 0) . 2 Photon: quantized radiation field The equal-time commutation rules for the radiation field are h i [Aµ (x), Aν (y)] = A˙ µ (x), A˙ ν (y) = 0, h i A˙ µ (x), Aν (y) = igµν δ (3) (x − y). We can define the photon Feynman propagator as igµν ∆0F (x − y) h0| Tˆ [Aµ (x)Aν (y)] |0i Z d4 k −igµν −ik·(x−y) = e (2π)4 k 2 + i Z −igµν d4 k 1 − ξ −ikµ kν −ik·(x−y) (arbitrary ξ) ⇒ + . 4 k 2 + i 2 e 2 ξ (2π) (k + i) ≡ I Feynman gauge: ξ = 1 and Landau gauge: ξ → ∞. I The longitudinal polarization state could be cured by introduing a fictitious photon mass µ → 0. A generic experiment I MpM A generic experiment is understood diagramatically |a,ini I I |b,outi The amplitude hb, out|a, ini describes the probability that |ai will evolve in time and be measured in the |bi state. For the incoming state |i, ini, the transition probability to a final state |f, outi is wf ←i = |hf, out|i, ini|2 . I There is a unitary operator, S † S = SS † = 1, S-matrix: hf, out|i, ini = hf, in| S |i, ini = hf, out| S |i, outi , I where S = 1 + iτ and S † = 1 − iτ , where the τ -matrix contains the information on the interactions. The τ -matrix is consist of the energy-momentum conservation and the invariant matrix element M: hf | iτ |ii = (2π)4 δ (4) (Pi − Pf ) · iM (i → f ) . Total decay rate Consider a reaction of decay a → 1 + 2 + · · · + nf (eg., Ne2P4 → Neg.s. + γ). The transition probability per unit time is |Sf i |2 . T In a cubic box of volume V = L3 with infinitely high potential well, the differential transition probability is Y d3 pf 1 1 (4) 2 dwf ←i = δ (p − p ) |M | . a f f i 2Ef (2π)3nf −4 2Ea f The lifetime τa = Γa −1 is the inverse of the total decay width X X Γa = Γa→{nf } = w{nf }←a wf ←i = nf = 1 1 2Ea (2π)3nf −4 nf Z d3 pnf (4) d3 p1 ··· δ (pf − pi ) |Mf i |2 . 2E1 2Enf Differential cross section Consider a scattering process a + b → 1 + 2 + · · · + nf (eg., Ne3S2 + γ → Ne2P4 + 2γ). The transition rate (transition probability per unit time) density to one definite final state is wf ←i w ¯f ←i = lim = (2π)4 δ (4) (Pi − Pf ) |Mf i |2 . V →∞ V The differential cross sectin (in Lab.) is defined as the transition rate density per target density (nt ) per incident flux (F ) dσf i nf w ¯f ←i Y d3 pf 0 = . nt F (2π)3 2ωpf 0 f =1 The target density nt = 2ωp2 and the flux F = 2ωp1 vrel yield dσf i nf Y d3 pf 0 1 = (2π)4 δ (4) (Pi − Pf ) |Mf i |2 . 3 2ωp1 2ωp2 vrel (2π) 2ωpf 0 f =1 Interaction picture I I I I Let |Ωi be the ground state of the interacting theory. R R Let Hint (t) = d3 xHint = − d3 xLint be the interacting Hamiltonian and H = H0 + λHint with 0 ≤ λ ≤ 1. Let φ(x) = eiHt φ (x) e−iHt be an Heisenberg picture field and for t 6= t0 , φ (t, x) = eiH(t−t0 ) φ (t0 , x) e−iH(t−t0 ) . For λ = 0, H becomes H0 and we can define an interaction picture field as φ (t, x)|λ=0 = eiH0 (t−t0 ) φ (t0 , x) e−iH0 (t−t0 ) ≡ φI (t, x) . I The full Heisenberg picture field φ in terms of φI : n o φ (t, x) = eiH(t−t0 ) eiH0 (t−t0 ) φI (t, x) e−iH0 (t−t0 ) e−iH(t−t0 ) ≡ U † (t, t0 ) φI (t, x) U (t, t0 ) , where we have defined the unitary operator U (t, t0 ) ≡ eiH0 (t−t0 ) e−iH(t−t0 ) . Unitary time-evolution operator I I The initial condition is U (t0 , t0 ) = 1. ¨ The Schrodinger equation: i I ∂ U (t, t0 ) = eiH0 (t−t0 ) (H − H0 ) e−itH(t−t0 ) ∂t = eiH0 (t−t0 ) (Hint ) e−itH(t−t0 ) z }| { iH0 (t−t0 ) −itH0 (t−t0 ) itH0 (t−t0 ) −itH(t−t0 ) e e = e (Hint ) e {z } | = HI (t)U (t, t0 ) . We expand U ∼ exp (−iHI t) as a power series in λ: Z t U (t, t0 ) = 1 + (−i) dt1 HI (t1 ) t0 2 Z t Z t (−i) dt1 dt2 Tˆ [HI (t1 ) HI (t2 )] + · · · + 2 t0 t0 Z t dt0 HI t0 . ≡ Tˆ exp −i t0 Interacting ground state I I I The interacting ground state |Ωi is not |0i; hΩ|0i = 6 0. E0 ≡ hΩ| H |Ωi with the zero of energy H0 |0i = 0. When H |ni = En |ni, X e−iHT |0i = e−iEn T |ni hn|0i n = E −iE0 T |Ωi hΩ|0i + X e−iEn T |ni hn|0i. n6=0 | I {z →0 } Since En > E0 for all n 6= 0, as we send T → ∞ (1 − i) −1 −iHT |Ωi = lim e−iE0 T hΩ|0i e |0i . T →∞(1−i) I Since T is very large, we can shift it T → ±t0 : −1 |Ωi = lim e−iE0 (t0 +T ) hΩ|0i U (t0 , −T ) |0i T →∞(1−i) hΩ| = lim T →∞(1−i) −1 h0| U (T, t0 ) e−iE0 (T −t0 ) h0|Ωi . Two-point correlation in the interacting system The normalization of the interacting ground state is −1 1 = hΩ|Ωi = |h0|Ωi|2 e−iE0 (2T ) h0| U (T, t0 ) U (t0 , T ) |0i . Now we have the two-point correlation function: hΩ| Tˆ {φ(x)φ(y)} |Ωi n h R io T h0| Tˆ φI (x)φI (y) exp −i −T dtHI (t) |0i n h R io = lim T T →∞(1−i) h0| Tˆ exp −i −T dtHI (t) |0i We need to evaluate the expressions of the form h0| Tˆ {φI (x1 ) φI (x2 ) · · · φI (xn )} |0i . Note that h0| Tˆ {φI (x)φI (y)} |0i is just the Feynman propagtor. Normal ordering From now on we drop the subscript I. We decompose φ(x) into the positive- and negative-frequency parts: φ(x) = φ+ (x) + φ− (x), Z Z d3 p d3 p 1 1 −ip·x − p p φ+ (x) = a e , φ (x) = ap † e+ip·x . p 3 3 2E 2E (2π) (2π) p p These decomposed fields satisfy φ+ (x) |0i = 0 and h0| φ− (x) = 0. ˆ is defined as A normal ordring operator N ˆ ap ak † ≡ ak † ap ⇒ N ˆ φ+ (x)φ− (y) = φ− (y)φ+ (x). N So we, with ak † , ap ± = ak † ap ± ap ak † , have identities ˆ {φ(x)φ(y)} |0i = 0 h0| N ˆ {φ(x)φ(y)} + φ(x)φ(y). Tˆ {φ(x)φ(y)} = N Contractions and propagators The contraction ab is defined by the commutators: I The contraction for the Klein-Gordan fields is defined by + [φ (x), φ− (y)] , for x0 > y 0 ; φ(x)φ(y) ≡ [φ+ (y), φ− (x)] , for y 0 > x0 . ˆ (φ(x)φ(y)) |0i + h0| φ(x)φ(y) |0i h0| Tˆ (φ(x)φ(y)) |0i = h0| N | {z } =0 φ(x)φ(y) = DF (x − y) = h g Aµ (x)Aν (x) = ∆F µν (x − y) = I µ ν The contraction for the Dirac field is defined by + 0 0 ¯− ¯ ψ + (x), ψ− (y) , for x0 > y 0 ; ψ(x)ψ(y) ≡ ¯ ψ (y), ψ (x) , for y > x . F ¯ ¯ ¯ |0i ˆ ψ(x)ψ(y) h0| Tˆ ψ(x)ψ(y) |0i = h0| N |0i + h0| ψ(x)ψ(y) ¯ ψ(x)ψ(y) = SF (x − y) = Wick’s theorem and connected diagrams I For n field operators, we have an identity ˆ [φ (x1 ) φ (x2 ) · · · φ (xn )] Tˆ [φ (x1 ) φ (x2 ) · · · φ (xn )] = N + {all possible contractions} . I which is knwon as Wick’s theorem. The two-point correlation has the structure hΩ| Tˆ {φ(x)φ(y)} |Ωi = Numerator = f fcf c cc c cc f fcf Numerator , Denominator + x y x × exp + Denominator = exp + ⇓ hΩ| Tˆ {φ(x)φ(y)} |Ωi = x y + x + ··· y + ··· + ··· , + ··· y connected connected . S-matrix S-matrix is simply the time-evolution operator, exp (−iHt): hf, out| S |i, outi = lim hf, out| e−iH(2T ) |i, outi . T →∞ To compute this quantity we consider the external states (|Ωi): |i, outi ∝ lim T →∞(1−i) e−iHT |i, 0i . The S-matrix will be of the form lim T →∞(1−i) ∝ hf, 0| e−iH(2T ) |i, 0i lim T →∞(1−i) Z hf, 0| Tˆ exp −i T dtHI (t) |i, 0i −T Then the τ -matrix (cf., S = 1 + iτ ) elements becomes hf, out| iτ |i, outi = (2π)4 δ (4) (Pi − Pf ) · iM (i → f ) Z T ˆ = lim hf, 0| T exp −i dtHI (t) |i, 0i T →∞(1−i) −T connected amputated . Coulomb interaction The M matrix element of the Coulomb interaction in the leading order is p2 0 p1 0 iM = −ie0 γ µ −ie0 γ ν k p1 p2 −igµν = u ¯ p1 0 (−ie0 γ µ ) u (p1 ) u ¯ p2 0 (−ie0 γ ν ) u (p2 ) 2 k µ ν −igµν 2 0 0 = (−ie0 ) u ¯ p1 γ u (p1 ) u ¯ p γ u (p2 ) . 2 k2 This is known as the first part of the Møller scattering. Bhabha scattering The Bhabha scattering is a deformed Møller scattering: p2 0 p1 0 µ iM = ν k p1 p2 −igµν 0 = u ¯ p1 0 (−ie0 γ µ ) u (p1 ) v p (−ie0 γ ν ) v¯ (p2 ) 2 k2 −igµν = (−ie0 )2 u ¯ p1 0 γ µ u (p1 ) v p2 0 γ ν v¯ (p2 ) . 2 k I I I The electrons-2 travels in reverse-time order T −1 . The CP T symmetry → (CP )−1 . The negative-energy electron is known as positron. Compton scattering The Compton scattering contains two diagrams, p0 p0 k0 k0 µ p+k iM = µ + p − k0 ν ν k p p k i p / + k/ + m0 = u ¯ p0 µ ∗ k 0 (−ie0 γ µ ) (−ie0 γ ν ) ν (k) u (p) (p + k)2 − m0 2 0 i p / − k/ + m0 0 ν ∗ 0 +¯ u p (−ie0 γ ) ν (k) k (−ie0 γ µ ) u (p) . µ 2 2 (p − k) − m0 Compton scattering The numerators and denominators can be simplified as follows: I Since p2 = m0 2 and k 2 = 0, the denominators are (p + k)2 − m0 2 = 2p · k in iM1 2 p − k 0 − m0 2 = −2p · k in iM2 . I For numerators, we use a bit of Dirac algebra: ν ν ν ν p / + m0 γ u(p) = 2p − γ p / + γ m0 u(p) = 2pν u(p) − γ ν p / − m0 u(p) = 2pν u(p), We obtain iM = ie0 2 µ ∗ k 0 ν (k) # " 0 γ µ k/γ ν + 2γ µ pν −γ ν k/ γ µ + 2γ ν pν 0 + u(p). ×¯ u