Advanced Particle Physics: VI. Probing the weak interaction

3.4 V-A coupling of leptons and quarks

Reminder u γ µ (1− γ 5 )u = u γ µu L = (u L + u R )γ µu L = u Lγ µu L l ν l ν l l ν l ν

In V-A theory the weak interaction couples left-handed lepton/quark currents (right-handed anti-lepton/quark currents) with an universal coupling strength:

2 GF gw = 2 2 8MW

Weak transition appear only inside weak-isospin doublets: Not equal to the mass eigenstate

Lepton currents: Quark currents: ⎛ν ⎞ ⎛ u ⎞ 1. ⎜ e ⎟ j µ = u γ µ (1− γ 5 )u 1. ⎜ ⎟ j µ = u γ µ (1− γ 5 )u ⎜ − ⎟ eν e ν ⎜ ⎟ du d u ⎝e ⎠ ⎝d′⎠

⎛ν ⎞ ⎛ c ⎞ 5 2. ⎜ µ ⎟ j µ = u γ µ (1− γ 5 )u 2. ⎜ ⎟ j µ = u γ µ (1− γ )u ⎜ − ⎟ µν µ ν ⎜ ⎟ sc s c ⎝ µ ⎠ ⎝s′⎠

⎛ν ⎞ ⎛ t ⎞ µ µ 5 3. ⎜ τ ⎟ j µ = u γ µ (1− γ 5 )u 3. ⎜ ⎟ j = u γ (1− γ )u ⎜ − ⎟ τν τ ν ⎜ ⎟ bt b t ⎝τ ⎠ ⎝b′⎠

3.5 CKM matrix to describe the quark mixing

One finds that the weak eigenstates of the down type quarks entering the weak isospin doublets are not equal to the their mass/flavor eigenstates:

⎛d′⎞ ⎛V V V ⎞ ⎛d ⎞ d V u ⎜ ⎟ ⎜ ud us ub ⎟ ⎜ ⎟ ud ⎜ s′⎟ = ⎜Vcd Vcs Vcb ⎟ ⋅ ⎜ s ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ W ⎝b′⎠ ⎝Vtd Vts Vtb ⎠ ⎝b⎠ Cabibbo-Kobayashi-Maskawa mixing matrix The quark mixing is the origin of the flavor number violation of the weak interaction.

Until the early 70s, only 3 quark flavor were known. The weak transition between quarks was described by a quark doublet:

⎛ u ⎞ ⎛ u ⎞ Mixing angle θ = Cabibbo-Angle ⎜ ⎟ = ⎜ ⎟ c ⎝d′⎠ ⎝cosθc ⋅ d + sinθc ⋅ s⎠

2 The mixing described automatically the suppression of ∆S=1 transitions (~sin θc)

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Missing FCNC and GIM mechanism

FCNC in the 3 quark model: K 0 → µ +µ −

W µ − d 0 Theoretically one predicts large BR, K u in contradiction with experimental s + limits for this decay: W µ BR (K → µ + µ − ) L = (7.2 ± 0.5) ⋅ 10 −9 M ~ sinθc cosθc BR (K L → all )

Proposal by Glashow, Iliopoulos, Maiani, 1970: W µ − There exists a fourth quark which builds d together with the s quark a second doublet: K 0 c s + ⎛ c ⎞ ⎛ c ⎞ W µ ⎜ ⎟ = ⎜ ⎟ s′ − sinθc ⋅ d + cosθc ⋅ s ⎝ ⎠ ⎝ ⎠ M ~ −sinθc cosθc

0 Additional Feynman-Graph for K →µµ Prediction of a fourth quark which compensates the first one:

Properties of the CKM-Matrix

⎛V V V ⎞ ⎜ ud us ub ⎟ VCKM = ⎜Vcd Vcs Vcb ⎟ ⎜ ⎟ ⎝Vtd Vts Vtb ⎠

Unitary N × N Matrix: → N2 Parameters: N=3

+ 9 VCKMVCKM = 1

N ( N - 1 )/2 Euler angles 3 (rotation angles)

