• Structure of the Standard Model

Electroweak physics (II)

(The SM: structure and phenomenology) Andrea Romanino

SISSA/ISAS

Plan (II)

• Structure of the Standard Model

• The Higgs sector

• The gauge sector

• The flavour sector

• Tests

• The neutrino sector

• Beyond the Standard Model

The Standard Model (SM) of particle physics

(in compact notation) i=1,2,3: family index

masses neutrino

Λ

) )(

(

flavour Ψ

Ψ

breaking symmetry

) (

|

|

gauge 4

Ψ 1 Ψ ˆ

2

H L H η L

H H V H

D

F F D

i

L

j ij i

i j ij µ

µν a µνa i

i

SM

+ λ

− +

−

=

The SM as a renormalizable theory

A precise definition requires specifying:

• The fermion content: Ψ

• The gauge group G and its action on Ψ

• The symmetry breaking (scalar) sector: H

• The globally symmetric lagrangian L ( Ψ , H )

( ) ( ) ψ ψ ψ ( ) ψ ( ) ψ ψ ψ

ψ ψ

ψ , , , ,

½,0) (

0,½) (

+

→ +

→

+ 

 



The fermion content

spinors Dirac

) , , , , , , , ,

, , , (

Ψ = e µ τ ν

ν

ν

d s b u c t

) , , ,

, ,

, ,

,

:

(notation e_i ↔ e µ τ ν_i ↔ν_e ν_µ ν_τ u_i ↔ u c t d_i ↔ d s b

A 4-component Dirac spinor ψ has two 2-

components with definite chirality ( ): γ

ψ

ψ 2

1

,

∓ γ

=

A gauge symmetry can mix fields with same Lorentz quantum numbers ⇒ can act independently on ,

(chiral symmetry)

ψ ψ

SO(3,1)=SU(2)xSU(2) →

(or)

 

 



= 

 

 



= 

, 0

0

R

L

ψ ε

ψ _ψ ^ψ

( ) ( )

↔

L

ψ ψ ψ

ψ , ψ ψ , ψ ψ

½,0) 0,½) (

(





 





GeV 300

~ Λ Λ ,

1

2 / )

1 ( 2

/ ) 1

( 2

2 weak

=

+

=

− +

−

=

=

F

µ L L µ L

µ L µ

cµ

†µ µ c

F c

G

d u

e ν

d u

e ν

j

j G j

L

γ γ

γ γ

γ

γ

The gauge group G

Works well at E « Λ

However, it must fail at E » Λ:

) (unitarity /

1

~ vs

~ G

E

σ E

σ

(1 family only)

M E E

g

~ E σ

A~ g

M M E

g E

~ M σ

A~ g

» 1 @

«

@

2 4

2 2

4 4 2

2 2

νe ν_e

e e

At E > Λ it should be possible to account for the 4-fermion operator in terms of renormalizable operators

νe ν_e

e e

νe ν_e

e e

2 h.c.

weakren

= g j

W

+ L

symmetry gauge

of SSB vector

massive ⇒

µ+

W

2 2

2 8

F

M G ₌ g

ψ W

T ψ

W

j

→ γ

) prediction (NC

2

, ]

,

[ ^{1 2 3 3} σ³

T iT

T

T = =

• , are mixed by the generators → define

• Denote also

• then

• : G ⊇ SU(2),

• Q (charge/e) must also be a generator; Q = diag(0,1) on L (independent of T’s)

• Define ; is also a generator and commutes with the T’s:

From the lepton charged current to G



 



= 

L L

e L ν

νL e_L

2

2

1 µ

µ µ

iW W ^± =W ^∓

L W

T W

T L

g

L_weak^ren = γ^µ ( ¹ _µ¹ + ² _µ²)

, 2 2

2 2

1 1 σ

σ T

T = =





singlet

~

doublet

~ eR

L

T3

Q Y = −

Action of G on the quarks

• L doublet, singlets of SU(2)

• On Q: ok

•

• triplets of SU(3)



 



= 

L L

d

Q u u_R ,d_R



 



= 









− −









= −

−

= 0 1

0 1

6 1 2

/ 1 0

0 2

/ 1 3

/ 1 0

0 3

/ 2 T3

Q Y

; 3 on

; 1 3 on

2 uR Y dR

Y = = −

L R

L

R u d d u , , ,

Left-handed fermions quantum numbers (summary)

3 / 2 1

3

3 / 1 1

3

6 / 1 2

3

1 1

1

2 / 1 2

1

U(1) SU(2)

SU(3)

−

−

R i R i i R i

i

u d

Q e

L

U(1) x

SU(2) x

SU(3)

G =

0 )

3 3

6 2

( 3 anomaly

Grav.