p 2p · k −2p · k 0 Renormalization The laws of nature are constructed in such a way as to make the universe as interesting as possible. – Freeman Dyson. Soft Bremsstrahlung Bremsstrahlung = Bremsen (to break) + Strahlung (radiation). p0 p0 k k +Ze = p +Ze + p For the soft photon radiation, |k| |p0 − p|, M0 p0 , p − k ≈ M0 p0 + k, p ≈ M0 p0 , p 0 ∗ p · p · ∗ 0 0 iM = −ie0 u ¯ p M0 p , p u(p) e0 − . p0 · k p·k Soft Bremsstrahlung The differential cross section is then dσ p → p0 + γ = dσ p → p0 2 Z d3 k 1 X 2 p0 · (λ) p · (λ) × e0 0 − . p ·k p·k (2π)3 2k λ=1,2 The differential probability becomes d3 k X e0 2 dP p → p + γ(k) = (2π)3 λ 2k 0 0 p p 2 λ · − . p0 · k p · k The total probability, for soft photons 0 ≤ k ≤ |q| = |p0 − p|, and the differential cross section with fictitious photon mass µ, are Z |q| 1 −q 2 P ≈ dk I v, v0 ∼ log( 2 ) → ∞, k µ 0 2 2 α0 −q −q dσ p → p0 + γ ∝ log log . 2 2 π µ m0 2 −q →∞ Radiative corrections + + + There are three classes of the radiative corrections; Vertex corrections, Self-energies, and Polarizations. ··· Vertex corrections iM = 0 0 = −ie0 u ¯ p0 Γµ p0 , p u(p) · A˜cl iM (2π) δ p0 − p0 µ p −p . To lowest order, Γµ = γ µ . We may express Γµ in a symmetrical form: µ µ Γµ = γ µ A + p0 + pµ B + p0 − pµ C. Vertex corrections Using the Gordon identity p0 µ + pµ iσ µν qν u ¯ p γ u(p) = u ¯ p + u(p), 2m0 2m0 0 0 µ we have σ µν qν Γµ p0 , p = γ µ F1 q 2 + i F2 q 2 , 2m0 where the unknown functions F1 and F2 are called form factors. To lowest order, F1 = 1 and F2 = 0. 0 ˜cl I When A µ (x) = (2π) δ q φ (q) , 0 , iM = −ie0 u ¯ p0 Γ0 p0 , p u(p) · φ˜ (q) † −ie0 F1 (0)φ˜ (q) · 2m0 ξ 0 ξ = e0 F1 (0)φ (x) . = q→0 V (x) Vertex corrections I cl When Acl µ (x) = 0, A (x) , i iσ iν qν 0 F2 u(p) A˜icl (q) . iM = +ie0 u ¯ p γ F1 + 2m0 Again the expression in brackets vanishes at q = 0, so in this limit −1 k 0† ˜ k (q) , iM = −i (2m0 ) · e0 ξ σ [F1 (0) + F2 (0)] ξ B 2m0 ˜ k (q) = −iijk q i A˜j (q). This is just that of a where B cl magnetic moment interaction V (x) = − hµi · B (x), where e0 †σ hµi = [F1 (0) + F2 (0)] ξ 0 ξ, m0 2 e0 µ = g S, 2m0 where the Lande´ g-factor is g = 2 [F1 (0) + F2 (0)] = 2 + 2F2 (0). Vertex corrections To one-loop order, the vertex function Γµ = γ µ + δΓµ : p0 k0 = k + q δΓµ = p−k k q p where u ¯ p0 δΓµ p0 , p u(p) = Z 0 u ¯ (p0 ) k/γ µ k/ + m0 2 γ µ − 2m0 (k + k 0 )µ u(p) d4 k 2 2ie0 . (2π)4 (k − p)2 + i k 0 2 − m0 2 + i (k 2 − m0 2 + i) Vertex corrections The Feynman parameterization Z 1 1 1 = dx AB [xA + (1 − x)B]2 0 Z 1 1 = dxdyδ (x + y − 1) [xA + yB]2 0 simplifies the denominator D to D = l2 − ∆ + i l ≡ k + yq − zp, ∆ ≡ −xyq 2 + (1 − z)2 m0 2 > 0. The numerator will be # " µ − z 2 m0 2 γ · − 12 l2 + (1 − x)(1 − y)q 2 + 1 − 2z µ + p0 + pµ · m0 z(z − 1) u(p). N =u ¯ p0 +q µ · m0 (z − 2)(x − y) Vertex corrections Using the Gordon identity again, we have an entire expression u ¯ p0 δΓµ p0 , p u(p) = Z 1 Z 2 d4 l 2 dxdydzδ (x + y + z − 1) 3 2ie0 4 D (2π) 0 1 ×¯ u p0 γ µ · − l2 + (1 − x)(1 − y)q 2 + 1 − 4z + z 2 m0 2 2 µν iσ qν 2 2m0 z(1 − z) u(p). + 2m0 There are two classes of integrations: Z Z d4 l d4 l 1 l2 and n n. (2π)4 (l2 − ∆) (2π)4 (l2 − ∆) | {z } | {z } convergent divergent for n≤3 Pauli-Villars regularizatoin We introduce ad hoc a cut-off Λ(→ ∞) in the photon propagators: (k 2 1 1 1 → 2 − 2 2 2 2 − p ) + i (k − p ) + i (k − p ) − Λ2 + i so the denominator is altered as ∆ → ∆Λ = −xyq 2 + (1 − z)2 m0 2 + zΛ2 . Then the divergent integral is replaced by convergent ones: Z l2 ∆Λ d4 l l2 i − = log , ∆ (2π)4 (l2 − ∆)3 (l2 − ∆Λ )3 (4π)2 which looks like (∞ − ∞Λ ) ∝ log (∆Λ /∆). I How can this result affect on F1 q 2 and F2 q 2 with F1 (0) = 1? The convergent form factor F2 The form factor F2 is corrected to order α0 (= e0 2 /4π~c) Z α0 1 2m0 2 z(1 − z) 2 F2 q = dxdydzδ(x + y + z − 1) 2π 0 m0 2 (1 − z)2 − q 2 xy is convergent especially for q 2 = 0, such that F2 q 2 = 0 = = α0 2π Z α0 π Z 1 dxdydzδ (x + y + z − 1) 0 1 Z 1−z dz 0 dy 0 2m0 2 z (1 − z) m0 2 (1 − z)2 z α0 = . 