Remaining parameters are phases: 6 2N-1 are unmeasurable phase diff 5 Observable phases 1

( N – 1 )2 observable parameters 4

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Parameterization of CKM Matrix: 3 Angles + 1 Phase

PDG choice where cij = cosθ ij , sij = sinθ ij

⎛d'⎞ ⎛1 0 0 ⎞⎛ c 0 s e−iδ ⎞⎛ c s 0⎞⎛d⎞ ⎜ ⎟ ⎜ ⎟⎜ 13 13 ⎟⎜ 12 12 ⎟⎜ ⎟ ⎜s' ⎟ = ⎜0 c23 s23 ⎟⎜ 0 1 0 ⎟⎜− s12 c12 0⎟⎜s ⎟ ⎜ iδ ⎟ ⎜b'⎟ ⎜0 s c ⎟⎜ s e 0 c ⎟⎜ 0 0 1⎟⎜b⎟ ⎝ ⎠ ⎝ − 23 23 ⎠⎝− 13 13 ⎠⎝ ⎠⎝ ⎠

Wolfenstein Parameterization λ, A, ρ, η

→ hierarchy expressed by orders of λ = sinθc≈ 0.22 2 ⎛ λ 3 ⎞ ⎜ 1 − λ Aλ ( ρ − iη )⎟ ⎜ 2 ⎟ 2 ⎜ λ 2 ⎟ 4 VCKM = ⎜ − λ 1 − Aλ ⎟ + O()λ ⎜ 2 ⎟ ⎜ Aλ3 (1 − ρ − iη ) − Aλ2 1 ⎟ ⎜ ⎟ ⎝ ⎠

Modulus of the matrix elements: Vij

d s b ⎛d'⎞ ⎛Vu ud Vus Vub ⎞ ⎛d⎞ ⎜ ⎟ ⎜ CKM ⎟ ⎜ ⎟ ⎜s'⎟ = ⎜CVcd MatrixVcs Vcb ⎟ ⎜s⎟ ⎜ ⎟ b' ⎜t ⎟ ⎜ ⎟ ⎝ ⎠ ⎝Vtd Vts Vtb ⎠ ⎝b⎠ PDG 2006

In leading order only the elementsVub andVtd are complex.

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3.6 Test of V-A structure in particle decays a) Muon decay − − µ →ν µ + e ν e k ν Muon lifetime µ − p′ − µ p e

GF 5 α 5 − M = []uν (k)γ α (1− γ )uµ (p) []ue (p′)γ (1− γ )vν (k′) W 2 ν k′ e

Analogous to the QED calculations of chapter III

one finds after a lengthy calculation: mµ 2 2 5 1 dΓ GF mµ = Γ = dE′ = 3 1 2 τ ∫ dE′ 192π 2 ′ ′ 0 M = ∑ M = 64GF (k ⋅ p )(k ⋅ p) 2 Spins Measurement of the muon lifetime thus provides a determination of the 1 2 Using dΓ = M dΦ one obtains the fundamental coupling G 2E F

electron spectrum in the muon rest frame: −6 τ µ = (2.19703 ± 0.00004) ⋅10 s 2 −5 -2 dΓ GF 2 2 4E′ Gµ = (1.16639 ± 0.00001) ⋅10 GeV = 3 mµ E′ (3 − ) dE′ 12π mµ Fermi constant measured in muon

with E’ = electron energy decays is often called Gµ

ν Test of V-A structure in the muon decay µ µ− e− Most general form of the matrix element for L / R L / R W− G M = F ⋅ g i u (e)Γi v (ν ) u (ν )Γi u (µ) ∑ λ′λ ( λ′ λi′ e )( λi µ λ ) 2 i =S,V,T L / R λ=L,R L / R ⎧±λ for S,V λ′=⎨ −λ for T ⎩ Chirality λi , λi′ determined by Γi ⎧λ′ i = S,T λi′= ⎨ ⎩ − λ′ i = V ⎧ λ i = V λi = ⎨ ⎩− λ i = S,T

Possible current-current couplings:

’ There are in general 10 complex i \ λ λ RR RL LR LL i amplitudes gλ′λ S x x x x V Pure V-A coupling: gLL = 1 V x x x x i all other gλ′λ = 0 T x x

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i Experimental determination of g λ ′λ from Couplings in muon decay energy spectra and spin correlation of the decay electrons from the polarized muons

Idea: 90% C.L.

