Weak Interactions and

FYS3500 - spring 2020

Weak Interactions and Electroweak Unification*

Alex ReadUniversity Of OsloDepartment of Physics

*Martin and Shaw, Nuclear and Particle Physics, 3rd Ed., Chapter 6 (Last update 01.04.2020 16:39)

Part II Electroweak unification and the Higgs boson

FYS3500 Spring 2020 Alex Read, U. Oslo, Dept. Physics

Electroweak interactions and gauge invariance

❖ Gauge invariance and spontaneous breaking of gauge invariance is at the heart of electroweak unification and the BEH mechanism.

Electroweak interactions and gauge invariance

❖ Gauge invariance and spontaneous breaking of gauge invariance is at the heart of electroweak unification and the BEH mechanism.

❖ Gauge principle: Propose a gauge (phase) transformation of the wavefunction and add an interaction so that the gauge remains unobservable.

Gauge invariance and EM❖ Example, fields in electromagnetism:

B = ∇ × A , E = − ∇ ϕ −1c

∂ A∂t

B = ∇ × A , E = − ∇ ϕ −1c

∂ A∂t

❖ Gauge transformation of potential and vector-potential : ϕ A

B = ∇ × A , E = − ∇ ϕ −1c

∂ A∂t

❖(ϕ, A ) → (ϕ, A )′ = (ϕ −

∂α∂t

, A + ∇ α)

B = ∇ × A , E = − ∇ ϕ −1c

∂ A∂t

❖(ϕ, A ) → (ϕ, A )′ = (ϕ −

∂α∂t

, A + ∇ α)❖ where is an arbitrary doubly-differentiable scalar

functionα(t, x )

Gauge invariance

❖ B ′ = ∇ × A ′ = ∇ × ( A + ∇ α) = ∇ × A + ∇ × ∇ α

Gauge invariance

❖ B ′ = ∇ × A ′ = ∇ × ( A + ∇ α) = ∇ × A + ∇ × ∇ α

❖ ∇ × ∇ α ≡ 0 ⟹ B ′ = ∇ × A = B

Gauge invariance

❖ B ′ = ∇ × A ′ = ∇ × ( A + ∇ α) = ∇ × A + ∇ × ∇ α

❖ ∇ × ∇ α ≡ 0 ⟹ B ′ = ∇ × A = B

❖E ′ = − ∇ (ϕ −

∂α∂t ) −

∂∂t

( A + ∇ α) = − ∇ ϕ −1c

∂ A∂t

+1c ( ∇

∂α∂t

−∂∂t

∇ α)

Gauge invariance

❖ B ′ = ∇ × A ′ = ∇ × ( A + ∇ α) = ∇ × A + ∇ × ∇ α

❖ ∇ × ∇ α ≡ 0 ⟹ B ′ = ∇ × A = B

❖E ′ = − ∇ (ϕ −

∂α∂t ) −

∂∂t

( A + ∇ α) = − ∇ ϕ −1c

∂ A∂t

+1c ( ∇

∂α∂t

−∂∂t

∇ α)❖

∇∂α∂t

−∂∂t

∇ α = 0 ⟹ E ′ = = − ∇ ϕ −1c

∂ A∂t

Gauge invariance

❖ B ′ = ∇ × A ′ = ∇ × ( A + ∇ α) = ∇ × A + ∇ × ∇ α

❖ ∇ × ∇ α ≡ 0 ⟹ B ′ = ∇ × A = B

❖E ′ = − ∇ (ϕ −

∂α∂t ) −

∂∂t

( A + ∇ α) = − ∇ ϕ −1c

∂ A∂t

+1c ( ∇

∂α∂t

−∂∂t

∇ α)❖

∇∂α∂t

−∂∂t

∇ α = 0 ⟹ E ′ = = − ∇ ϕ −1c

∂ A∂t

❖ The electric and magnetic fields are unaffected by the gauge transformation: gauge invariance, or gauge symmetry∴

Global gauge invariance❖ Consider the double-slit experiment: The intensity at the

screen is proportional to the phase difference from the slits to the screen.

❖ ψ = ei( p ⋅ x −Et)

❖ Introduce a global phase (gauge) transformation ψ′ = e−ieαψ

❖ ψ = ei( p ⋅ x −Et)

❖ Introduce a global phase (gauge) transformation ψ′ = e−ieαψ

❖ Since the global phase doesn’t affect the outcome.

