The Standard Model in the LHC era III: Hidden Symmetry ...lapth.cnrs.fr/~boudjema/cours/Fawzi_Taza_2.pdfhidden symmetry (Nambu-Goldstone) Many degenerate vacua The vacuum caracterises

The Standard Modelin the LHC era

III: Hidden Symmetry, Goldstones and Higgses

Fawzi BOUDJEMA LAPTh-Annecy, France

x x

EPAM, Taza, Maroc, March. 2011 F. BOUDJEMA, The Standard Model – p. 1/21

Introduction

Lme= −me

(

eReL + eLeR

)

Breaks SU(2) × U(1)

one SU(2) and one U(1) object can not combine like this

Y charge not conserved YeR= −1, YeL

= −1/2

must use EL

Construct a singlet eR ?? EL

?? = Dim = 1, Y = −1/2


Gauge boson masses

LMW= +M2

WW iµW

iµ

Breaks SU(2) × U(1)

Mass means longitudinal polarisation

ǫLµ =kµ

MZ−MZ

sµs.k

s2 = 0

ZLµ =

∂µφ3

MZO(MZ/EZ) ZL

µν = ∂µZνL − ∂νZ

µL = 0

ZLµ = ∂µφ . . .????

Is the mass encoded in a scalar field?EPAM, Taza, Maroc, March. 2011 F. BOUDJEMA, The Standard Model – p. 3/21

Auxiliary fields, Abelian case M2

BB2

Lφ =1

2

(

∂µφ− gvBµ

)(

∂µφ− gvBµ

)

Lφ invariant under the local scale (not phase) transformation

Bµ → Bµ∂µβ φ → φ + gvβ|z

χ

Dim[v] = 1

global Scale invariance is because the free scalar Lagrangian is massless

Lfree

φ =1

2∂µφ∂µφ invariant φ → φ + χ


Auxiliary fields, Abelian case M2

BB2

Lφ =1

2

(

∂µφ− gvBµ

)(

∂µφ− gvBµ

)

Lφ invariant under the local scale (not phase) transformation

Bµ → Bµ∂µβ φ → φ + gvβ|z

χ

Dim[v] = 1

global Scale invariance is because the free scalar Lagrangian is massless

Lfree

φ =1

2∂µφ∂µφ invariant φ → φ + χ

Pity, this is a special kind of transformation, it is not unitary an is revealing about the state

of the vacuum.

The vacuum is not invariant under U

φ′ = φ + χ = U†φU

If U(χ)|0 >= |0 > then < φ′ >=< φ >EPAM, Taza, Maroc, March. 2011 F. BOUDJEMA, The Standard Model – p. 4/21

Fabri-Picasso Theorem

If L is invariant under a continuous group, there is aconserved current ∂.j = 0, the associated charge Q =

∫d3xj0

is such that

AQ|O >= 0

manifest symmetry Wigner-Weyl

Particles with degeneratemasses

.




∫d3xj0

is such that

AQ|O >= 0



.

BQ|O > 6= 0(does not exist)

hidden symmetry(Nambu-Goldstone)

Many degenerate vacua




∫d3xj0

is such that

AQ|O >= 0



.

BQ|O > 6= 0(does not exist)

hidden symmetry(Nambu-Goldstone)

Many degenerate vacua

The vacuum caracterises the symmetry

The Hamiltonian is symmetric but not the backgroundAn example is a ferromagnet below the Curie Temperature. Rotational is broken in the ground state (all

spins aligned in the same direction) but the dynamics described but the fully symmetric Heisenberg

spin-spin Hamiltonian.


Enter Goldstone, the theorem

If B is realised (vacuum is not invariant)There must be massless particles

These particles are observable when the symmetry isglobal





if these particles interact with gauge particles, they areabsorbed





if these particles interact with gauge particles, they areabsorbed

For a symmetry that leaves the Lagrangian invariant, foreach generator that corresponds to an invariantvacuum, there is a Goldstone


Goldstone Potential at work

Consider a complex scalar field φ(x), with Lagrangian

L = ∂µφ†∂µφ − V (φ) , V (φ) = λ

φ†φ − µ2

2λ

!2

.

L is invariant under global phase transformations of the scalar field

φ(x) −→ φ′(x) ≡ exp iθφ(x) .

In order to have a ground state, potential should be bounded from below: λ > 0




L = ∂µφ†∂µφ − V (φ) , V (φ) = λ

φ†φ − µ2

2λ

!2

.


µ2 < 0:

|φ|

V(φ)

Trivial minimum at φ0 = 0.Describes massive scalar particle of

massp

−µ2




L = ∂µφ†∂µφ − V (φ) , V (φ) = λ

φ†φ − µ2

2λ

!2

.