0 )

3 3

6 2

( 6 U(1)

0 )

Tr(

SU(2)) SU(2)x(not

) 3

( 2 / 3

xU(1) SU(2)

0 )

Tr(

xSU(3) SU(2)

2 ,

) 2 ( SU

0 )

Tr(

SU(3)) SU(3)x(not

0 )

2 ( 2 / 3

xU(1) SU(3)

0 )

Tr(

xSU(2) SU(3)

vectorlike )

3 ( SU

) ) (

) ( (

3) 3 (

) 3 (

) 3 (

3

3 2 a

2 C

2 a b

3 2 a

) ) (

2 a (

2 3

= +

+ +

+

= +

+ +

+

=

+

=

=

=

= +

+

=

R R R

R R R

R R

d e Q u

L

d e Q u

L

Q ab L

ab

u d Q

AB

Y Y

Y Y

Y

Y Y

Y Y

Y σ

Y Y

δ λ

δ }

σ σ

{

λ

Y Y

Y δ

σ

1

Anomaly cancellation

Y T

t τ

} τ τ { τ Tr

T

(nice, but why?)

• G must be broken, or else

– All vector bosons would be massless – All fermions would be massless:

• A G-symmetric mass term combines left-handed fermions with conjugated quantum numbers under G:

• The representation of G on the SM fermions is chiral (= no such term is allowed)

• Note: the electroweak symmetry protects the fermion protects masses

( )

) h.c.

( + = +

=

m ψ

ψ

m ψ

ψ

ψ

ψ m

Spontaneous symmetry breaking

Fermion masses and Higgs quantum numbers

• Electron mass term:

• = a doublet with Y=½ under SU(2)xU(1)

• The only possible renormalizable gauge invariant coupling of is (Yukawa interaction), where H is a scalar doublet with Y=½ under SU(2)xU(1) (Higgs doublet)

• One Higgs doublet is sufficient to embed all mass terms in gauge-invariant interactions

• The most general gauge invariant Yukawa lagrangian is then , where SU(2) and

SU(3) contractions have been understood ( ) )

h.c.

(

e

e

m

L e e

e

R

LH

e

L e

Li Ri Li

Ri Li

Ri

e d d u u

e

H Q u

H Q d

H L

e

Eij λ λ

λ + +

pq q p

ε H Q

QH

The Higgs sector

Most general gauge invariant ren. lagrangian for H :

2

2 ( )

) 2 (

) (

) (

H H

H H

µ H

H V

H H

V H

D H

D L

H †

†

†

† µ

µ † H

+ λ

=

−

=

breaking symmetry

k Electrowea 0

2 < 0 ⇒

H

µ

)

~ scale a

at but breaking,

symmetry k

Electrowea 0

0

(

µ

H

m

> 0 λH

breaking symmetry

) (

|

|

flavour Ψ

Ψ

gauge 4

Ψ 1 Ψ ˆ

2

V H H

D

H F F

D i L

µ

j i ij

µν a µνa

i i

SM

− +

+

−

= λ

The renormalizable part of the SM

is now fully specified

The Higgs sector (II)

• Unbroken generators

• 3 broken generators ↔ 3 massive vectors ↔ 3

unphysical Goldstone bosons ↔ 1 real physical Higgs

2 2

2 | | (174GeV) ,

0 0 ,

=

=

 >



 



= 

H

v µ v v

H

Q b T

a

ib v b

H

T









−

= −

=

3 2 1

0 2

, 2 real, 2

,

, = = 1

+

=

σ Y

T b

a T

b aY

T

Goldstone bosons and Higgs

• To identify the Goldstone bosons:

– Write down the 4x4 Higgs mass matrix and find the massless eigenstates or

– Perform infinitesimal SU(2) x U(1) rotations of the vacuum

– CP: ; H has CP = +1, has CP = -1

 ⇒













−

= +

=

3 2 1

2

iG

G v iG

ε H

T G ε i H

δ















 + +

=

+

2

iG

v h H G

0 3 2

1 ,

2

G G G

G

iG

G

H

H

Higgs mass and self-coupling

• Higgs potential in unitary gauge:

• The Higgs self-coupling is proportional to its mass (non decoupling effects)

2 2

2

2 | µ | 2 v

m

= = λ

const.