I ∝ ψ*′ ψ′ = ψ*ψ

Local gauge invariance❖ Now try , i.e. a local phase (gauge) transformationα = α( x )

❖ Now the phase difference would also depend on , affecting the intensity pattern, but there is no experimental support for this.

❖ ∇ (phase′ ) = ∇ (i( p ⋅ x − Et) − ieα( x )) = i p − ie ∇ α( x )

❖ Have to introduce something in addition to restore gauge invariance: Let p → p + e A , and make use of A → A + ∇ α( x )

❖ ∇ (phase′ ) = i p + ie A − ieα( x ) = i p = ∇ (phase)

❖ Adding the interaction with the photon field has restored the local gauge invariance.

Standard model gauge interactions❖ QED: , gives a photonψ ⟶ e−iqα( x )ψ

❖ QCD: , 8 color matrices give 8 gluons

ψ ⟶ eiαa( x )⋅Taψ Ta = (3 × 3)

❖ Electroweak: , Pauli matrices

ψ ⟶ eig′ α( x )+ig τ⋅ Λ( x )ψ τ = 3(2 × 2)

ψ ⟶ eiαa( x )⋅Taψ Ta = (3 × 3)

ψ ⟶ eig′ α( x )+ig τ⋅ Λ( x )ψ τ = 3(2 × 2)

❖ : coupling to weak hypercharge, i.e. a bosong′ B0

ψ ⟶ eiαa( x )⋅Taψ Ta = (3 × 3)

ψ ⟶ eig′ α( x )+ig τ⋅ Λ( x )ψ τ = 3(2 × 2)

❖ : coupling to weak isospin, i.e., bosonsg W±, W0

ψ ⟶ eiαa( x )⋅Taψ Ta = (3 × 3)

ψ ⟶ eig′ α( x )+ig τ⋅ Λ( x )ψ τ = 3(2 × 2)

❖ : coupling to weak isospin, i.e., bosonsg W±, W0

❖ Choice of symmetry determines dynamics of the system!!

Electroweak unification/mixing

Gauge interactions

Electroweak unification/mixing❖ Balancing act: don’t want weak neutral currents as strong

as the charged ones, and want to recover the photon (QED).

Gauge interactions

❖Mixing hypothesis: ( γ

Z0) = ( cos θW sin θW

−sin θW cos θW) ( B0

Gauge interactions Physical interactions

W0)❖ is the Weinberg angle, defined by θW

cos θW = mW /mZ

❖ Unification (massless photon) when e = g sin θW = g′ cos θW

cos θW = mW /mZ

❖ Unification (massless photon) when e = g sin θW = g′ cos θW

❖or , where

e2 2ϵ0

= gW sin θW = gZ cos θW

gW ≡g

2 2ϵ0, gZ ≡

2 2ϵ0

Aside: Gauge symmetry removes divergences

❖ This diagram alone is divergent but after adding the and contributions of the same order the cross

section is finite.γ Z0

❖ Showing the electroweak theory is as “renormalizable” (divergence free) as QED was worthy of a Nobel Prize ('t Hooft and Veltman, 1999)

Question: Can you draw the additional and diagrams of the same order?

Other predictions❖ Theory free of “anomalies” relates lepton and quark charges : l a

Ql + 3∑a

Qa = 0

Ql + 3∑a

Qa = 0

❖ Partial explanation for complete generations (so far zero evidence of a 4’th).

Ql + 3∑a

Qa = 0

❖Low energy couplings: GW ≡ GF =

(ℏc)2 2g2W

, GZ =(ℏc)2 2g2

Ql + 3∑a

Qa = 0

(ℏc)2 2g2W

, GZ =(ℏc)2 2g2

❖, measured to in low-energy

neutrino scattering

= sin2 θW sin2 θW = 0.227 ± 0.014

Ql + 3∑a

Qa = 0

(ℏc)2 2g2W

, GZ =(ℏc)2 2g2

❖, measured to in low-energy

neutrino scattering

= sin2 θW sin2 θW = 0.227 ± 0.014

❖ Remember that it was challenging to discover neutral currents - this is partly why.

Other predictions❖ Combined with measurements such as muon lifetime, and

taking into account higher-order diagrams, the masses of the and bosons were predicted (ca. 80 and 91 GeV/c2).W Z

❖ and bosons were discovered by the UA1 and UA2 experiments at CERN in 1983 (not 1993 as M&S write, e.g. on page 230).

❖ Precision measurements of and at TeVatron (USA) and LEP (CERN) in the 1990’s are consistent with the electroweak theory.