1−z 2π We get a correction to the g-factor of the electron: ae ≡ g−2 α0 = 2 2π ≈ 0.0011614. (α0 $α) Experiments give ae exp = 0.0011597, which differs by ≈ 0.15 %. Infrared divergence The divergent form factor F1 q 2 is corrected to F1 q 2 α0 = 1+ 2π " Z 1 dxdydzδ (x + y + z − 1) ! m0 2 (1 − z)2 × log m0 2 (1 − z)2 − q 2 xy m0 2 1 − 4z + z 2 + q 2 (1 − x) (1 − y) + m0 2 (1 − z)2 − q 2 xy + µ2 z # m0 2 1 − 4z + z 2 − , m0 2 (1 − z)2 + µ2 z 0 where µ is the fictitious photon mass. In the limit µ → 0, we may obtain 2 2 α0 −q −q 2 F1 −q → ∞ = 1 − log log . 2 2π m0 µ2 What did we have made mistake? I The S-matrix theory hΩ| Tˆφ (x1 ) φ (x2 ) · · · |Ωi = X connected amputated is based on the completeness of the normalized interacting ground state |Ωi: 1 = |Ωi hΩ| I I I from the free vacuum |0i. Let H |λ0 i = λ0 |λ0 i, but P |λ0 i = 0. q Let |λp i be the boosts of |λ0 i with Ep (λ) ≡ |p|2 + mλ 2 . The desired completeness relation will be X Z d3 p 1 1 = |Ωi hΩ| + 3 2E (λ) |λp i hλp | . (2π) p λ I Accordingly we need to normalize |Ωi again: ⇒Renormalization. The particle dispersion H multiparticle continuum 1-particle in motion bound state m 1-particle at rest P The eigenvalues of P µ = (H, P) of particle mass m. Renormalization Assume x0 > y 0 and drop off hΩ| φ(x) |Ωi hΩ| φ(y) |Ωi (= 0). The two-point correlation function is X Z d3 p 1 hΩ| φ(x)φ(y) |Ωi = 3 2E (λ) hΩ| φ(x) |λp i hλp | φ(y) |Ωi . (2π) p λ The matrix element hΩ| φ(x) |λp i = hΩ| eiP ·x φ(0)e−iP ·x |λp i = hΩ| φ(0) |λp i e−ip·x p0 =Ep = hΩ| φ(0) |λ0 i e−ip·x p0 =Ep . The two-point correlation function becomes for x0 > y 0 X Z d4 p i −ip·(x−y) hΩ| φ(x)φ(y) |Ωi = |hΩ| φ(0) |λ0 i|2 . 4 p2 − m 2 + i e (2π) λ λ ¨ en-Lehmann ´ Kall representation For both cases of x0 > y 0 and y 0 > x0 , we have Z ∞ dM 2 ˆ hΩ| T φ(x)φ(y) |Ωi = ρ M 2 DF x − y; M 2 , 2π 0 where ρ M 2 is a positive spectral density, X ρ M2 = (2π) δ M 2 − mλ 2 |hΩ| φ(0) |λ0 i|2 . λ ρ M2 1-particle states bound states 2-particle states m2 (2m)2 M2 Field-strength renormalization The spectral density is ρ M 2 = 2πδ M 2 − m2 Z + nothing else for M 2 . (2m)2 , where Z is refered as field-strength renormalization. The Fourier transform of the two-point correlation becomes Z Z ∞ i dM 2 4 ip·x ˆ ρ M2 2 d xe hΩ| T φ(x)φ(0) |Ωi = 2π p − M 2 + i 0 Z ∞ 2 dM i iZ = 2 + ρ M2 2 . 2 p − m + i p − M 2 + i ∼(2m)2 2π p2 m2 isolated pole (2m)2 poles from bound states branch cut The electron self-energy I The electron two-point correlation function is ¯ |Ωi = hΩ| Tˆψ(x)ψ(y) fyf fyfyf x y + I x x y y + ··· The free-field propagator: p I + i p / + m0 . = 2 p − m0 2 + i The lowest order electron self-energy: fyf p−k i p i p / + m0 / + m0 = 2 [−iΣ2 (p)] 2 . p − m0 2 p − m0 2 The electron self-energy We have the explicit form of the electron self-energy: Z −i d4 k µ i (k/ + m0 ) , −iΣ2 (p) = (−ie0 )2 2 γ k 2 − m 2 + i γµ 2 (2π) (p − k) − µ2 + i 0 where we regulate it by adding a small photon mass µ. We use the Feynman parametrization and shift the momentum l = k − xp to get Z 1 Z d4 l −2xp / + 4m0 2 , −iΣ2 (p) = −e0 dx 3 2 (2π) (l − ∆ + i)2 0 where ∆ = −x(1 − x)p2 + xµ2 + (1 − x)m0 2 . We regulate it by the Pauli-Villars procedure: 1 1 1 → − . 2 2 2 2 (p − k) − µ + i (p − k) − µ + i (p − k) − Λ2 + i 2 The electron self-energy Introducing ∆Λ = −x(1 − x)p2 + xΛ2 (1 − x)m0 2 → xΛ2 , Λ→∞ we have Σ2 (p) = α0 2π Z 1 dx 2m0 − xp / log 0 xΛ2 2 (1 − x)m0 + xµ2 − x(1 − x)p2 The logarithm of x has a branch cut begining at the point where (1 − x) m0 2 + xµ2 − x (1 − x) = 0, or r 1 1 m0 p2 − (m0 + µ)2 p2 − (m0 − µ)2 . x= + 2 − 2 ± 2 2 2p 2p 2p The branch cut of Σ2 p2 begins at p2 = (m0 + µ)2 , two-particle threshold. 