What Is Electron?

Home , George Johnstone Stoney

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. < ηλεκτρoν¯ , amber.

1838 Richard Laming: P Atom = core matter + surrounding (unit electric charge). 1846 William Weber: Electricity = P ﬂuid(+) + P 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 carried energy, I moved from cathod to anode, and I 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 2 2 I S = s(s + 1)~ , the square of the spin magnitude, 1 h −34 where s = ± 2 and ~ = 2π = 1.0546 × 10 Js. I µ = −gµBs, the spin magnetic moment, e~ −20 where µB = 2mc = 0.927 × 10 emu and g is the Lande´ g-factor, for free-electron g = 2.0023. Size: A point particle, no larger than 10−22 m,

α~ −15 I re = mc = 2.818 × 10 m, the classical electron radius, 2 where α = e = 1 = 0.00730, the ﬁne structure 4π~c 137.04 constant. ~ −13 I λC = mc = 3.862 × 10 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 ∂ψ i = Hψ ~ ∂t where the Hamiltonian H is a linear Hermitian operator. For an isolated free particle, the Hamiltonian is p2 H = 2m and the quantum mechanical transcriptions are ∂ H → i , p → ~∇ ~∂t i leads to a relativistically incorrect equation ∂ψ 2 i = − ~ ∇2ψ. ~ ∂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,

[length] = [time] = [energy]−1 = [mass]−1 .

2 I The mass of a particle is equal to the rest energy (mc ) and to its inverse Compton wavelength (mc/~). I The thermal unit system is the same as the natural unit system with the additional Boltzmann constant kB = 1.

I In this system, [energy] = [temperature]. I Especially, 1 eV = 11605 K.

I The atomic Hartree unit system is deﬁned by the constants 2 −1 ~ = e = m = 1, but c = α . 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 Homogeneous Lorentz group if λ = 0,

I Poincare´ group or inhomogeneous Lorentz group if λ 6= 0. Relativistic notions

I x is the four-vector of space and time. µ I x (µ = 0, 1, 2, 3) are the contravariant components of this vector. I xµ are the covariant components effected by the Minkowski metric tensor,  1 0 0 0  µν  0 −1 0 0  gµν = g =   .  0 0 −1 0  0 0 0 −1

µ 0 ν P3 ν 0 I x = x , x and xµ = gµνx = ν=0 gµνx = x , −x . µ 2 2 I The scalar product is deﬁned by x · x ≡ x xµ = t − x . 2 µ I The equation for the lightcone: x ≡ x xµ = 0. µ I The displacement vector is naturally raised, x , while the derivative operator is naturally lowered ∂ ∂ ∂ = = , ∇ . µ ∂xµ ∂x0 Lightcone x0

future timelike

elsewhere

spacelike past Relativistic quantum mechanics

µ I Momentum vectors are similarly deﬁned p = (E, px, py, pz) I and the scalar product is deﬁned by

µ 2 2 p · p = p pµ = E − p · p = m .

µ I Likewise p · x = p xµ = Et − p · x. I The quantum mechanical transcriptions will be (~ = 1) ∂ E = i , p = −i∇ ∂x0 or pµ = i∂µ.

I For a relativistic free particle, we may try a relativistic p Hamiltonian H = p2 + m2 (c = 1) to obtain

∂ψ p i = −∇2 + m2ψ (?) ∂t The causality violation

The amplitude for a free particle to propagate from x0 to x:

U(t) = hx|e−iHt |x i √ 0 −it p2+m2 = hx|e |x0i √ 1 Z 2 2 = d3p peit p +m eip·(x−x0) (2π)3 Z ∞ √ 1 −it p2+m2 = 2 dp p sin (p |x − x0|) e . 2π |x − x0| 0

At the point x2 t2 (well outside the lightcone), the phase p function px − t p2 + m2 has a stationary point at p = i √ mx . x2−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 ∂µφr(x): Z σ 4 S = L (φr, ∂µφr) d x, 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 4 δS = − ∂µ δ0φrd x + F (σ) − F (σ0), σ0 ∂φr ∂ (∂µφr) Z ∂L ν µν ∂L F (σ) = ∂ φr − g L δxν − δφr dσµ. σ ∂ (∂µφr) ∂ (∂µφr) 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 µ the total variation δφr = δ0φr + ∂µφrδx ,

F (σ) = F (σ0) = 0 Z σ ∂L ∂L 4 δS = − ∂µ δ0φrd x = 0. σ0 ∂φr ∂ (∂µφr) This equation is satisﬁed if the integrand vanishes at every point: ∂L ∂L − ∂µ = 0. ∂φr ∂ (∂µφr) 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 = x02 = 0 invariant,

0µ µ ν µ λν µλ x = Λ νx ;Λ νΛ = g , and which, in addition, satisﬁes

µ µ 0 Λ ν real, det (Λ ν) > 0, Λ 0 > 0. The inhomogeneous Lorentz transformation involves displacements, such that x0 = Lx:

0µ µ ν µ L : x = Λ νx + λ , where λµ is a four-vector independent of x. The ﬁeld component φr(x) transforms according to the proper Lorentz transformation: 0 s φr (Lx) = Sr φs(x). Lorentz group For the Lorentz transformation of the ﬁeld

0 −1 φr (x) = U(L)φr(x)U (L),

we observe that the operators U(L) form a representation of the Lorentz group:

U(L2L1) = U(L2)U(L1).

The inﬁnitesimal Lorentz transformations are deﬁned by

ν ν ν Λµ = gµ + αµ , λµ = αµ,

ν where αµ and αµ are inﬁnitesimals of ﬁrst order. The relation

µ λν µλ Λ νΛ = g

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 µνα + P µα ,M µν = −M νµ. 2 µν µ

Lie’s theorem asserts that such operators Xr satisfy

X t [Xr,Xs] = crs Xt, t t where the coefﬁcients crs are called the structure constants of the group. This relation takes the form [P µ,P ν] = 0, h i −i M µν,P λ = gµνP ν − gνλP µ, −i [M µν,M ρσ] = gµρM νσ − gµσM νρ + gνσM µρ − gνρM µσ. Poincare´ group 0 The transformed ﬁeld components φr under the opertator U may be written

0 −1 φr = UφrU = (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] . s The infinitesimal part of the transformation matrix Sr : 1 S s = δ s + Σ s, Σ s = Σ sµνa , r r r r 2 r µν sµν sνµ where the coefficients Σr = −Σr . The increment of φr becomes 1 δφ (x) = [Σ sµνφ (x) + (xµ∂ν − xν∂µ) φ (x)] α −∂µφ (x)α . r 2 r s r µν r µ Momentum operators We obtain the defining relations for the momentum operators: µν sµν µ ν ν µ i [M , φr(x)] = Σr φs(x) + (x ∂ − x ∂ ) φr(x), µ µ i [P , φr(x)] = −∂ φr(x). Under an infinitesimal Lorentz transfromation the plane σ suffers a displacement: µ µ ν ν δx = α νx + α ,

and the ﬁeld at the displaced point x + δx is φr + δφr, with 1 δφ (x) = Σ sµνφ (x)α . r 2 r s µν The generating operator F (σ) yields Z µν ρ 1 rµ sνρ F (σ) = T (ανρx + αν) − π Σr φsανρ dσµ, σ 2 where rµ ∂L µν rµ ν µν π = ,T = π ∂ φr − g L. ∂ (∂µφr) Momentum operators We write F (σ) in the form 1 F (σ) = M µνα + P µα , 2 µν µ Z µν ρµ ν ρν µ rρ sµν M = (T x − T x − π Σr φs) dσρ σ Z µ ρµ P = T 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 δ φ = [Σ sµνφ + (xµ∂ν − xν∂µ) φ ] α − ∂µφ α 0 r 2 r s r µν r µ = i [F (σ), φr] ,

which must hold for arbitrary values of the ten parameters αµν and αµ. Momentum operators

So we obtain the set of equations

µν sµν µ ν ν µ i [M , φr] = Σr φs + (x ∂ − x ∂ ) φr, µ µ i [P , φr] = −∂ 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 (total angular momentum), Z µν ρµ ν ρν µ L = (T x − T x ) dσρ (orbital angular momentum), σ Z µν rρ sµν N = − π Σr φsdσρ (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 4 µ µ ∂µf d x = − f dσµ + f dσµ = 0, σ0 σ σ0 if the conservation law holds, so that we have

µ ∂µf = 0.