InteractionsZ0

❖ Basic vertices

a = quark

InteractionsZ0

❖ Basic vertices

eqa, gZa = quark

InteractionsZ0

❖ Basic vertices

❖ Assume that couples to Z0 uu, cc, tt, d′ d′ , s′ s′ , b′ b′

eqa, gZa = quark

InteractionsZ0

❖ Basic vertices

eqa, gZa = quark

(a, b, c)T ≡ (abc)

(AB)† ≡ (A*B*)T = B*T A*T = B†A†

InteractionsZ0

❖ Basic vertices

d′ d′ + s′ s′ + b′ b′ = [Vi,j(d, s, b)T]†Vi,j(d, s, b)T

= (d, s, b)V†i,jVi,j(d, s, b)T

= (d, s, b)(d, s, b)T = dd + ss + bb

eqa, gZa = quark

(a, b, c)T ≡ (abc)

(AB)† ≡ (A*B*)T = B*T A*T = B†A†

InteractionsZ0

❖ Basic vertices

= (d, s, b)(d, s, b)T = dd + ss + bb

❖ We can use the strong quark eigenstates for both and interactions

eqa, gZa = quark

(a, b, c)T ≡ (abc)

(AB)† ≡ (A*B*)T = B*T A*T = B†A†

InteractionsZ0

❖ Basic vertices

= (d, s, b)(d, s, b)T = dd + ss + bb

❖ In principle in any diagram where there is a there is a similar corresponding diagram with a , i.e. we could always write .

γZ0 γ/Z0

eqa, gZa = quark

(a, b, c)T ≡ (abc)

(AB)† ≡ (A*B*)T = B*T A*T = B†A†

InteractionsZ0

❖ Basic vertices

= (d, s, b)(d, s, b)T = dd + ss + bb

❖ In principle in any diagram where there is a there is a similar corresponding diagram with a , i.e. we could always write .

γZ0 γ/Z0

❖ Sometimes (in 2 slides) it is practical to keep them separate.

eqa, gZa = quark

(a, b, c)T ≡ (abc)

(AB)† ≡ (A*B*)T = B*T A*T = B†A†

InteractionsZ0

❖ Lepton-quark symmetry almost as straightforward as for

-interactionsW

InteractionsZ0

-interactionsW

https://en.wikipedia.org/wiki/W_and_Z_bosons

x ≡ sin2 θw

InteractionsZ0

-interactionsW

x ≡ sin2 θw

14.34.84.84.8

l-q symmetry

InteractionsZ0

-interactionsW

❖ Remember that and are mixtures of

B0(g′ ) and W0(g′ /tan θ)

x ≡ sin2 θw

14.34.84.84.8

l-q symmetry

InteractionsZ0/γ❖ Consider at low energy

( )e+e− → μ+μ−

E ≪ mZc2

( )e+e− → μ+μ−

E ≪ mZc2

❖ By dimensional arguments and

σγ ≈ α2EM(ℏc)2/E2

σZ ≈ G2ZE2/(ℏc)4

( )e+e− → μ+μ−

E ≪ mZc2

α2EM(ℏc)6

= ( GZ

(ℏc)3 )2 E4

= ( 2g2Z

ℏcm2Zc4 )

≈ ( EmZc2 )

( )e+e− → μ+μ−

E ≪ mZc2

α2EM(ℏc)6

= ( GZ

(ℏc)3 )2 E4

= ( 2g2Z

ℏcm2Zc4 )

≈ ( EmZc2 )

15mzc2

( )e+e− → μ+μ−

E ≪ mZc2

α2EM(ℏc)6

= ( GZ

(ℏc)3 )2 E4

= ( 2g2Z

ℏcm2Zc4 )

≈ ( EmZc2 )

❖ M&S 6.65b ( ) is for high-energy interactions ( ) !!

σZ /σγ ≈ 1/cos4 θW ≈ 1E ≫ mZc2

15mzc2

BEH Mechanism*

BEH Mechanism*❖ (*) Sometimes still called the Higgs Mechanism

❖ No parity violation (more in Chap. 7) together with and masses and gauge symmetry

W± Z0

❖ Don’t abandon gauge principle - introduce a scalar field that has a non-zero value in the vacuum.

W± Z0

❖ This is well-known in superconductivity, but here there is clearly a physical medium. Old-school physicists told the young people proposing this (4 different groups had more or less the same ideas in the early 1960’s) that they didn’t understand physics!

W± Z0

❖ This is well-known in superconductivity, but here there is clearly a physical medium. Old-school physicists told the young people proposing this (4 different groups had more or less the same ideas in the early 1960’s) that they didn’t understand physics!