2 I µ2 Where is the simple pole at p2 = m2 ? . The electron self-energy p pp The two-point correlation function is written as = + + = = = Hence + ··· i p i p i p / + m0 / + m0 / + m0 + 2 (−iΣ) 2 2 2 2 p + m0 p + m0 p + m0 2 i p i p i p / + m0 / + m0 / + m0 + 2 (−iΣ) 2 (−iΣ) 2 + ··· p + m0 2 p + m0 2 p + m0 2 ! !2 Σ p Σ p i i i / / + + + ··· p p / − m0 p / − m0 p / − m0 / − m0 p / − m0 i . p / − m0 − Σ p / p − m − Σ p / / 0 =0 p /=m gives us the simple pole at the physical mass, m = m0 + Σ p / . Mass renormalization In the vicinity of the pole, p / − m0 − Σ p / has the form   2 dΣ p /   + O p − m . p − m · 1 − / / dp / p=m / When we write the two-point correlation function as Z iZ p + m / 2 ¯ |Ωi = d4 xeip·x hΩ| Tˆψ(x)ψ(0) , p2 − m2 + i we obtain the mass renormalization constant to be dΣ p / Z2 −1 = 1 − . dp / p /=m Mass renormalization To order α0 , the mass shift is δm = m − m0 = Σ2 p / = m ≈ Σ2 p / = m0 . Then the mass shift is Z 1 xΛ2 α0 m0 dx (1 − x) log , δm = 2π (1 − x)2 m0 2 + xµ2 0 which is ultraviolet divergent O log Λ2 for Λ → ∞. The correction for Z2 in order α0 is calculated to be dΣ2 δZ2 = dp / p/=m Z xΛ2 α0 1 dx −x log = 2π 0 (1 − x)2 m2 + xµ2 x (1 − x) m2 +2 (2 − x) . (1 − x)2 m2 + xµ2 ¶The small correction to mass m0 is infinite! Mass renormalizaiton One can show that the exact vertex should be read iσ µν qν F2 q 2 . Z2 Γµ p0 , p = γ µ F1 q 2 + 2m The left-hand side of the exact vertex function becomes Z2 Γµ = (1 + δZ2 ) (γ µ + δΓµ ) = γ µ + δΓµ + γ µ δZ2 , while in the right-hand side F1 q 2 becomes F1 q 2 = 1 + δF1 q 2 + δZ2 = 1 + δF1 q 2 − δF1 (0) , if δZ2 = −δF1 (0). Define another rescaling factor Z1 by the relation Γµ (q = 0) = Z1 −1 γ µ , where Γµ is the complete amputated vertex function. Mass renormalization However, the divergent part of the vertex correction is Z α0 1 δF1 (0) = dxdydzδ (x + y + z − 1) 2π 0 # " 1 − 4z + z 2 m2 zΛ2 × log + (1 − z)2 m2 + zµ2 (1 − z)2 m2 + zµ2 Z α0 1 = dz (1 − z) 2π 0 " # 1 − 4z + z 2 m2 zΛ2 × log + . (1 − z)2 m2 + zµ2 (1 − z)2 m2 + zµ2 We can show that δF1 (0) + δZ2 = 0. To find F1 (0) = 1, we must provide the identity Z1 = Z2 , so that the vertex rescaling exactly compensates the electron field-strength renormalization. ¶The understanding of mass is postponed. Vacuum polarization Photon is dressed in order e0 2 k+q iΠ2 µν (q) ≡ µ q ν q k 2 Z = (−1) (−ie0 ) d4 k i i µ ν tr γ γ . k/ − m k/ + /q − m (2π)4 gpg gpg g gcg gcgcg Generally the polarized photon propagator is defined by iΠµν (q) ≡ q q µ . ν As for the electron self-energy the polarization decomposes q µ q ν = + + + ··· Ward-Takahashi identity The guage invariance of radiation field leads the charge conservation (qµ Mµ (q) = 0) in such a way that    p p+q  p+q  µ     − qµ ·   = e0  q      p p p+q    .  This identity is known as the Ward-Takahashi identity: −iqµ Γµ (p + q, p) = S −1 (p + q) − S −1 (p) . We defined Z1 and Z2 by the relations Γµ (p + q, p) → Z1 −1 γ µ as q → 0 and S(p) ∼ iZ2 . p /−m Setting p near mass shell and expanding the Ward-Takahashi identity about q = 0, we find −iZ1 −1 /q = −iZ2 −1 /q ⇒ Z1 = Z2 . Charge renormalization I I The Ward-Takahashi identity tells us that qµ Πµν = 0. In other words, Πµν ∝ g µν − q µ q ν /q 2 . I Furthermore, we can expect Πµν (q) will not have a pole at q 2 = 0. I It is convenient to write Πµν (q) = q 2 g µν − q µ q ν Π q 2 , where Π q 2 is regular at q 2 = 0. gpg The exact photon two-point correlation function is µ ν = = −igσν −igµν 2 ρσ −igµν + i q g − q ρ q σ Π (q) + ··· 2 2 q q q2 −igµρ ρ σ 2 2 −igµν −igµρ ρ + ∆ν Π q 2 + ∆σ ∆ν Π q + · · · 2 2 q q q2 where ∆ρν ≡ δνρ − q ρ qν /q 2 and ∆ρσ ∆σν = ∆ρν . Charge renormalization I gpg We can simplify further µ ν = = qµ qν −i −i qµ qν gµν − 2 + 2 q 2 (1 − Π (q 2 )) q q q2 −igµν (∵ qµ Mµ (q) = 0) . 2 q (1 − Π (q 2 )) I As long as Π q 2 is regular at q 2 = 0, the exact propagator alway has a pole at q 2 = 0. I In other words, the photon remains absolutely massless at all orders in the perturbation theory. I The residue of the q 2 = 0 pole is 1 ≡ Z3 . 1 − Π(0) Charge renormalization I Since the scattering amplitude will be shifted by ··· e0 2 gµν Z3 e0 2 gµν · · · → · · · ··· , q2 q2 we will have the charge renormalization p e = Z3 e 0 . I Considering a scattering process with nonzero q 2 in leading order α0 , −igµν igµν e0 2 e0 2 ≈ . q2 1 − Π (q 2 ) q2 1 − [Π2 (q 2 ) − Π2 (0)] I The quantity in (· · · ) has an interpretation of a q 2 -dependent electric charge, so we have α0 → α q 2 = α0 e0 2 /4π ≈ . 