Applying this results to the integrands of M µν and P µ, we obtain

µν νµ ρµν T − T + ∂ρH = 0, µν ∂µT = 0, ρµν rρ sµν ρνµ with H = π Σr φs = −H . Conservation laws

Deﬁning the symmetrical momentum tensor

µν µν ρµν Θ = T + ∂ρG ,

where 1 G = (H + H + H ) , ρµν 2 ρµν µνρ νµρ we obtain the following tensor properties

Θµν = Θνµ, Z ν µν P = Θ dσµ, σ µ ∂ Θµν = 0. Commutation rules The generating operator is given by Z rµ F (σ) = − π δφrdσµ, 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} .

sµ So that for the three operators φr(x), π (y), and δφs(y), we have two possibilities

sµ s µ (a) [φr(x), δφs(y)] = 0, [φr(x), π (y)] = −δr δ (x, y), sµ s µ (b) {φr(x), δφs(y)} = 0, {φr(x), π (y)} = −δr δ (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 1 1 1 L = φ˙2 − (∇φ)2 − m2φ2 2 2 2 1 1 = (∂ φ)2 − m2φ2, 2 µ 2 we obtain the Klein-Gordon equation

∂2 − ∇2 + m2 φ = 0, ∂µ∂ + m2 φ = 0, or + m2 φ = 0. ∂t2 µ

Noting that the conjugate to φ(x) is π(x) = φ˙(x), we can construct the Hamiltonian: Z Z 1 1 1 H = d3xH = d3x π2 + (∇φ)2 + m2φ2 . 2 2 2 The Klein-Gordon attempts

In the momentum space representation, the Klein-Gordon ﬁeld is expanded as

Z d3p φ (x, t) = eip·xφ (p, t) (2π)3

so that the Klein-Gordon equation will be

∂2 ∂2 + |p|2 + m2 φ (p, t) = 0 or + ω 2 φ (p, t) = 0, ∂t2 ∂t2 p

which is a simple harmonic oscillator (SHO) equation which can be easily solved by introducing the annihilation and creation operators such that

h †i 3 (3) 0 ap, ap0 = (2π) δ p − p . The Klein-Gordon attempts

We will expand the ﬁeld φ (x) and π (x) in terms of the annihilation and creation operators as

Z 3 d p 1 † ip·x φ (x) = 3 p ap + a−p e ; (2π) 2ωp Z 3 r d p ωp † ip·x π (x) = (i) ap − a−p e . (2π)3 2

These expansions yield the ﬁeld commutator relation

Z 3 3 0 r 0 d pd p −i ωp0 φ (x) , π x = 6 (2π) 2 ωp h † i h †i i(p·x+p0·x0) × a−p , ap0 − ap, ap0 e = iδ(3) x − x0 . The Klein-Gordon attempts Then the Hamiltonian will be   Z 3 d p  † 1 h †i H = ωp ap ap + ap, ap  . (2π)3  2  | {z } =0 The total momentum operator is written as Z Z 3 3 d p † P = − d xπ (x) ∇φ (x) = pap ap. (2π)3

† ⇒ The operator ap creates momenum p and energy p 2 ωp = |p| + m . † † ⇒ The state ap aq · · · |0i has momentum p + q + ··· . ⇒ We call these excitations particles. q 2 2 ⇒ We will refer to ωp as Ep = + |p| + m , since it is the positive energy of the particle. The Klein-Gordon attempts

† The one-particle state |pi ∝ ap |0i is normalized with the 0 Lorentz invariance with a boost p i = γ (pi + βE) and 0 p 2 E = γ (E + βpi), where β = vi/c and γ = 1 − β :

dp0 δ(3) (p − q) = δ(3) p0 − q0 i dpi dE = δ(3) p0 − q0 γ 1 + β dpi γ = δ(3) p0 − q0 (E + βp ) E i E0 = δ(3) p0 − q0 . E We deﬁne

p † 3 (3) |pi ≡ 2Epap |0i → hp|qi = 2Ep (2π) δ (p − q) . Time-evolution of the Klein-Gordon ﬁelds The Heisenberg picture of the ﬁelds φ(x) = φ (x, t) = eiHtφ (x) e−iHt π(x) = π (x, t) = eiHtπ (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

H iHt −iHt −iEpt H† iHt † −iHt † iEpt ap ≡ e ape = ape , ap ≡ e ap e = ap e .

H H† † Omitting the superscript H, ap → ap and ap → ap , the ﬁelds are expanded by the operators: Z 3 d p 1 −ip·x † ip·x φ (x, t) = ape + ap e ; 3 p 0 (2π) 2Ep p =Ep ∂ 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 − m2φ2 2 2 2 has resolved the relativistic inconsistency of the Schrodinger¨ equation. † I However, the quantization a, a = 1 ; electron. I Dirac (1928) suggested another Lagrangian:

µ † 0 L = ψ¯ i∂/ − m ψ = ψ¯ (iγ ∂µ − m) ψ. (ψ¯ ≡ ψ γ )

† I The canonical momentum conjugate to ψ is iψ , I and thus the Hamiltonian is Z Z H = d3xψ¯ (−iγ · ∇ + m) ψ = d3xψ† −iγ0γ · ∇ + mγ0 ψ Z = d3xψ¯ [−iα · ∇ + mβ] ψ. α ≡ γ0γ, β ≡ γ0 Dirac matrices The Dirac matrices follows the algebra i {γµ, γν} ≡ γµγν + γνγµ = 2gµν × 1 → Sµν = [γµ, γν] . 4×4 4 Deﬁne i γ5 ≡ iγ0γ1γ2γ3 = − µνρσγ γ γ γ . 4 µ ν ρ σ There are 5 standard classes of the γ-matrices 1 scalar 1 γµ vector 4 µν i µ ν σ = 2 [γ , γ ] tensor 6 γµγ5 pseudo-vector 4 γ5 pseudo-scalar 1 16 Explicitly, we have 0 1 0 σi −1 0 γ0 = ; γi = ; γ5 = . 1 0 −σi 0 0 1 Dirac spinor Pauli spin matrices are deﬁned by the Dirac algebra

γj ≡ iσj ⇒ γi, γj = −2δij.

The Lorentz algebra are then 1 Sij = ijkσk, 2 the two-dimensional representation of the rotation group. The boost and rotation generators are

i i σi 0 S0i = γ0, γi = − , 4 2 0 −σi i 1 σk 0 1 Sij = γi, γj = − ijk ≡ ijkΣk, 4 2 0 σk 2

which transform the four-component ﬁeld ψ, a Dirac spinor. Dirac equation

The action principle yields the Dirac equation

µ (iγ ∂µ − m) ψ(x) = 0.

This implies the Klein-Gordon equation shown by acting µ (−iγ ∂µ − m) on the left

µ ν µ ν 2 (−iγ ∂µ − m)(iγ ∂ν − m) ψ = γ γ ∂µ∂ν + m ψ 1 = {γµ, γν} ∂ ∂ + m2 ψ 2 µ ν = ∂2 + m2 ψ = 0.

Since the canonical momentum conjugate to ψ is iψ†, the Hermitian conjugate form of the Dirac equation is

µ −i∂µψγ¯ − mψ¯ = 0. 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

µ 2 2 0 (γ pµ − m) u(p) = p/ − m u(p) = 0, p = m , p > 0, µ 2 2 0 (γ pµ + m) v(p) = p/ + m v(p) = 0, p = m , p > 0.

In the rest frame, with σµ ≡ (1, σ) and σ¯µ = (1, −σ), the column vectors u(p) and v(p) are in the form √ √ p · σξs p · σηs us(p) = √ , vs(p) = √ , s = 1, 2, p · σξ¯ s − p · ση¯ s

where ξs and ηs are the bases of the two-component spinors. Spin sums The solutions are normalized accoring to

r s rs s† s rs u¯ (p)u (p) = +2mδ , u (p)u (p) = +2Epδ , r s rs s† s rs v¯ (p)v (p) = −2mδ , v (p)v (p) = +2Epδ .