❖ In EM a heated ferromagnet has no net magnetic field. As it is cooled below the critical (Curie) temperature, small domains will be spontaneously magnetized in some random direction. At high temperature the symmetry of EM is manifest, but it appears to be broken at low temperature.

BEH Mechanism❖ Have 4 vector bosons and want to give mass to 3 of them.

❖ Introduce scalar field with 4 components

❖ 3 are absorbed by the and , allowing them to have 3 degrees of polarization (the photon has only 2) and mass.

W± Z0

❖ The Higgs boson is a quantum excitation of the remaining component of the scalar field; the potential is postulated to have a Mexican hat form.

W± Z0

❖ The Higgs boson is a quantum excitation of the remaining component of the scalar field; the potential is postulated to have a Mexican hat form.

❖ The direction is determined spontaneously, breaking/hiding the original symmetry seen from the top of the hat.

Bonus from BEH for fermions

Bonus from BEH for fermions❖ The fermions couple to the Higgs field with a strength proportional to

their mass: .gHff = 2gWmf

❖ So the fermion masses are not predicted, but since we have measured the masses we can test whether the coupling is related to like this.mf

❖ We still have not predicted the masses, but at least we have predicted how the fermions get mass.

❖ …apart from the neutrino masses: They are so incredibly much smaller that it seems unlikely to be the same mechanism, and anyway it not possible to give them masses the same way due again to parity violation (more on this in Chap. 7).

❖ By the way, until Weinberg and Salaam applied BEH to electroweak interactions a few years later, the BEH mechanism was being explored as a way to understand the strong interaction in hadrons (massive vector bosons like the )!ρ±,0

Higgs boson decays

First order Second order

Higgs boson decays

❖ One of the in

decays must be virtual

V = Z0, W±

H → VV

Higgs boson decays

❖ One of the in

decays must be virtual

V = Z0, W±

H → VV

❖ (M&S write for )H → VV *

Alex Read - Higgs boson measurements Nobel Symposium, 15.05.201321

Higgs production @ LHC

Compiled by LHCXSWG

Higgs production at LHC

❖ Modest increases in yield for increasing in pp-collisions

s ≡ Ecm

Higgs production at LHC

❖ Modest increases in yield for increasing in pp-collisions

s ≡ Ecm

❖ We have data at TeVs = 7,8,13

Candidate H ! ��

CandidateH ! ZZ

⇤ ! (e+e�)(µ+µ�)

Candidate H ! W+W�(⇤) ! e+�eµ��µ

July, 2012

A new boson, “Higgs-like”

Combination of all channels and data available at the time

2 experiments with 5σ at ~same mass

The most sensitive channels making the impact

A new boson, “Higgs-like”

Combination of all channels and data available at the time

2 experiments with 5σ at ~same mass

The most sensitive channels making the impact

Rest of 2012: The signals grew...

Animations: https://cds.cern.ch/record/2230893?ln=en

Rest of 2012: The signals grew...

Animations: https://cds.cern.ch/record/2230893?ln=en

State of the art (ATLAS only)

https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/HIGGS/

Mass measurements

Mass measurements Cross-section versuscenter-of-mass energy s

Test of (scalar)JP = 0+

Cross-sections andBranching fractions

Test of coupling strength versusmass of fermion or vector boson

Test of coupling strength versusmass of fermion or vector boson Coupling strengths and

limits on exotic decayshttps://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/HIGGS/

Higgs summary❖ So far all observations consistent with simplest possible model of the BEH mechanism.

❖ All -collision production mechanisms observed.pp

❖ Coupling pattern to vector bosons and 3rd-generation fermions confirmed.

❖ Need more data to test second generation: by 2037?cc and μ+μ−

❖ Need more data to confirm (a fundamental prediction of the model): challenging but possible by 2037.

H → HH

❖ Speculative models rely on Higgs boson as a “portal” to dark matter - no signs yet.

H → HH

❖ Speculative models rely on Higgs boson as a “portal” to dark matter - no signs yet.

❖ Most fanatic proponents of supersymmetry (see Chap. 10) interpret as indirect evidence of supersymmetry.mH ≈ 125 GeV ∼ mZ

Lists of concepts

❖ Gauge invariance

❖ Gauge symmetry

❖ Gauge principle

❖ Electroweak unification

❖ Weak mixing angle

❖ Weinberg angle

❖ BEH mechanism

❖ BEH Mechanism

❖ Vector boson masses

❖ Higgs boson

❖ Fermion masses

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