2 1 − Π (q ) 1 − [Π2 (q 2 ) − Π2 (0)] The divergent polarization Π2 In order e0 2 , the polarization is badly ultraviolet divergent: " # Z 4k / k + q + m i / d i ( k + m) / iΠ2 µν (q) = − (−ie0 )2 tr γ µ 2 γν k − m2 (2π)4 (k + q)2 − m2 Z ν µ d4 k k µ (k + q) + k ν (k + q) − g µν k · (k + q) − m2 2 = −4e0 . (2π)3 (k 2 − m2 ) (k + q)2 − m2 Introducing a Feynman parameter, we combine the denominator as Z 1 1 1 = dx , (l2 + x (1 − x) q 2 − m2 )2 0 (k 2 − m2 ) (k + q)2 − m2 where l ≡ k + xq. In terms of l, the numerator will be Numerator = 2lµ lν − g µν l2 − 2x (1 − x) q µ q ν + + g µν m2 + x (1 − x) q 2 + (terms linear in l) . Wick rotation I The momentum (contour) integral in the Minkowski metric space-time, g µµ = (+1, −1, −1, −1), is difficult. I So Wick suggested a rotation of the time coordinate t → −ix0 , i.e., the Euclidean four-vector product: x2 = t2 − |x|2 → − x0 2 − |x|2 = − |xE |2 . l0 l0 − q − |l|2 + ∆ + i q |l|2 + ∆ + i ⇒ q + |l|2 + ∆ − i Wick rotation q + |l|2 + ∆ − i Dimensional regularization I I I I For sufficiently small dimension d, any loop-momentum integral will converge. Therefore the Ward identity can be proved. The final expression for Π2 should have well-defined limit as d → 4. A typical d-dimensional Euclidean space integral reads Z Z d Z ∞ dΩd d lE 1 lE d−1 = dl · 2 , E (2π)2 lE 2 + ∆ 2 (2π)d 0 lE 2 + ∆ where the area of a unit sphere in d dimensions is identified as √ d Z 2 ( π) dΩd = Γ d2 and the second factor of the integral becomes d Z ∞ ld−1 1 1 2− 2 Γ 2 − d2 Γ dlE 2 = 2 ∆ Γ (2) 0 lE 2 + ∆ d 2 . Dimensional regularization I I I I I Near d = 4, define = 4 − d, and use the approximation 2 d =Γ = − γ + O () , Γ 2− 2 2 where γ ≈ 0.5772 is the Euler-Mascheroni constant. The integral is then Z 4 1 d lE 1 2 → − log ∆ − γ + log (4π) + O () (2π) lE 2 + ∆ 2 →0 (4π)2 In d dimensions, gµν g µν = d. Thus, lµ lν of the numerators in the integrands should be replaced by d1 l2 g µν . The Dirac matrices in d = 4 − should be modified to γ µ γ ν γµ = − (2 − ) γ ν γ µ γ ν γ ρ γµ = 4g νρ − γ ν γ ρ γ µ γ ν γ ρ γ σ γµ = −2γ σ γ ρ γ ν + γ ν γ ρ γ σ . Evaluation of Π2 The unpleasant terms with l2 in the numerator gives Z 2− d 2 dd lE − d2 + 1 g µν lE 2 1 1 d = (−∆g µν ) . Γ 2− 2 2 d/2 2 2 ∆ (2π) (4π) lE + ∆ Evaluating remaining terms and using ∆ = m2 − x (1 − x) q 2 are iΠ2 µν (q) = q 2 g µν − q µ q ν · iΠ2 q 2 , where Π2 q 2 = → →0 Γ 2 − d2 dxx (1 − x) − ∆2−d/2 (4π)d/2 0 Z 1 2α0 2 − dxx (1 − x) − log ∆ − γ + log (4π) . π 0 8e0 2 Z 1 This satisfies the Ward identity, but it is still logarithmically divergent. The electron charge shift I In order α0 the electric charge shift is computed as 2α0 e2 − e0 2 = δZ3 → Π2 (0) ≈ − → ∞. 2 e0 3π →0 I The bare charge is infinitely larger than the observed charge. I This bare charge is not observable. I What can be observed is α0 α0 α q2 ≈ ≡ , 2 ˆ 2 (q 2 ) 1 − [Π2 (q ) − Π2 (0)] 1−Π where the difference Z m2 2α0 1 2 ˆ dxx (1 − x) log , Π2 q = − π 0 m2 − x (1 − x) q 2 which is independent of in the limit → 0. Classical picture In nonrelativistic limit, the attractive Coulomb potential reads Z d3 q iq·x −e2 . V (x) = e ˆ 2 − |q|2 (2π)3 |q|2 1 − Π ˆ 2 for q 2 m2 , we obtain Expanding Π 4α2 (3) α − δ (x) 2 r 15m Z ∞ 1 ie2 QeiQr ˆ 2 −Q2 . 1 + Π dQ Q2 + µ2 (2π)2 r −∞ V (x) = − = When r−1 m(= λC ), we can approximate the potential as ! α α e−2mr V (r) = − 1+ √ + ··· r 4 π (mr)3/2 → vacuum polarizations–virtual dipoles screening. Short-distance limit For small distance or −q 2 m2 , we have Z 2α 1 2 ˆ ≈ Π2 q dxx (1 − x) π 0 2 2 m −q + log (x (1 − x)) + O × log 2 m q2 2 2 α −q 5 m = log − +O . 2 3π m 3 q2 The effective coupling constant in this limit is therefore αeff q 2 = α 1− α 3π log −q 2 Am2 , A = exp (5/3) . The effective electric charge becomes much larger at small distances, as we penetrate the screening cloud of virtual electron-positron pairs. Renormalized quantum electrodynamics The original QED Lagrangian is 1 ¯ µ ψAµ . L = ψ¯ i∂/ − m0 ψ − Fµν F µν − e0 ψγ 4 The renormalization scheme modifies the electron and photon propagators as p gpg = = iZ2 + ··· p /−m −iZ3 gµν + ··· q2 To absorb Z2 and Z3 into L, we substitute ψ → Z2 1/2 ψ and Aµ → Z3 1/2 Aµ . The Lagrangian becomes 1 ¯ µ ψAµ , L = Z2 ψ¯ i∂/ − m0 ψ − Z3 Fµν F µν − e0 Z2 Z3 1/2 ψγ 4 with the physical electric charge e0 Z2 Z3 1/2 = eZ1 . Renormalization Group and Higgs mechanism It seems that scientists are often attracted to beautiful theories in the way that insects are attracted to flowers. – Steven Weinberg. Cutoff problem I Our coupling constant (fine-structure constant) is not a constant, but it is running as −1 α q 2 ∝ log −q 2 . as q → ∞, the ultraviolet divergence. I The divergences are removed by the physical parameter-fitting (m and e) from the experiments. I The ad hoc Pauli-Villars cutoff Λ, for example, has been introduced for eliminating very large momentum contributions from the theory. I In other words, the small distance scale physics are eliminated and replaced by those parameters. I However, we do not have any precise information for short distance physics. Renormalization group flows I For a scaling parameter b < 1, but b ≈ 1, we rescale distances and momenta in accoring to k 0 = k/b, I I x0 = xb, so that the variable k 0 is integrated over |k 0 | < Λ. The field is also rescaled as h i1/2 φ0 = b2−d (1 + ∆Z) φ. Our model system with an effective Lagrangian Z Z 1 02 02 1 0 04 0 0 2 d d 0 1 d xLeff = d x ∂µ φ + m φ + λ φ 2 2 4! will yield the rescaled parameters 2 2 m0 = m2 + ∆m (1 + ∆Z)−1 b−2 −→ m∗ 2 , λ0 = (λ + ∆λ) (1 + ∆Z)−2 bd−4 −→ λ∗ . Renormalization scale We introduce an arbitrary momentum scale M (renormalization scale) and impose the renormalization condition at a spacelike momentum p with p2 = −M 2 . Then we may have hΩ| φ0 (p) φ0 (−p) |Ωi = iZ p2 at p2 = −M 2 . The n-point Green’s function is defined by G(n) (x1 , · · · , xn ) = hΩ| Tˆφ (x1 ) · · · φ (xn ) |Ωiconnected . If we shift M by δM , then correspondingly we obtain M → M + δM, λ → λ + δλ, φ → (1 + δη) φ, G (n) → (1 + nδη) G(n) . The Callan-Symanzik equation If we think of G(n) as a function of M and λ, we can write as dG(n) = ∂G(n) ∂G(n) δM + δλ = nδηG(n) . ∂M ∂λ It is convenient to introduce dimensionless parameters β≡ M δλ; δM γ≡− M δη, δM so to arrive at the equation ∂ ∂ M +β + nγ G(n) (x1 , · · · , xn ; M, λ) = 0. ∂M ∂λ Since G is renormalized, β and γ cannot depend on Λ, these functions cannot depend on M . We concluded that ∂ ∂ M + β (λ) + nγ (λ) G(n) ({xi } ; M, λ) = 0. ∂M ∂λ This is known as the Callan-Symanzik equation. Solutions of the Callan-Symanzik equations f fxf The generic form of the two-point Green’s function is G(2) (p) = = + loops + + ··· 2 i i i Λ i + 2 A log + finite + 2 ip2 δZ 2 + · · · 2 2 p p −p p p The M dependence comes entirely from the counterterm δZ . By neglecting the β term, we find ∂ 1 δZ . γ= M 2 ∂M Because the counterterm must be δZ = A log Λ2 + finite, M2 to lowest order we have γ = A. Solutions of the Callan-Symanzik equations In a similar manner we obtain ∂ β(λ) = M ∂M 1 X −δλ + λ δZi 2 Since δλ = −B log ! . i Λ2 + finite, M2 to lowest order we have β(λ) = −2B − λ X Ai . i So β and γ are not depending on the renormalization scale M . The QED solutions There is a γ term for each field and a β term for each coupling. ∂ ∂ M + β (e) + nγ2 (e) + mγ3 (e) G(n,m) ({xi } ; M, e) = 0, ∂M ∂e where n and m are, respectively, the number of electron and photon fieldsin G(n,m) and γ2 and γ3 are the rescaling functions of the electron and photon fields. I β ∝ the shift in the coupling constant and I γ ∝ the shift in the field renormalization, when the renormalization scale M is increased. Using the methods described before we obtain, to lowest order, β(e) = e2 e2 e3 , γ (e) = , γ (e) = . 2 3 12π 2 16π 2 12π 2 Running coupling in QED If M ∼ O(m), then the renormalized value er is close to e. For the static potential V (x), we have the Callan-Symanzik equation ∂ ∂ M + β (er ) V (q; M, er ) = 0. ∂M ∂er Since the dimension of the Fourier transformed potential V (q) is (mass)−2 , we trade M and q: ∂ ∂ − β (er ) + 2 V (q; M, er ) = 0. q ∂q ∂er The potential will be in the form V (q, er ) = 1 V (¯ e (q; er )) , q2 where e¯ (q) is the solution of the renormalization group equation d d log ¯ (q; er ) q e M = β (¯ e) , e¯ (M ; er ) = er . Running coupling in QED Since the potential, in leading order, is e2 , q2 we can identify V (¯ e) = e¯2 + O e¯4 . We immediately obtain V (q) ≈ V (q, er ) = e¯2 (q; er ) . q2 By solving the renormalization group equation for e¯ and using β (e) = e3 /12π 2 , we find 12π 2 1 1 q − = log . 