The u’s and v’s are orthogonal to each other:

u¯r(p)vs(p) =v ¯r(p)us(p) = 0,

but ur† (p) vs (p) = vr† (−p) us (p) = 0. Then the completeness relations are

X s s µ u (p)¯u (p) = γ · p + m = γ pµ + m = p/ + m, s X s s µ v (p)¯v (p) = γ · p − m = γ pµ − m = p/ − m. s The quantized Dirac ﬁeld The Dirac ﬁeld operators are expanded by plane waves

Z 3 d p 1 X s s −ip·x s † s ip·x ψ(x) = 3 p apu (p)e + bp v (p)e ; (2π) 2Ep s Z 3 ¯ d p 1 X s s −ip·x s † s ip·x ψ(x) = 3 p bpv¯ (p)e + ap u¯ (p)e , (2π) 2Ep s where the creation and annihilation operators obey the anticommutation relations

n r s †o n r s †o 3 (3) rs ap, aq = bp, bq = (2π) δ (p − q) δ .

The equal-time anticommutation relations for ψ and ψ† are then

n † o (3) ψa (x) , ψb (y) = δ (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

s s ap |0i = bp |0i = 0.

The Hamiltonian, with dropping the inﬁnities, are written

Z 3 d p X s † s s † s H = 3 Ep ap ap + bp bp . (2π) s The momentum operator is

Z Z d3p X P = d3xψ† (−i∇) ψ = p as †as + bs †bs . (2π3) p p p p s

s † s † Thus both ap and bp create particles with energy +Ep and p s † momentum p. The one-particle states |p, si ≡ 2Epap |0i is 3 (3) rs defined so that hp, r|q, si = 2Ep (2π) δ (p − q) δ is Lorentz invariant. Conservations of the Dirac field The Dirac field transforms according to 0 −1 ψ(x) → ψ (x) = Λ 1 ψ Λ x . 2 The change in the field at a fixed point is i µν i 3 (Λ 1 ' i − ωµνS = 1 − θΣ , i.e. infinitesimal rotation angle θ 2 2 2 about z-axis) 0 −1 δψ = ψ (x) − ψ(x) = Λ 1 ψ Λ x − ψ(x) 2 i = 1 − θΣ3 ψ (t, x + θy, y − θx, z) − ψ(x) 2 i = −θ x∂ + y∂ + Σ3 ψ(x) ≡ θ∆ψ. y x 2 The conserved Noether currents are 0 ∂L 0 i 3 j = ∆ψ = −iψγ¯ x∂y − y∂x + Σ ψ, ∂ (∂0ψ) 2 Z 1 J = d3xψ† x × (−i∇) + Σ ψ. 2 1 Spin-2 Dirac field At t = 0, for simplicity,

Z Z 3 3 0 3 d pd p 1 −ip0·x ip·x Jz = d x 6 p e e (2π) 2Ep2Ep0 X r0 † r0 0 r0 r0 † 0 × ap0 u p + b−p0 v −p r,r0 Σ3 × ar ur (p) + br †vr (−p) . 2 p −p

s† The commutator rules for a0 yields

1 X Σ3 X σ3 J as† |0i = us†(0) ur(0) ar† |0i = ξs† ξr ar |0i ; z 0 2m 2 0 2 0 r r

1 the eigenvalues of Jz are ± 2 . 1 ⇒ The Dirac ﬁeld conveys spin- 2 . Conserved quantities of the Dirac ﬁeld

µ µ I A current j (x) = ψ¯(x)γ ψ(x) is conserved by the Dirac equation

µ µ µ ∂µj = ∂µψ¯ γ ψ + ψγ¯ ∂µψ = imψ¯ ψ + ψ¯ (−imψ) = 0.

I The charge associated with this current is

Z 3 d p X s † s s † s Q = 3 ap ap − bp bp (2π) s is conserved: there is a unit charge e. µ5 µ 5 I An axial vector current j (x) = ψ¯(x)γ γ ψ(x) is conserved µ5 5 ∂µj = 2imψγ¯ ψ. 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.

5 µ µ 5 µν ψψ¯ iψγ¯ ψ ψγ¯ ψ ψγ¯ γ ψ ψσ¯ ψ ∂µ

P +1 −1 (−1)µ −(−1)µ (−1)µ(−1)ν (−1)µ T +1 −1 (−1)µ (−1)µ −(−1)µ(−1)ν −(−1)µ C +1 +1 −1 +1 −1 +1 CPT +1 +1 −1 −1 +1 −1

µ I The free Dirac Lagrangian L0 = ψ¯ (iγ ∂µ − m) ψ is invariant under C, P , and T separately.

I The perturbation δL must be a Lorentz scalar.

I All Lorentz scalar combinations of ψ¯ and ψ are invariant under the combined symmetry CPT . 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 3 H ? 0 d p 1 −ip·(x−y) = h0|φ(x)φ(y)|0i = D (x − y) = 3 e . y x (2π) 2Ep

2 I When x − y is purely in the time direction, (x − y) > 0; x0 − y0 = t, x − y = 0:

Z ∞ 2 √ 0 4π p −i p2+m2t D (x − y) = 3 dp p e (2π) 0 2 p2 + m2 Z ∞ 1 p 2 2 −iEt = 2 dE E − m e 4π m ∼ e−imt. t→∞ Propagators and causality

2 I When x − y is purely spatial direction, (x − y) < 0; x0 − y0 = 0, x − y = r:

Z 3 0 d p 1 ip·r D (x − y) = 3 e (2π) 2Ep 2π Z ∞ p2 eipr − e−ipr = 3 dp (2π) 0 2Ep ipr −i Z ∞ peipr = 2 dpp 2 (2π) r −∞ p2 + m2 ∼ e−mr. 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 3 d p 1 −ip·(x−y) ip·(x−y) = 3 e − e (2π) 2Ep preserves causality under the Lorentz transformation by taking (x − y) → −(x − y) on the second term to cancel each other. I The amplitude integral can convey the frequency integral through the residue theorem: 0 0 0 0 I dp e−ip0(x −y ) Z dp e−ip0(x −y ) 0 = ℘ 0 2 2 2 2 2π p − m 2π p0 − Ep = −2πi Res| + Res| p0=+Ep p0=−Ep

1 0 0 0 0 = −i e−iEp(x −y ) − e+iEp(x −y ) . 2Ep Propagators and causality The amplitude for x0 > y0 is then Z 3 Z 0 d p dp −1 −ip·(x−y) h0|[φ(x), φ(y)]|0i = 3 e . x0>y0 (2π) 2πi p2 − m2 Let us define a function 0 0 DR(x − y) ≡ θ x − y h0|[φ(x), φ(y)]|0i , which satisfies the Klein-Gordon Green’s function equation 2 2 (4) ∂ + m DR(x − y) = −iδ (x − y), 2 2 or −p + m D˜R(p) = −i. which is known as the retarded Green’s function, explicitly Z 4 d p i −ip·(x−y) DR(x − y) = e . (2π)4 p2 − m2 For x0 < y0, we have the advanced Green’s function 0 0 DA(x − y) = θ y − x h0|[φ(x), φ(y)]|0i = −DR(x − y). Propagators and causality Let us define the Klein-Gordon Feynman propagator as

0 ˆ DF (x − y) ≡ h0|T {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−y0) Z d4p i = e−ip·(x−y). (2π)4 p2 − m2 + i

where Tˆ {· · · } is the time-ordering operator, and the integrand 0 + of the last line has the poles p = ± (Ep − i), for → 0 . Similiarly, we can also deﬁne the Dirac Feynman propagator as

0 ˆ ¯ SF (x − y) ≡ h0|T ψ(x)ψ(y) |0i Z d4p 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 We have understood the spin and dynamics of electron as a free Dirac ﬁeld. I However, a free particle is not measurable so we need interaction to really observe it. I An electron is subjected to the electromagnetic interaction with the Lagrangian such that

LQED = LDirac + LMaxwell + Lint 1 = ψ¯ i∂/ − m ψ + F µνF − e ψγ¯ µψA , 0 4 µν 0 µ

where Aµ is the electromagnetic vector potential, Fµν = ∂µAν − ∂νAµ is the skew-symmetrical electromagnetic ﬁeld tensor, and e0 < 0 is the electron charge. I Introducing Dµ ≡ ∂µ + ie0Aµ we have a simpler form 1 L = ψ¯ iD/ − m ψ − F µνF . QED 0 4 µν Electromagnetic interaction

I The QED Lagrangian is invariant under the gauge transformations iα(x) ψ(x) → e ψ(x),Aµ(x) → Aµ(x) + ∂µα(x).