µ2 < 0:

|φ|

V(φ)

Trivial minimum at φ0 = 0.Describes massive scalar particle of

massp

−µ2

µ2 > 0:

2ϕ

|φ|ϕ

1

V(φ)

< 0|φ|0 >= v/√

2.Because of U(1), infinite number of

degenerate states, φ0(x) = v√2

exp iθ.

By choosing a particular solution, θ = 0

for example, as the ground state,

the symmetry gets spontaneously broken.EPAM, Taza, Maroc, March. 2011 F. BOUDJEMA, The Standard Model – p. 7/21

Higgs Kibble Mechanism

Such non symmetric backgrounds in QFT are introduced by a scalar potential

that prefers stability rather than zero energy.

V = λ(|φ|2−v2/2)2

(λ > 0)

< 0|φ|0 >= v/√

2

Massive QED as an example: take a charged scalar field, charge e

φ = (φ1 + iφ2)/√

2 = (h+ v)eiθ/v/√

2interaction Dµφ = (∂µ + ieAµ)φ

Invariance Aµ → Aµ − 1e∂µχ φ→ eiχφ ≡ θ

v → θv + χ


Higgs Potential and U(1) mass

L = −1

4FµνF

µν +1

2(ev)2︸︷︷︸

m2γ

(

Aµ +1

ev∂µθ

)2

+1

2∂µh∂

µh− λ

4(h2 + 2vh)2 +

1

2

(

eAµ +1

v∂µθ

)

(h2 + 2vh) .

This Lagragian is completely GI and yet mγ 6= 0

choose a gauge such that θ → 0: No Goldstone.

there remains a Higgs, h with mh =√

2λv2

Number of degrees of freedom is unchanged


Higgs mechansim and SU(2) × U(1)

We now need to give mass to W± and the Z but not thephoton.⊚ Need three Goldstones with a vacuum that remainsinvariant under U(1)em. most simple choice is a doublet Φ → YΦ = −1/2(Q|0 >= |0 >)

Φ =

(

01√2(v +H)

)

eiωj τj

2v

LHiggs = (DµΦ)†(DµΦ) − V (Φ†Φ), V (Φ†Φ) = λ(

Φ†Φ − v2

2

)2

.

⋆ same Higgs gives mass to all fermionsEPAM, Taza, Maroc, March. 2011 F. BOUDJEMA, The Standard Model – p. 10/21

Gauge Invariant Masses and Higgs

If one takes the physical (unitary) gauge θi(x) = 0 , the kinetic piece of the scalarLagrangian takes the form:

(Dµφ)† Dµφθi

=0−→ 1

2∂µH∂µH + (v + H)2

g2

4W †

µWµ +g2

8 cos2 θW

ZµZµ

ff

.

The vacuum expectation value of the neutral scalar has generated a quadratic term for theW± and the Z, i.e., those gauge bosons have acquired masses:

MZ cos θW = MW =1

2v g .

MZ = 91.1875 ± 0.0021 GeV , MW = 80.390 ± 0.005 GeV .

From these experimental numbers, one obtains the electroweak mixing angle

sin2 θW = 1 − M2

W

M2

Z

= 0.223 .

M2

W c2WM2

Z

= ρ = 1.EPAM, Taza, Maroc, March. 2011 F. BOUDJEMA, The Standard Model – p. 11/21

Contact with Fermi

Zµ = cos θWW 3µ − sin θWBµ = 1√

g2+g′2

(gW 3

µ − g′Bµ

)

Aµ = sin θWW 3µ + cos θWBµ

MW = gv2

and MZ =

√

g2 + g′2 v2

= MW

cos θW

LCC = g

2√

2

∑

i ψiγµ(1 − γ5)

(T+W+

µ + T−W−µ

)ψi

e−

νe

νµ νµ

W−µ−

e−

νe

Fermi Theory

µ

Standard Model

GF√2

= g2

8M2

W

= 12v2


Lepton masses

Remember: Construct a singlet eR ?? EL

Lm = −ye

(

eR

(

Φ† EL

))

uni.gauge︷︸︸︷=⇒

(

− yev√2

)

︸︷︷︸

me

(

1 +H

v

)

ee

ye is a Yukawa coupling not a gauge coupling as in the caseof the gauge boson.


quark masses

Lm = −(

− yuuRΦ†QL +yd drΦ

†QL

)

Φ = iτ2Φ∗

md,u = yd,uv√2


Neutral currents: 1

L0 =∑

i

QiψiγµψiAµ (QED)