V-A at µ vertex ⇒ LH νµ

ν µ e− ν e Configuration w/ max e- momentum possible

V+A at µ vertex ⇒ RH νµ

ν µ e− ν e Due to angular momentum conservation not possible V-A theory is confirmed

b) Pion decay

ν ,ν + + d + µ e π → µ ν µ W π + + + µ+, e+ π → e ν e u

Naïve expectation: Qualitative explanation within V-A theory: Assuming the same decay dynamics π the decay rate to e+ should be much , + + larger than to µ+ as the phase space ν µ ν e µ e is much bigger. Jπ = 0

Measurement: (PDG) Angular momentum conservation forces the lepton into the “wrong” Γ(π + → e+ν ) e = (1.230 ± 0.004) ⋅10−4 helicity state: suppressed ~ β=v/c Γ(π + → µ +ν ) µ i.e. for vanishing lepton masses the Large suppression due to a dynamic pion could not decay into leptons. effect.

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Determination of decay rates:

+ k π q νµ,νe

µ+, e+ p

Quarks in pion are bound G GF µ 5 F µ 5 M = ⋅ ()π ⋅ []u γ (1− γ )v M = ⋅ ()pµ + kµ ⋅ fπ ⋅ [uνγ (1− γ )vµ ] 2 µ ν µ 2

As the pion spin sπ=0, q is the only relevant 4-vector: q µ = pµ + k µ 2 ( π )µ = qµ⋅fπ (q )

Pion form factor: 2 2 2 2 q = mπ : fπ (q ) = fπ (mπ ) = fπ

2 2 + + GF 2 2 mµ Γ(π → µ ν µ ) = ⋅ fπ ⋅ mπ mµ (1− 2 ) 8π mπ 2 2 + + GF 2 2 me Γ(π → e ν µ ) = ⋅ fπ ⋅ mπ me (1− 2 ) 8π mπ

Γ(π + → e+ν ) ⎛ m2 ⎞⎛ m2 − m2 ⎞ µ = ⎜ e ⎟⎜ π e ⎟ = 1.275 ⋅10−4 + + ⎜ 2 ⎟⎜ 2 2 ⎟ Γ(π → µ ν µ ) ⎝ mµ ⎠⎝ mπ − mµ ⎠

The prediction of the V-A theory is confirmed by the experimental observation. The pion decay rates, although in agreement with the V-A theory, are not a proof of the V- A coupling. V or A coupling together with LH neutrinos would result to the same rates.

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3.7 Neutrino scattering in V-A theory ν k p′ − a) Neutrino-electron scattering µ µ

− − q ν µ e → µ ν e − − e p k′ ν e ν e e

G M = F []u (k′)γ (1− γ 5 )u (p) [u (p′)γ α (1− γ 5 )u (k)] 2 ν α e µ ν

2 1 2 2 ′ ′ 2 2 M = ∑ M = K = 64GF (k ⋅ p)(k ⋅ p ) = 16GF ⋅ s 2 Spins 2 Limit me≈mµ≈0 s =(k + p) = 2kp= 2k′p′

2 dσ 1 2 G s (ν e− ) = M = F Using the phase space factor of chapter II: dΩ µ 64π 2s 4π 2 G2s σ (ν e− ) = F µ π

b) Anti-Neutrino-electron scattering (V-A) k p′ e− ν e − − ν e e → ν ee − p e k′ ν e Crossing: s ⇔ t (u)