I The equation of motion for ψ is iD/ − m0 ψ(x) = 0, which is jus the Dirac equation coupled to the electromagnetic field. I The equation of motion for Aν is µν ν ν ∂µF = e0ψγ¯ ψ = e0j , which is the inhomogeneous Maxwell equations, with the current density jν = ψγ¯ νψ. I The quantization of Aµ fields are depending on the choice of gauges, such as the Coulomb gauge ∇ · A = 0 or the µ Lorentz gauge ∂µA = 0. Maxwell field In the relativistic notations, the maxwell field is defined by F µν = ∂µAν − ∂νAν, jµ = (ρ, j) ∂A ⇒ E = −∇A0 − , B = ∇ × A ∂t and written as  0 −E1 −E2 −E3  1 3 2 µν νµ  E 0 −B B  F = −F =   .  E2 B3 0 −B1  E3 −B2 B1 0 The four-vector potential Aµ does not determined uniquely for a gauge transformation Aµ(x) → Aµ(x) + ∂µα(x), but it yields the Lorentz invariant Maxwell equation µ µ µ A − ∂ (∂ · A) = j . Radiation field

I We modify the Lagrangian LMaxwell to 1 1 L = − F µνF − ξ (∂ · A)2 Maxwell 4 µν 2 so that the Maxwell’s equation are replace by

Aµ − (1 − ξ)∂µ (∂ · A) = 0

µ and the conjugate momenta π to Aµ are ∂L πρ = Maxwell = F µ0 − ξgµ0 (∂ · A) , ∂ (∂0Aµ)

where (∂ · A) is a scalar ﬁeld such that (∂ · A) = 0. I For radiation ﬁeld, we conveniently choose ξ = 1 (Feynman gauge) to yield the Maxwell equation

µ A = 0. Dixitque Deus,

et facta est lux. – Genesis 1:3 Radiation ﬁeld µ The solutions of A = 0 are the plane waves: Z d3k 1 Aµ(x) = 3 √ (2π) 2Ek 3 X h (λ) (λ) −ik·x (λ)† (λ)∗ +ik·xi × a (k)εµ (k)e + a (k)εµ (k)e , λ=0 where ε(λ) are the bases of polarization vectors, which satisﬁes

(λ) (λ)∗ X εµ (k)εν (k) (λ) (λ0)∗ λλ0 = gµν, ε (k) · ε (k) = g . ε(λ)(k) · ε(λ)∗(k) λ Real photons convey only the transverse polarizations εµ = (0, ε), where k · ε = 0. For k k zˆ, the right- and left-handed 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 (3) A˙ µ(x),Aν(y) = igµνδ (x − y).

We can deﬁne the photon Feynman propagator as

0 ˆ igµν∆F (x − y) ≡ h0|T [Aµ(x)Aν(y)]|0i Z d4k −ig = µν e−ik·(x−y) (2π)4 k2 + i Z d4k −ig 1 − ξ −ik k (arbitrary ξ) ⇒ µν + µ ν e−ik·(x−y). (2π)4 k2 + i ξ (k2 + i)2

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

a → 1 + 2 + ··· + nf (eg., Ne2P4 → Neg.s. + γ). The transition probability per unit time is |S |2 w = fi . f←i T In a cubic box of volume V = L3 with inﬁnitely high potential well, the differential transition probability is 3 1 1 (4) 2 Y d pf dwf←i = 3n −4 δ (pf − pa) |Mfi| . f 2Ea 2Ef (2π) f −1 The lifetime τa = Γa is the inverse of the total decay width X X Γa = Γ = w a→{nf } {nf }←a nf nf Z 3 3 1 1 d p1 d pnf (4) 2 = ··· δ (pf − pi) |Mfi| . 3nf −4 2Ea (2π) 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 4 (4) 2 w¯f←i = lim = (2π) δ (Pi − Pf ) |Mfi| . V →∞ V The differential cross sectin (in Lab.) is defined as the transition rate density per target density (nt) per incident flux (F )

nf 3 0 w¯f←i Y d pf dσfi = 3 . ntF (2π) 2ω 0 f=1 pf

The target density nt = 2ωp2 and the ﬂux F = 2ωp1 vrel yield

nf 3 0 1 Y d pf 4 (4) 2 dσfi = 3 (2π) δ (Pi − Pf ) |Mfi| . 2ωp 2ωp vrel (2π) 2ω 0 1 2 f=1 pf Interaction picture

I Let |Ωi be the ground state of the interacting theory. R 3 R 3 I Let Hint(t) = d xHint = − d xLint be the interacting Hamiltonian and H = H0 + λHint with 0 ≤ λ ≤ 1. iHt −iHt I Let φ(x) = e φ (x) e be an Heisenberg picture field iH(t−t ) −iH(t−t ) and for t 6= t0, φ (t, x) = e 0 φ (t0, x) e 0 . I For λ = 0, H becomes H0 and we can define an interaction picture field as

iH0(t−t0) −iH0(t−t0) φ (t, x)|λ=0 = e φ (t0, x) e ≡ φI (t, x) .

I The full Heisenberg picture ﬁeld φ in terms of φI : n o iH(t−t0) iH0(t−t0) −iH0(t−t0) −iH(t−t0) φ (t, x) = e e φI (t, x) e e † ≡ U (t, t0) φI (t, x) U (t, t0) , where we have deﬁned the unitary operator

iH0(t−t0) −iH(t−t0) U (t, t0) ≡ e e . Unitary time-evolution operator

I The initial condition is U (t0, t0) = 1. I The Schrodinger¨ equation: ∂ i U (t, t ) = eiH0(t−t0) (H − H ) e−itH(t−t0) ∂t 0 0 iH0(t−t0) −itH(t−t0) = e (Hint) e z }| { iH0(t−t0) −itH0(t−t0) itH0(t−t0) −itH(t−t0) = e (Hint) e e e | {z } = HI (t)U (t, t0) .

I We expand U ∼ exp (−iHI t) as a power series in λ: Z t U (t, t0) = 1 + (−i) dt1HI (t1) t0 (−i)2 Z t Z t + dt1 dt2Tˆ [HI (t1) HI (t2)] + ··· 2 t0 t0 Z t 0 0 ≡ Tˆ exp −i dt HI t . t0 Interacting ground state

I The interacting ground state |Ωi is not |0i; hΩ|0i= 6 0. I E0 ≡ hΩ|H |Ωi with the zero of energy H0 |0i = 0. I When H |ni = En |ni, X e−iHT |0i = e−iEnT |ni hn|0i n X = E−iE0T |Ωi hΩ|0i + e−iEnT |ni hn|0i. n6=0 | {z } →0

I Since En > E0 for all n 6= 0, as we send T → ∞ (1 − i) −1 |Ωi = lim e−iE0T hΩ|0i e−iHT |0i . T →∞(1−i)

The normalization of the interacting ground state is

−1 2 −iE0(2T ) 1 = hΩ|Ωi = |h0|Ωi| e h0|U (T, t0) U (t0,T )|0i .

Now we have the two-point correlation function:

hΩ|Tˆ {φ(x)φ(y)}|Ωi ˆ n h R T io h0|T φI (x)φI (y) exp −i −T dtHI (t) |0i = lim n h io T →∞(1−i) ˆ R T h0|T 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 3 Z 3 + d p 1 −ip·x − d p 1 † +ip·x φ (x) = 3 p ape , φ (x) = 3 p ap e . (2π) 2Ep (2π) 2Ep These decomposed ﬁelds satisfy φ+(x) |0i = 0 and h0| φ−(x) = 0. A normal ordring operator Nˆ is deﬁned as

† † + − − + Nˆ apak ≡ ak ap ⇒ Nˆ φ (x)φ (y) = φ (y)φ (x).