+g

2 cos θW

∑

i

ψiγµ(gi

V − giAγ

5)ψiZµ (NC)

giV = T3(i) − 2Qis

2W ; , gi

A = T3(i)

LNC =g

2 cos θW

∑

i

ψiγµ(giL(1 − γ5) + gi

R(1 + γ5))ψiZµ

gV = gL + gR gA = gL − gR


Phenomenology at the Z pole (1)

e

e

f

f(γ),Z

dσfZ

dΩ=

9

4

s/M2ZΓeeΓff

(s−M2Z)2 + s2Γ2

Z/M2Z

[

(1 + cos2 θ)(1 − PeAe)

+ 2 cos θAf (−Pe + Ae)

]

Γ(Z → ff) =αMZ

3

1

s2W c2W(gf 2

V + gf 2A ) Nf

c

Af =2gf

V gfA

(gfV )2 + (gf

A)2=

(gfL)2 − (gf

R)2

(gfL)2 + (gf

R)2, Aℓ =

2(1 − 4s2W )

1 + (1 − 4s2W )2∼ 0.15


Phenomenology at the Z pole (2)

AfFB ≡ σf

F − σfB

σfF + σf

B

=3

4AeAf ,

AfLR ≡ 1

Pe

σf (−|Pe|) − σf (+|Pe|)σf (−|Pe|) + σf (+|Pe|)

= Ae

AfFB ≡

(

σfF (−|Pe|) − σf

B(−|Pe|))

−(

σfF (+|Pe|) − σf

B(+|Pe|))

σfF (−|Pe|) + σf

B(−|Pe|) + σfF (+|Pe|) + σf

B(+|Pe|)

=3

4PeAf .


Z Lineshape: scan at the pole

σhad = 12πΓeeΓhad

M2ZΓ2

Z

Rℓ ≡ Γhad

Γℓ∼ 21

Rb,c ≡ Γbb,cc

Γhad

Γinv = ΓZ − Γhad,e,µ,τ

Nν =Γinv/Γℓ

(Γν/Γℓ)SM

Ecm [GeV]

σ had

[nb

]

σ from fit

QED unfolded

measurements, error barsincreased by factor 10

ALEPH

DELPHI

L3

OPAL

σ0

ΓZ

MZ

10

20

30

40

86 88 90 92 94


Measurements of key quantities at the Z peak

Quantity Group(s) Value

MZ [GeV] LEP 91.1876 ± 0.0021

ΓZ [GeV] LEP 2.4952 ± 0.0023

Γ(had) [GeV] LEP 1.7444 ± 0.0020

Γ(inv)[MeV] LEP 499.0 ± 1.5Γ(ℓ+ℓ−)[MeV] LEP 83.984 ± 0.086

σhad [nb] LEP 41.541 ± 0.037

Re LEP 20.804 ± 0.050

Rµ LEP 20.785 ± 0.033

Rτ LEP 20.764 ± 0.045

AFB(e) LEP 0.0145 ± 0.0025

AFB(µ) LEP 0.0169 ± 0.0013

AFB(τ) LEP 0.0188 ± 0.0017

Rb LEP + SLD 0.21664 ± 0.00065

Rc LEP + SLD 0.1718 ± 0.0031

Rs,d/R(d+u+s) OPAL 0.371 ± 0.023

AFB(b) LEP 0.0995 ± 0.0017

AFB(c) LEP 0.0713 ± 0.0036

AFB(s) DELPHI,OPAL 0.0976 ± 0.0114

Ab SLD 0.922 ± 0.020

Ac SLD 0.670 ± 0.026

As SLD 0.895 ± 0.091

ALR(hadrons) SLD 0.15138 ± 0.00216

ALR(leptons) SLD 0.1544 ± 0.0060

Aµ SLD 0.142 ± 0.015

Aτ SLD 0.136 ± 0.015


Wmass

√s

[GeV]

σ(e

+e

−→

W+W

−(γ

)) [

pb

] LEP

only νe exchange

no ZWW vertex

GENTLE

YFSWW3

RACOONWW

Data

√s

≥ 189 GeV: preliminary

0

10

20

160 170 180 190 200

+

e-W+

W-W+e

Z γ,

W-e-

e+

MW is also measured at Tevatron via pp→ W ∗


top mass at tevatron: Run I

Production: qq → g → ttDecay t→ Wb almost 100% of the time


Documents

The Standard Model in the LHC era III: Hidden Symmetry ...lapth.cnrs.fr/~boudjema/cours/Fawzi_Taza_2.pdfhidden symmetry (Nambu-Goldstone) Many degenerate vacua The vacuum caracterises