2 er 2 e¯2 M This simplifies to e¯2 (q) = er 2 . 1 − (er 2 /6π 2 ) log (q/M ) Running coupling in QED By setting M 2 = exp (5/3) m2 and er ≈ e, with α = reproduce e2 4π , we α α ¯ (q) = 1− α 3π , A = exp(5/3). q2 log − Am 2 There is a renormalization scale M , which replaces the ad hoc Pauli-Villars cutoff Λ. ? The electric charge is the result of the virtual vacuum polarization by the existence of interacting electron. Evolution of mass If LM is the massless Lagrangian renormalized at the scale M , the new massive Lagrangian will be in the form 1 L = LM − m2 φM 2 . 2 We treat mass term by replacing m2 → ρm M 2 and expanding the Lagrangian about the free field one L0 reads: 1 1 L = L0 − ρm M 2 φM 2 − λM 4−d φM 4 , 2 4 which is the Landau-Ginzburg theory for ferromagnetism! The Callan-Symanzik equation will give us β = − (4 − d) λ + 3λ2 16π 2 and for the condition β = 0 λ∗ = 16π 2 (4 − d) . 3 Mass from a phase transition The corresponding renormalization group equation would be d ¯ ρ¯m . ρ¯m = −2 + γφ2 λ d log p ¯ = λ∗ , The solution is, for the coupling λ 2−γ 2 (λ∗ ) φ M ρ¯m = ρm . p The solution gives a nontrivial relation ξ ∼ ρm −ν , where the exponent ν is given formally by the expression ν= 1 , 2 − γφ2 (λ∗ ) explicitly, the Wilson-Fisher relation in statistical physics ν −1 = 2 − 1 (4 − d) . 3 β-decay I I The radioactivity discovered by Becquerel is β-decay. This is the neutron decay process: n → p + ν¯ + e− , p-udu ν¯ W− e− n-udd I I I β-decay violates the CP gauge symmetry. A non-Abelian gauge theory, SU (2) × U (1), is required. No massive bosons and fermions are allowed to satisfy the SU (2) × U (1) gauge symmetry. Massless Dirac field Let a Dirac field ψ is massless, but it is a doublet of Dirac fields ψL ψ= . ψR The kinetic energy term may be written as σ · ∂ψL + ψR † iσ · ∂ψR . L = ψL † i¯ The left-handed fields may coupled to a non-Abelian gauge field Aa µ , which defines the corresponding field tensor as F a µν = ∂µ Aa ν − ∂ν Aa µ + gf abc Ab µ Ac ν , through the minimal substitution Dµ = ∂µ − igAa µ ta r to yield 1 − γ5 µ a a ¯ L = ψiγ ∂µ − igA µ t r ψ. 2 Here ta follows the commutation relation ta , tb = if abc tc . Higgs coupling I We may assign the left-handed components of quarks and leptons to doublets of an SU (2) gauge symmetry like u ν QL = , LL = . d e I Since these fields are massless, we introduce a U (1) gauge symmetric field φ, which is known as Higgs field, Dµ φ = (∂µ − igAa µ τ a ) φ, a I where τ a = σ2 . If the vacuum expectation value of φ has broken symmetry 1 0 hφi = √ , 2 v then the gauge boson masses arise from 1 2 µ 0 2 a b |Dµ φ| = g (0 v) τ τ Aa µ Ab + · · · v 2 Higgs mechanism After a symmetrization we find the mass term ∆L = g 2 v 2 a aµ A µA . 8 All three gauge bosons receive the mass mA = gv 2 . When Higgs field transforms under φ follows SU (2) × U (1) gauge symmtry, a a φ → eiα τ eiβ/2 φ, two bosons acquire masses and one boson remains massless: W ± mW ± = gv , p2 Z 0 mZ 0 = g 2 + g 0 2 v2 , A mA = 0. Higgs mechanism Similarly, the electron fields e¯L and eR follows the mass term 1 ∆Le = − √ λe v¯ eL eR + h.c. + · · · , 2 by which the massless electron acquires mass me = √1 λe v. 2 ? The electron mass is the result of the spontaneous continuous symmetry breaking of Higgs field. Summary I Relativistic quantum field theory I I I Field quantization I I I I I I I S-matrix theory Perturbation expansion Photon as gauge particle Elementary processes Renormalization ⇒ Physically observed parameters: I I I I Klein-Gordon fields Dirac fields Propagator and causality Interacting field theory I I Intrinsic spin Pauli’s principle spin-magnetic momentum (definite), electron mass (cancelled divergences), electric charge of electron (leaving divergence). Renormalization Group and Higgs mechanism. I I The origin of the charge of electron, The origin of the electron mass. James Clerk Maxwell The work of James Clerk Maxwell changed the world forever. by Albert Einstein