† † † So we, with ak , ap ± = ak ap ± apak , have identities h0|Nˆ {φ(x)φ(y)}|0i = 0

Tˆ {φ(x)φ(y)} = Nˆ {φ(x)φ(y)} + φ(x)φ(y). 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 > y0; φ(x)φ(y) ≡ [φ+(y), φ−(x)] , for y0 > x0.

h0|Tˆ (φ(x)φ(y))|0i = h0|Nˆ (φ(x)φ(y))|0i + h0|φ(x)φ(y)|0i | {z } =0

φ(x)φ(y) = DF (x − y) = h

µ ν µν A (x)A (x) = ∆F (x − y) = g µ ν I The contraction for the Dirac ﬁeld is deﬁned by ψ+(x), ψ¯−(y) , for x0 > y0; ψ(x)ψ¯(y) ≡ ψ¯+(y), ψ−(x) , for y0 > x0.

h0|Tˆ ψ(x)ψ¯(y)|0i = h0|Nˆ ψ(x)ψ¯(y)|0i + h0|ψ(x)ψ¯(y)|0i ¯ ψ(x)ψ(y) = SF (x − y) = F Wick’s theorem and connected diagrams

I For n ﬁeld operators, we have an identity

Tˆ [φ (x1) φ (x2) ··· φ (xn)] = Nˆ [φ (x1) φ (x2) ··· φ (xn)] + {all possible contractions} .

which is knwon as Wick’s theorem. I The two-point correlation has the structure Numerator hΩ|Tˆ {φ(x)φ(y)}|Ωi = , Denominator Numerator = f + fcf + ··· x y x y connected × exp c + cc + ··· Denominator = exp c + cc + ··· , ⇓ hΩ|Tˆ {φ(x)φ(y)}|Ωi = f + fcf + ··· . x y x y 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 e−iHT |i, 0i . T →∞(1−i) The S-matrix will be of the form lim hf, 0|e−iH(2T ) |i, 0i T →∞(1−i) Z T ∝ lim hf, 0|Tˆ exp −i dtHI (t) |i, 0i T →∞(1−i) −T Then the τ-matrix (cf., S = 1 + iτ) elements becomes 4 (4) hf, out|iτ |i, outi = (2π) δ (Pi − Pf ) · iM (i → f) Z T = lim hf, 0|Tˆ 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 0 0 p1 p2

µ ν −ie0γ −ie0γ iM = k

p1 p2 −ig =u ¯ p 0 (−ie γµ) u (p ) µν u¯ p 0 (−ie γν) u (p ) 1 0 1 k2 2 0 2 −ig = (−ie )2 u¯ p 0 γµu (p ) µν u¯ p 0 γνu (p ) . 0 1 1 k2 2 2 This is known as the ﬁrst part of the Møller scattering. Bhabha scattering The Bhabha scattering is a deformed Møller scattering: 0 0 p1 p2

µ ν iM = k

p1 p2 −ig =u ¯ p 0 (−ie γµ) u (p ) µν v p 0 (−ie γν)v ¯ (p ) 1 0 1 k2 2 0 2 −ig = (−ie )2 u¯ p 0 γµu (p ) µν v p 0 γνv¯ (p ) . 0 1 1 k2 2 2

−1 I The electrons-2 travels in reverse-time order T . −1 I The CPT symmetry → (CP ) . I The negative-energy electron is known as positron. Compton scattering

The Compton scattering contains two diagrams,

p0 k0 p0 k0 µ p − k0 p + k µ ν iM = + ν k p k p / 0 ∗ 0 µ i p/ + k + m0 ν =u ¯ p µ k (−ie0γ ) (−ie0γ ) ν (k) u (p) 2 2 (p + k) − m0 /0 0 ν i p/ − k + m0 ∗ 0 µ +¯u p (−ie0γ ) ν (k) µ k (−ie0γ ) u (p) . 2 2 (p − k) − m0 Compton scattering The numerators and denominators can be simpliﬁed as follows:

2 2 2 I Since p = m0 and k = 0, the denominators are 2 2 (p + k) − m0 = 2p · k in iM1 02 2 p − k − m0 = −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) ν ν = 2p u(p) − γ p/ − m0 u(p) = 2pνu(p), We obtain 2 ∗ 0 iM = ie0 µ k ν (k) " # γµkγ/ ν + 2γµpν −γνk/0γµ + 2γνpν ×u¯ p0 + u(p). 2p · k −2p · k0 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 +Ze = +

p p

For the soft photon radiation, |k| |p0 − p|,

0 0 0 M0 p , p − k ≈ M0 p + k, p ≈ M0 p , p p0 · ∗ p · ∗ iM = −ie u¯ p0 M p0, p u(p) e − . 0 0 0 p0 · k p · k Soft Bremsstrahlung The differential cross section is then dσ p → p0 + γ = dσ p → p0 2 Z d3k 1 X p0 · (λ) p · (λ) × e 2 − . 3 2k 0 p0 · k p · k (2π) λ=1,2 The differential probability becomes 3 2 0 2 0 d k X e0 p p dP p → p + γ(k) = λ · − . 3 2k p0 · k p · k (2π) λ The total probability, for soft photons 0 ≤ k ≤ |q| = |p0 − p|, and the differential cross section with ﬁctitious photon mass µ, are Z |q| 2 1 0 −q P ≈ dk I v, v ∼ log( 2 ) → ∞, 0 k µ 2 2 0 α0 −q −q dσ p → p + γ ∝ log 2 log 2 . −q2→∞ π µ m0 Radiative corrections

+ + + ···

There are three classes of the radiative corrections; Vertex corrections, Self-energies, and Polarizations. Vertex corrections

iM =

00 0 0 µ 0 ˜cl 0 iM (2π) δ p − p = −ie0 u¯ p Γ p , p u(p) · Aµ p − p .

To lowest order, Γµ = γµ. We may express Γµ in a symmetrical form:

Γµ = γµA + p0µ + pµ B + p0µ − pµ C. Vertex corrections Using the Gordon identity

p0µ + pµ iσµνq u¯ p0 γµu(p) =u ¯ p0 + ν u(p), 2m0 2m0

we have µν µ 0 µ 2 σ qν 2 Γ p , p = γ F1 q + i F2 q , 2m0

where the unknown functions F1 and F2 are called form factors. To lowest order, F1 = 1 and F2 = 0. ˜cl 0 I When Aµ (x) = (2π) δ q φ (q) , 0 ,

0 0 0 iM = −ie0u¯ p Γ p , p u(p) · φ˜ (q) 0† = −ie0F1(0)φ˜ (q) · 2m0ξ ξ q→0

V (x) = e0F1(0)φ (x) . Vertex corrections cl cl I When Aµ (x) = 0, A (x) , iν 0 i iσ qν ˜i iM = +ie0 u¯ p γ F1 + F2 u(p) Acl (q) . 2m0 Again the expression in brackets vanishes at q = 0, so in this limit 0† −1 k k iM = −i (2m0) · e0ξ σ [F1(0) + F2(0)] ξB˜ (q) , 2m0 ˜k ijk i ˜j where B (q) = −i q Acl (q). This is just that of a magnetic moment interaction V (x) = − hµi · B (x), where

e0 0† σ hµi = [F1(0) + F2(0)] ξ ξ, m0 2 e µ = g 0 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 = γµ + δΓµ: p0

k0 = k + q

p − k δΓµ = k q

p where

u¯ p0 δΓµ p0, p u(p) = Z 4 0 µ 0 2 µ 0 µ 2 d k u¯ (p ) kγ/ k/ + m0 γ − 2m0 (k + k ) u(p) 2ie0 4 . 2 02 2 2 2 (2π) (k − p) + i k − m0 + i (k − m0 + i) Vertex corrections The Feynman parameterization

1 Z 1 1 = dx 2 AB 0 [xA + (1 − x)B] Z 1 1 = dxdyδ (x + y − 1) 2 0 [xA + yB]

simpliﬁes the denominator D to

D = l2 − ∆ + i l ≡ k + yq − zp, 2 2 2 ∆ ≡ −xyq + (1 − z) m0 > 0.