2 1 2 2 2 2 2 2 M = ∑ M = 16GF ⋅t = 4GF ⋅ s (1− cosθ ) 2 Spins − e 2 dσ G s (νe− ) = F (1− cosθ )2 θ − 2 ν e e dΩ 16π 2 − GFs ν e σ (νe ) = 3π

ν ν ν e Beside the charged current contribution there Z is of course a neutral current contribution W which is ignored in this reasoning. e e − − νee →νee e ν

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Result of V-A structure σ cc For the charged current (CC) contribution to νe = 3 the (anti) neutrino electron scattering one finds cc σνe

dσ G2s (ν e− ) = F Different angular distribution of (anti) dΩ e 4π 2 neutrino scattering can be understood from a helicity discussion dσ G2s (νe− ) = F (1− cosθ)2 dΩ 16π 2

− − − − νee →νee νee →νee

− ν − ν e e initial e e

ν − − e e final ν e e state Allowed → isotropic Forbidden (angular mom.) → (1-cosθ) suppression

c) (Anti) neutrino-quark scattering

ν d → µ−u − µ νµu → µ d − − 2 ν νµ µ dσ G s µ µ (ν d) = F dΩ µ 4π 2 2 GFs σ (ν µd) = d u π u d dσ dσ (ν u ) = (ν u) dΩ µ dΩ µ

+ ν u → µ d dσ G2s µ (ν u) = F (1+ cosθ )2 dΩ µ 16π 2 + + νµ µ 2 G s νµd → µ u σ (ν u) = F µ 3π dσ dσ d (ν µd ) = (ν µd) u dΩ dΩ d ν µ u θ µ +

Neutrinos only interact w/ d and anti-u quarks Anti-neutrinos only interact w/ u and anti-d quarks

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d) Neutrino-nucleon (iso-scalar) scattering

Q2 ν x = y = ν = E − E′ QPM: 2Mν E µ − ν µ dσ(ν N → µ −X) µ = d dσ(νd) G2xs dy = F u dy π d 2σ(νN) ⎛ dσ (ν q ) ⎞ dσ (νu) G2xs = f (x)⎜ i i ⎟ = F (1− y)2 ∑ i dy π dxdy i ⎝ dy ⎠sˆ=xs

d 2σ(νN) G2xs = F ⋅ []Q(x) + Q(x)(1− y)2 Unter Vernachlässig- dxdy 2π ung der Massen gilt: 1− cosθ ν µ + y = µ 2 dσ(ν N → µ + X) 1 µ u 1− y ≈ (1+ cosθ ) = 2 dy d d 2σ(νN) G2xs = F ⋅ []Q(x) + Q(x)(1− y)2 dxdy 2π

Total cross section after integration over x and y (0…1):

G2ME ⎡ 1 ⎤ ( N) F ν Q Q σ ν = ⋅ ⎢ I + I ⎥ 2π ⎣ 3 ⎦

G2ME ⎡ 1 ⎤ ( N) F ν Q Q σ ν = ⋅ ⎢ I + I ⎥ 2π ⎣ 3 ⎦

with Q = xQ(x)dx I ∫

σ 1+ 3Q Q R = νN = I I σνN 3 + QI QI

If nucleon consists only of valence quarks (⎯Q=0): R=1/3 , because of V-A structure 0.34 Measurement: R = ⇒ Q Q ≈ 0.15 0.67 I I

⇒ There are sea quarks !

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3.8 Problems with V-A theory

νe − → e −ν G2s • Cross section for e in 4-fermion ansatz: σ(νe− ) = F i.e. cross section goes to infinity if s→∞: violates unitarity π • Lee and Wu (1965) introduced a massive exchange boson. Effect of propagator:

GF GF 1 2 2 − GFMw a 1 q2 M 2 σ (νe ) a 2 2 − W π

Energy behavior of cross section becomes better but still violates unitarity at very high s. W pair production:

− • In addition there are boson production processes e W of the kind: Violates ν In the V-A theory they also violate unitarity !! e unitarity

+ W We need a new theory: Standard Model e

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