Introduction to the

Lecture 13

The Standard Model

Up to this point we have been gathering all the materials needed from which to construct the Standard Model. The Standard Model with the with the gauge group:

SU(3) SU(2) U(1) C ⊗ L ⊗ Y It defines the fundamental of and . It is an application of sponta- neous symmetry breakdown. We divide our approach to the model into two sectors:

I.) SU(2) U(1) L ⊗ Y referred to as • Glashow-Salam-Weinberg Theory (c. 1970) or Electro-Weak Theory or Flavour Dynamics

describes electromagnetic and weak •

SU(2)L U(1)Y U(1)EM remains • ⊗ −−−−−−−→SSBreakdown it is a chiral theory: • ψL, φL transforms as an SU(2)L doublet ψR, φR transforms as an SU(2)L singlet does not distinguish between quarks, colour blind • this implies universal couplings to qr,qb,qg

II.) SU(3)C

referred to as • describes strong interactions, nuclear binding • the theory is exact in that the gauge are strictly massless (unbroken) • this is a vector-like theory: universal coupling to L and R-handed states • flavour blind; universal coupling to flavour eigenstates: u,d,s,c,b,t •

1 Glashow-Salam-Weinberg Theory

S. Glashow, Nuc. Phys. B (1961) • A. Salam, Elementary Theory ed. Svartholm et al, Stockholm (1968) • S. Weinberg, Phys. Rev. Lett. 19 (1967) • Abers and Lee, Phys. Report 9 C (Review) • P. W. Higgs, Phys. Lett. (1964) • Brout and Englert, Phys. Rev. Lett. (1964) •

We can write the theory as the sum of Lagrangian densities:

= + + + [+ ] L Lgauge LHiggs Lfermion LYukawa Lghost Note that the Yukawa sector, , allows for and interactions; the “” LYukawa sector is needed to maintain covariance whilst quantising a non-Abelian gauge theory.

The gauge sector: gauge Weak interactions areL short range hence we need a massive exchange particle

1 e−mr massless massive ∼ r → ∼ r Parity violation in is obesrved(no L/R symmetry) Electromagnetice interactions forms a good gauge theory as it has a massless boson with an unbroken generator; is conserved leaving the boson () massless.

Many Lagrangians were constructed to explain the data; Glashow, Salam, and Weinberg wrote the right one.

gauge group: SU(2) U(1) L ⊗ Y generators: T 1, T 2, T 3 Y

gauge coupling: g g′ | {z } The Lagrangian may be written as

1 1 = W jµν W j Bµν B Lgauge −4 µν − 4 µν where

W j =∂ W j ∂ W j gεjlmW l W m j [1, 2, 3] µν µ ν − ν µ − µ ν ∈ B =∂ B ∂ B the U(1) field µν µ ν − ν µ Y 2 This theory describes four massless vector bosons but we need only one.

Recall: explicit mass terms are not allowed as they would destroy gauge invariance; we need a Higgs sector.

The Higgs sector: LHiggs

We need the generation of gauge boson masses by spontaneous symmetry breakdown: 1 massless gauge boson (photon) and 3 massive gauge bosons (weak interactions)

Minimal choice: introduce a scalar field as an SU(2) doublet

+ ~ 1 π1 + iπ2 φ φ = = 0 √ σ + iπ3 φ 2     ~ 1 where T (φ)= 2 transforms under the fundamental representation of SU(2)L.

~ 1 Assign hypercharge: Y (φ)= 2 .

To make the theory invariant under local transformations, we need

1 D = ∂ + ig ~σ W~ + ig′Y B µ µ 2 · µ µ U(1) coupling SU(2) coupling | {z } The Lagrangian is then | {z }

=(D φ~)†(Dµφ~) V (φ~†φ~) LHiggs µ − where

V (φ~†φ~)= µ2φ~†φ~ + λ(φ~†φ~)2

(It can be shown that globally symmetric SU(2) scalar theory is isomorphic to the SO(4) sigma-model. See tutorial for more details.)

Note that µ2 < 0 and λ> 0 leads to SSBreakdown.

1 0 µ2 φ~ = with v = − h i √ v λ 2   r Consider

3 1 1 0 1 1 0 T 3 + Y φ~ = + φ~ eiΛQ φ~ = φ~ h i 2 0 1 2 0 1 h i⇒ h i h i   −    Q  0 1 0 1 | {z } | @ 0{z 0 A }

1 0 1 0 1 v T 1 φ~ = = = φ~ h i 2√2 1 0 v 3 0 6 h i  −    √2   Hence, T 1 is a broken generator. The same is true for T 2 and T 3 Y meaning that we have − three Goldstone bosons which we may gauge away. The SU(2)L doublet can be written as (tutorial)

1 1 0 1 0 φ~ = exp( i T jθj) = U † − v √ H + v √ H + v 2   2   † 1 j j The term U = exp( i v T θ ) looks like a SU(2)L local group element. The Goldstone bosons, θj, play here the− role of the space-time dependent parameters.

Applying the gauge transformation U leads to the so-called unitary gauge:

1 0 φ~ φ~′ = Uφ~ = → √ H + v 2   We obtain † 1 H + v 0 1 = D Dµ + V (H + v)2 LHiggs 2 µ 0 H + v 2      Lquad  | {z } 0 g g′ 1 0 0 Dµ = ∂ + i ~σ W~ µ + i H + v µ 2 · 2 0 1 H + v         0 g ′ 3 v + H g ′ 3 = + i g Bµ gW µ + i g Bµ gW µ ∂µH 2 − 0 2 −      

† 0 g g 1 Dµ = (0, ∂ H) i W 1 + iW 2 (0, v + H) i W + (0, v + H) H + v µ − 2 µ µ − 2 √   ! 2  

1 µ2 λ = ∂ H∂µH (v + H)2 (v + H)4 + LHiggs 2 µ − 2 − 4 g2 1 + (v + H)2W +W −µ + g′Bµ gW 3µ g′B gW 3 (v + H)2 2 µ 8 − µ − µ   This Lagrangian defines interaction and mass terms

4 Note:

~σ W~ µ =σ1W 1 + σ2W 2 + σ3W 3 · µ µ µ W 3 W 1 iW 2 W 3 W + = µ µ µ = µ µ W 1 + iW 2 W−3 W − W 3  µ µ µ   µ µ  The charged masses can be read off directly from

g2v2 g2v2 2W +W −µ = (W 1)2 +(W 1µ)2 8 µ 4 µ  leading to

gv M ± = M 1,2 = W W 2   3 For the quadratic term in the W µ, Bµ bosons we find

2 2 ′ 3 v 3 g gg Wµ quad = Wµ , Bµ ′ − ′2 L 8 g g g Bµ  −     The mass matrix has two eigenvalues, one of which is 0 implying that the determinant is 0 which means that there is the massless gauge boson. After the field redefinition,

W 3 cos θ sin θ Z g′ µ W W µ , θ = sin W = ′2 Bµ sin θW cos θW Aµ (g2 + g )1/2    −    we get

2 v 2 ′2 Zµ 1 0 Zµ quad = g + g L 8 Aµ 0 0 Aµ        ′2 µ µ 2 g2+g 2 We see that A is massless and that Z is a massive vector boson with mass MZ = 4 v , and the other massive state in Higgs sector is the with mass M 2 = 2µ2 =2λv2. H −

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