The numerator will be " µ 1 2 2 2 2 # γ · − 2 l + (1 − x)(1 − y)q + 1 − 2z − z m0 0 0µ µ N =u ¯ p + p + p · m0z(z − 1) u(p). µ +q · 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 4 Z 1 2 d l 2 2ie0 4 dxdydzδ (x + y + z − 1) 3 (2π) 0 D 1 ×u¯ p0 γµ · − l2 + (1 − x)(1 − y)q2 + 1 − 4z + z2 m 2 2 0 µν iσ qν 2 + 2m0 z(1 − z) u(p). 2m0

There are two classes of integrations:

Z d4l 1 Z d4l l2 n and 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: 1 1 1 → − (k2 − p2) + i (k2 − p2) + i (k2 − p2) − Λ2 + i

so the denominator is altered as

2 2 2 2 ∆ → ∆Λ = −xyq + (1 − z) m0 + zΛ .

Then the divergent integral is replaced by convergent ones:

Z d4l l2 l2 i ∆ − = log Λ , 4 2 3 2 3 2 (2π) (l − ∆) (l − ∆Λ) (4π) ∆

which looks like (∞ − ∞Λ) ∝ log (∆Λ/∆). 2 2 I How can this result affect on F1 q and F2 q with F1(0) = 1? The convergent form factor F2 2 The form factor F2 is corrected to order α0(= e0 /4π~c) Z 1 2 2 α0 2m0 z(1 − z) F2 q = dxdydzδ(x + y + z − 1) 2 2 2 2π 0 m0 (1 − z) − q xy

is convergent especially for q2 = 0, such that

Z 1 2 2 α0 2m0 z (1 − z) F2 q = 0 = dxdydzδ (x + y + z − 1) 2 2 2π 0 m0 (1 − z) α Z 1 Z 1−z z α = 0 dz dy = 0 . π 0 0 1 − z 2π We get a correction to the g-factor of the electron:

g − 2 α0 ae ≡ = ≈ 0.0011614. 2 2π (α0$α)

exp Experiments give ae = 0.0011597, which differs by ≈ 0.15 %. Infrared divergence 2 The divergent form factor F1 q is corrected to

Z 1 2 α0 F1 q = 1 + dxdydzδ (x + y + z − 1) 2π 0 " ! m 2 (1 − z)2 × log 0 2 2 2 m0 (1 − z) − q xy 2 2 2 m0 1 − 4z + z + q (1 − x) (1 − y) + 2 2 2 2 m0 (1 − z) − q xy + µ z 2 2 # m0 1 − 4z + z − , 2 2 2 m0 (1 − z) + µ z

where µ is the ﬁctitious photon mass. In the limit µ → 0, we may obtain

2 2 2 α0 −q −q F1 −q → ∞ = 1 − log 2 log 2 . 2π m0 µ What did we have made mistake?

I Accordingly we need to normalize |Ωi again: ⇒Renormalization. The particle dispersion

H multiparticle continuum

1-particle bound state in motion m 1-particle at rest P

The eigenvalues of P µ = (H, P) of particle mass m. Renormalization Assume x0 > y0 and drop off hΩ|φ(x)|Ωi hΩ|φ(y)|Ωi (= 0). The two-point correlation function is

iP ·x −iP ·x hΩ|φ(x)|λpi = hΩ|e φ(0)e |λpi −ip·x = hΩ|φ(0)|λpi e 0 p =Ep −ip·x = hΩ|φ(0)|λ0i e 0 . p =Ep

The two-point correlation function becomes for x0 > y0

Z 4 X d p i −ip·(x−y) 2 hΩ|φ(x)φ(y)|Ωi = 4 2 2 e |hΩ|φ(0)|λ0i| . p − mλ + i λ (2π) Kall¨ en-Lehmann´ representation For both cases of x0 > y0 and y0 > x0, we have

Z ∞ 2 dM 2 2 hΩ|Tˆ φ(x)φ(y)|Ωi = ρ M DF x − y; M , 0 2π

where ρ M 2 is a positive spectral density,

2 X 2 2 2 ρ M = (2π) δ M − mλ |hΩ|φ(0)|λ0i| . λ

2 1-particle ρ M states

bound states

2-particle states

m2 (2m)2 M 2 Field-strength renormalization The spectral density is

2 2 2 2 2 ρ M = 2πδ M − m Z + nothing else for M . (2m) ,

where Z is refered as ﬁeld-strength renormalization. The Fourier transform of the two-point correlation becomes

Z Z ∞ 2 4 ip·x ˆ dM 2 i d xe hΩ|T φ(x)φ(0)|Ωi = ρ M 2 2 0 2π p − M + i Z ∞ 2 iZ dM 2 i = 2 2 + ρ M 2 2 . p − m + i ∼(2m)2 2π p − M + i

2 m2 (2m)

isolated poles from branch cut pole bound states The electron self-energy

I The electron two-point correlation function is

hΩ|Tˆ ψ(x)ψ¯(y)|Ωi = ¦ + ¦fyf¦ x y x y

+ ¦fyfyf¦ + ··· x y

I The free-ﬁeld propagator:

i p + m ¦p / 0 = 2 2 . p − m0 + i

I The lowest order electron self-energy: p−k i p + m i p + m ¦fyf¦ / 0 / 0 = 2 2 [−iΣ2 (p)] 2 2 . p − m0 p − m0 The electron self-energy We have the explicit form of the electron self-energy:

Z 4 2 d k µ i (k/ + m0) −i −iΣ2 (p) = (−ie0) 2 γ 2 2 γµ 2 , (2π) k − m0 + i (p − k) − µ2 + i

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 4 2 d l −2xp/ + 4m0 −iΣ2 (p) = −e0 dx 3 2 , 0 (2π) (l2 − ∆ + i)

2 2 2 where ∆ = −x(1 − x)p + xµ + (1 − x)m0 . We regulate it by the Pauli-Villars procedure: 1 1 1 → − . (p − k)2 − µ2 + i (p − k)2 − µ2 + i (p − k)2 − Λ2 + i The electron self-energy

2 2 2 2 Introducing ∆Λ = −x(1 − x)p + xΛ (1 − x)m0 → xΛ , Λ→∞ we have Z 1 2 α0 xΛ Σ2 (p) = dx 2m0 − xp/ log 2 2 2 . 2π 0 (1 − x)m0 + xµ − x(1 − x)p The logarithm of x has a branch cut begining at the point where

2 2 (1 − x) m0 + xµ − x (1 − x) = 0, or 1 m 2 µ2 1 r x = + 0 − ± p2 − (m + µ)2 p2 − (m − µ)2 . 2 2p2 2p2 2p2 0 0

2 2 2 The branch cut of Σ2 p begins at p = (m0 + µ) , two-particle threshold. 2 2 I Where is the simple pole at p = m ? The electron self-energy The two-point correlation function is written as

= ¦ + ¦p¦ + ¦p¦p¦ + ··· i p/ + m0 i p/ + m0 i p/ + m0 = 2 2 + 2 2 (−iΣ) 2 2 p + m0 p + m0 p + m0 i p/ + m0 i p/ + m0 i p/ + m0 + 2 2 (−iΣ) 2 2 (−iΣ) 2 2 + ··· p + m0 p + m0 p + m0 ! !2 i i Σ p/ i Σ p/ = + + + ··· p/ − m0 p/ − m0 p/ − m0 p/ − m0 p/ − m0 i = . p/ − m0 − Σ p/ Hence p/ − m0 − Σ p/ = 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

  dΣ p/ 2 p/ − m · 1 −  + O p/ − m . dp/ p/=m

When we write the two-point correlation function as Z iZ2 p/ + m d4xeip·x hΩ|Tˆ ψ(x)ψ¯(0)|Ωi = , p2 − m2 + i

we obtain the mass renormalization constant to be −1 dΣ p/ Z2 = 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 2 α0 xΛ δm = m0 dx (1 − x) log , 2 2 2 2π 0 (1 − x) m0 + xµ 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 α Z 1 xΛ2 = 0 dx −x log 2 2 2 2π 0 (1 − x) m + xµ 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 iσµνq Z Γµ p0, p = γµF q2 + ν F q2 . 2 1 2m 2 The left-hand side of the exact vertex function becomes

µ µ µ µ µ µ Z2Γ = (1 + δZ2)(γ + δΓ ) = γ + δΓ + γ δZ2,

2 while in the right-hand side F1 q becomes

2 2 2 F1 q = 1 + δF1 q + δZ2 = 1 + δF1 q − δF1 (0) ,

if δZ2 = −δF1(0). Deﬁne another rescaling factor Z1 by the relation

µ −1 µ Γ (q = 0) = Z1 γ ,

where Γµ is the complete amputated vertex function. Mass renormalization However, the divergent part of the vertex correction is

Z 1 α0 δF1(0) = dxdydzδ (x + y + z − 1) 2π 0 " # zΛ2 1 − 4z + z2 m2 × log + (1 − z)2 m2 + zµ2 (1 − z)2 m2 + zµ2 α Z 1 = 0 dz (1 − z) 2π 0 " # zΛ2 1 − 4z + z2 m2 × 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 2 Photon is dressed in order e0 k + q

µν µ ν iΠ2 (q) ≡ q q k

Z 4 2 d k µ i ν i = (−1) (−ie0) tr γ γ . (2π)4 k/ − m k/ + /q − m

Generally the polarized photon propagator is deﬁned by q q iΠµν (q) ≡ gpg. µ ν As for the electron self-energy the polarization decomposes q q gpg = g + gcg + gcgcg + ··· µ ν Ward-Takahashi identity The guage invariance of radiation ﬁeld leads the charge µ conservation (qµM (q) = 0) in such a way that     p  p + q  p + q µ        qµ· q  = e0  −  .      p  p p + q

This identity is known as the Ward-Takahashi identity: µ −1 −1 −iqµΓ (p + q, p) = S (p + q) − S (p) .

We deﬁned Z1 and Z2 by the relations

µ −1 µ iZ2 Γ (p + q, p) → Z1 γ as q → 0 and S(p) ∼ . p/ − m Setting p near mass shell and expanding the Ward-Takahashi identity about q = 0, we ﬁnd −1 −1 −iZ1 /q = −iZ2 /q ⇒ Z1 = Z2. Charge renormalization

µν I The Ward-Takahashi identity tells us that qµΠ = 0. µν µν µ ν 2 I In other words, Π ∝ g − q q /q . µν I Furthermore, we can expect Π (q) will not have a pole at q2 = 0.

I It is convenient to write

Πµν (q) = q2gµν − qµqν Π q2 ,

where Π q2 is regular at q2 = 0. The exact photon two-point correlation function is −ig −ig −ig gpg = µν + µν i q2gρσ − qρqσ Π(q) σν + ··· µ ν q2 q2 q2 −ig −ig −ig = µν + µρ ∆ρΠ q2 + µρ ∆ρ ∆σΠ2 q2 + ··· , q2 q2 ν q2 σ ν

ρ ρ ρ 2 ρ σ ρ where ∆ν ≡ δν − q qν/q and ∆σ∆ν = ∆ν. Charge renormalization

I We can simplify further −i qµqν −i qµqν gpg = gµν − + µ ν q2 (1 − Π(q2)) q2 q2 q2 −ig = µν ( q Mµ(q) = 0) . q2 (1 − Π(q2)) ∵ µ

2 2 I As long as Π q is regular at q = 0, the exact propagator alway has a pole at q2 = 0.

I In other words, the photon remains absolutely massless at all orders in the perturbation theory. 2 I The residue of the q = 0 pole is 1 ≡ Z . 1 − Π(0) 3 Charge renormalization

I Since the scattering amplitude will be shifted by e 2g Z e 2g ··· 0 µν · · · → · · · 3 0 µν ··· , q2 q2 we will have the charge renormalization p e = Z3e0.

2 I Considering a scattering process with nonzero q in leading order α0, 2 2 igµν e0 −igµν e0 2 2 ≈ 2 2 . q 1 − Π(q ) q 1 − [Π2 (q ) − Π2 (0)]

I The quantity in (··· ) has an interpretation of a q2-dependent electric charge, so we have

2 2 e0 /4π α0 α0 → α q = 2 ≈ 2 . 1 − Π(q ) 1 − [Π2 (q ) − Π2 (0)] The divergent polarization Π2 2 In order e0 , the polarization is badly ultraviolet divergent:

Z 4 " # µν 2 d k µ i (k/ + m) ν i k/ + /q + m iΠ2 (q) = − (−ie0) tr γ γ (2π)4 k2 − m2 (k + q)2 − m2 Z 4 µ ν ν µ µν 2 2 d k k (k + q) + k (k + q) − g k · (k + q) − m = −4e0 3 . (2π) (k2 − m2) (k + q)2 − m2

Introducing a Feynman parameter, we combine the denominator as 1 Z 1 1 = dx 2 , (k2 − m2) (k + q)2 − m2 0 (l2 + x (1 − x) q2 − m2)

where l ≡ k + xq. In terms of l, the numerator will be

Numerator = 2lµlν − gµνl2 − 2x (1 − x) qµqν + + gµν m2 + x (1 − x) q2 + (terms linear in l) . Wick rotation

I The momentum (contour) integral in the Minkowski metric space-time, gµµ = (+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:

2 2 2 02 2 2 x = t − |x| → − x − |x| = − |xE| .

l0 l0

q q − |l|2 + ∆ + i − |l|2 + ∆ + i ⇒ q Wick q + |l|2 + ∆ − i + |l|2 + ∆ − i rotation Dimensional regularization

I For sufficiently small dimension d, any loop-momentum integral will converge. I Therefore the Ward identity can be proved. I The final expression for Π2 should have well-defined limit as d → 4. I A typical d-dimensional Euclidean space integral reads Z d Z Z ∞ d−1 d lE 1 dΩd lE = · dlE , 2 2 2 d 2 2 (2π) lE + ∆ (2π) 0 lE + ∆ where the area of a unit sphere in d dimensions is identified as √ Z 2 ( π)d dΩ = d d Γ 2 and the second factor of the integral becomes

d Z ∞ d−1 2− 2 d d l 1 1 Γ 2 − 2 Γ 2 dlE = . 2 2 0 lE + ∆ 2 ∆ Γ (2) Dimensional regularization

I Near d = 4, deﬁne = 4 − d, and use the approximation d 2 Γ 2 − = Γ = − γ + O () , 2 2 where γ ≈ 0.5772 is the Euler-Mascheroni constant. I The integral is then Z d4l 1 1 2 E → − log ∆ − γ + log (4π) + O () . 2 2 →0 2 (2π) lE + ∆ (4π) µν I In d dimensions, gµνg = d. µ ν I Thus, l l of the numerators in the integrands should be 1 2 µν replaced by d l g . I The Dirac matrices in d = 4 − should be modiﬁed to µ ν ν γ γ γµ = − (2 − ) γ µ ν ρ νρ ν ρ γ γ γ γµ = 4g − γ γ µ ν ρ σ σ ρ ν ν ρ σ γ γ γ γ γµ = −2γ γ γ + γ γ γ . Evaluation of Π2 The unpleasant terms with l2 in the numerator gives

d 2 µν 2 2− d Z d l − + 1 g lE 1 d 1 2 E d = Γ 2 − (−∆gµν) . 2 2 2 d/2 (2π) lE + ∆ (4π) 2 ∆

Evaluating remaining terms and using ∆ = m2 − x (1 − x) q2 are

µν 2 µν µ ν 2 iΠ2 (q) = q g − q q · iΠ2 q ,

where

8e 2 Z 1 Γ 2 − d Π q2 = − 0 dxx (1 − x) 2 2 d/2 2−d/2 (4π) 0 ∆ 2α Z 1 2 → − 0 dxx (1 − x) − log ∆ − γ + log (4π) . →0 π 0 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

2 2 e − e0 2α0 2 = δZ3 → Π2(0) ≈ − → ∞. 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 α α α q2 ≈ 0 ≡ 0 , 2 2 1 − [Π2 (q ) − Π2(0)] 1 − Πˆ 2 (q ) where the difference Z 1 2 ˆ 2 2α0 m Π2 q = − dxx (1 − x) log 2 2 , π 0 m − x (1 − x) q which is independent of in the limit → 0. Classical picture In nonrelativistic limit, the attractive Coulomb potential reads

Z d3q −e2 V (x) = eiq·x . 3 2 2 (2π) |q| 1 − Πˆ 2 − |q|

ˆ 2 2 Expanding Π2 for q m , we obtain

α 4α2 V (x) = − − δ(3) (x) r 15m2 2 Z ∞ iQr ie 1 Qe ˆ 2 = 2 dQ 2 2 1 + Π2 −Q . (2π) r −∞ Q + µ

−1 When r 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 −q2 m2, we have

Z 1 2 2α Πˆ 2 q ≈ dxx (1 − x) π 0 −q2 m2 × log + log (x (1 − x)) + O m2 q2 α −q2 5 m2 = log − + O . 3π m2 3 q2

The effective coupling constant in this limit is therefore

2 α αeﬀ q = ,A = exp (5/3) . α −q2 1 − 3π log Am2

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 L = ψ¯ i∂/ − m ψ − F F µν − e ψγ¯ µψA . 0 4 µν 0 µ The renormalization scheme modiﬁes the electron and photon propagators as iZ ¦p¦ = 2 + ··· p/ − m −iZ g gpg = 3 µν + ··· q2 1/2 To absorb Z2 and Z3 into L, we substitute ψ → Z2 ψ and µ 1/2 µ A → Z3 A . The Lagrangian becomes 1 L = Z ψ¯ i∂/ − m ψ − Z F F µν − e Z Z 1/2ψγ¯ µψA , 2 0 4 3 µν 0 2 3 µ with the physical electric charge 1/2 e0Z2Z3 = 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 q2 ∝ log −q2 .

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 k0 = k/b, x0 = xb,

so that the variable k0 is integrated over |k0| < Λ. I The ﬁeld is also rescaled as h i1/2 φ0 = b2−d (1 + ∆Z) φ.

I Our model system with an effective Lagrangian Z Z 1 1 1 ddxL = ddx0 ∂ 0φ02 + m02φ02 + λ0φ04 eﬀ 2 µ 2 4! will yield the rescaled parameters

2 m02 = 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 iZ hΩ|φ (p) φ (−p)|Ωi = at p2 = −M 2. 0 0 p2

The n-point Green’s function is deﬁned by

(n) ˆ G (x1, ··· , xn) = hΩ|T φ (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 ∂G(n) ∂G(n) dG(n) = δM + δλ = nδηG(n). ∂M ∂λ It is convenient to introduce dimensionless parameters M M β ≡ δλ; γ ≡ − δη, δM δM so to arrive at the equation ∂ ∂ M + β + nγ G(n) (x , ··· , x ; M, λ) = 0. ∂M ∂λ 1 n 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) ({x } ; M, λ) = 0. ∂M ∂λ i This is known as the Callan-Symanzik equation. Solutions of the Callan-Symanzik equations The generic form of the two-point Green’s function is

G(2)(p) = f + loops + fxf + ··· i i Λ2 i i = + A log + ﬁnite + ip2δ + ··· p2 p2 −p2 p2 Z p2

The M dependence comes entirely from the counterterm δZ . By neglecting the β term, we ﬁnd 1 ∂ γ = M δ . 2 ∂M Z Because the counterterm must be Λ2 δ = A log + ﬁnite, Z M 2 to lowest order we have γ = A. Solutions of the Callan-Symanzik equations

In a similar manner we obtain ! ∂ 1 X β(λ) = M −δ + λ δ . ∂M λ 2 Zi i

Since Λ2 δ = −B log + ﬁnite, λ M 2 to lowest order we have X β(λ) = −2B − λ 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) + nγ (e) + mγ (e) G(n,m) ({x } ; M, e) = 0, ∂M ∂e 2 3 i

where n and m are, respectively, the number of electron and (n,m) photon ﬁeldsin G 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,

e3 e2 e2 β(e) = , γ (e) = , γ (e) = . 12π2 2 16π2 3 12π2 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: ∂ ∂ q − β (er) + 2 V (q; M, er) = 0. ∂q ∂er The potential will be in the form 1 V (q, e ) = V (¯e (q; e )) , r q2 r where e¯(q) is the solution of the renormalization group equation d q e¯(q; er) = β (¯e) , e¯(M; er) = er. d log M Running coupling in QED Since the potential, in leading order, is e2 V (q) ≈ , q2 we can identify V (¯e) =e ¯2 + O e¯4. We immediately obtain

e¯2 (q; e ) V (q, e ) = r . r q2 By solving the renormalization group equation for e¯ and using β (e) = e3/12π2, we ﬁnd 12π2 1 1 q 2 − 2 = log . 2 er e¯ M This simpliﬁes to

2 2 er e¯ (q) = 2 2 . 1 − (er /6π ) log (q/M) Running coupling in QED

2 2 e2 By setting M = exp (5/3) m and er ≈ e, with α = 4π , we reproduce α α¯ (q) = ,A = exp(5/3). α q2 1 − 3π log − Am2

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 = L − m2φ 2. M 2 M 2 2 We treat mass term by replacing m → ρmM and expanding the Lagrangian about the free ﬁeld one L0 reads: 1 1 L = L − ρ M 2φ 2 − λM 4−dφ 4, 0 2 m M 4 M which is the Landau-Ginzburg theory for ferromagnetism! The Callan-Symanzik equation will give us 3λ2 β = − (4 − d) λ + 16π2 and for the condition β = 0 16π2 λ = (4 − d) . ∗ 3 Mass from a phase transition The corresponding renormalization group equation would be d ρ¯ = −2 + γ 2 λ¯ ρ¯ . d log p m φ m

The solution is, for the coupling λ¯ = λ∗, 2−γ (λ ) M φ2 ∗ ρ¯ = ρ . m m p The solution gives a nontrivial relation −ν ξ ∼ ρm , where the exponent ν is given formally by the expression 1 ν = , 2 − γφ2 (λ∗) explicitly, the Wilson-Fisher relation in statistical physics 1 ν−1 = 2 − (4 − d) . 3 β-decay

I The radioactivity discovered by Becquerel is β-decay. − I This is the neutron decay process: n → p +ν ¯ + e , p-udu ν¯

e− W −

n-udd

I β-decay violates the CP gauge symmetry. I A non-Abelian gauge theory, SU(2) × U(1), is required. I 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

ψ ψ = L . ψR

The kinetic energy term may be written as

† † L = ψL iσ¯ · ∂ψL + ψR iσ · ∂ψR.

The left-handed fields may coupled to a non-Abelian gauge a field A µ, which defines the corresponding field tensor as

a a a abc b c F µν = ∂µA ν − ∂νA µ + gf A µA ν,

a a through the minimal substitution Dµ = ∂µ − igA µt r to yield

1 − γ5 L = ψiγ¯ µ ∂ − igAa ta ψ. µ µ r 2

Here ta follows the commutation relation ta, tb = if abctc. 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 Q = ,L = . L d L e

I Since these fields are massless, we introduce a U(1) gauge symmetric field φ, which is known as Higgs field, a a Dµφ = (∂µ − igA µτ ) φ, a σa where τ = 2 . I If the vacuum expectation value of φ has broken symmetry 1 0 hφi = √ , 2 v then the gauge boson masses arise from

1 0 µ |D φ|2 = g2 (0 v) τ aτ b Aa Ab + ··· µ 2 v µ Higgs mechanism

After a symmetrization we ﬁnd the mass term

g2v2 ∆L = Aa Aaµ. 8 µ gv All three gauge bosons receive the mass mA = 2 . When Higgs ﬁeld transforms under φ follows SU(2) × U(1) gauge symmtry, a a φ → eiα τ eiβ/2φ, two bosons acquire masses and one boson remains massless: ± gv W mW ± = 2 , 0 p 2 02 v Z mZ0 = g + g 2 , Am A = 0. Higgs mechanism

Similarly, the electron ﬁelds e¯L and eR follows the mass term 1 ∆Le = −√ λeve¯LeR + h.c. + ··· , 2

by which the massless electron acquires mass m = √1 λ v. e 2 e

? The electron mass is the result of the spontaneous continuous symmetry breaking of Higgs ﬁeld. Summary

I Relativistic quantum field theory I Intrinsic spin I Pauli’s principle I Field quantization I Klein-Gordon fields I Dirac fields I Propagator and causality I Interacting field theory I S-matrix theory I Perturbation expansion I Photon as gauge particle I Elementary processes I Renormalization ⇒ Physically observed parameters: I spin-magnetic momentum (definite), I electron mass (cancelled divergences), I electric charge of electron (leaving divergence). I Renormalization Group and Higgs mechanism. I The origin of the charge of electron, I The origin of the electron mass. James Clerk Maxwell

The work of James Clerk Maxwell changed the world forever. by Albert Einstein