8 2

) 2 (

)

(

2

0

= + + +

=

m h v h h

H V

h

V

^λ

^λ

Theoretical constraints on the Higgs mass

• Assume that the SM holds up to the scale Λ:

finite (perturbative) ⇒ upper limit on

> 0 ⇒ lower limit on

(if < 0, the absolute

minimum of the effective potential resides at or above Λ)

) Λ

H ( λ

) Λ

H ( λ

) Λ

H ( λ

m

m

) Λ

H ( λ

[Gino’s courtesy]

h V(h)

Λ 2 2 4

8 ) ( 2

) ) (

( h h

h h h µ

V ≈ + ^λH

• The lower limit can be relaxed if we live in a metastable vacuum

[Hambye, Riesselmann]

Experimental constraints on the Higgs mass

• Direct experimental limit:

> 114 GeV

• Indirect upper limit from EW precision tests (see below):

< 280 GeV @ 95% CL

(assumes no new physics contributions to the EW precision tests)

m

m

[LEP EW WG]

W, Z, and γ

• The vector boson masses arise from

•

• depend on the multiplet on which they act

• can be different because the corresponding generators cannot be linked by a gauge transformation

Y B ig T

igW t

G ig

D_µ = ∂_µ + _{s µ}^{A A} + _µ^{a a} + ' _µ

' , , g g g_s

Y T

t ^A, ^a ,

H D

H

D

' 2

2 )

(

2 _{3 2 2}

2 1

















+

 =















− ′

−

=

+

µ µ µ

µ

µ µ

µ

Z g

g iv gW

B g gW

iW W

iv g H

D

µ µ

µ µ

µ Wµ iW Z c W s B

W − ≡ −

+ ≡ 1 2 3

,

2

2 '

/

cosθ g g g

c_W ≡ _W = +

• Therefore,

• A is indeed the photon field

• Z couples to the neutral current (vector and axial couplings)

•

– Not guaranteed by gauge invariance nor by the breaking pattern – Peculiar of EW breaking by a doublet (triplets ruled out)

where 2 ,

) (

2 2 µ

Z µ µ µ

µ W

µ † M Z Z

W W

M H

D H

D = ^{+ −} +

2 1 iT T

T ^± = ±

2

2 '

' '

g g

c gg g gs

e _{W W}

= +

=

=

W µ µ W

µ µ

µ

µ T s Q Z

c i g ieQA

T g W

i T

g W i

D ( )

2 2

2 3 − +

+ +

+

∂

= ^{+ + − −} +ig_sG_µ^At ^A

level) (tree

cos² 1

2

2 =

≡

W Z

W

θ M

ρ M

W µ W µ

µ s W c B

A = ³ +

2 2 2 2

2 2 2

2 , '

2 g g v

M g v

M_W = _Z = ⁺

2 = 0 MA

Also,

4 2

1

2 v

G_F

=

(Custodial symmetry)

• In general, ρ = 1 can be traced back to a symmetry of the Higgs potential (custodial symmetry)

• ⇒ V(H) is symmetric under O(4) ~ , broken to the diagonal (custodial) SU(2)

• The whole lagrangian is symmetric for vanishing g’ and L-R

symmetric Yukawas. In particular, the W-H couplings are invariant under SU(2) ~ O(3), hence ρ = 1

• ρ = 1 can be reproduced if the symmetry breaking mechanism has a custodial SU(2) symmetry under which both the W’s and the G’s

22 22

12 12

|2

|H = h _r +h _i +h _r +h _r xSU(2)R

SU(2)_L

2 / ) Φ Φ ( ,

Φ Φ

, Φ

2 1 2

1 † † †

L UR H H Tr h U

h h

ε h  → =

























= 

∗

Gauge interactions

• Fermion gauge interactions:

• Gauge boson self-interactions: from

in terms of mass eigenstates:

µA A sµ s µ µ

em µ µ

W n µ µ

c

j Z ej A g j G

c W g

g j Ψ

i Ψ Ψ

D i

Ψ



 



 +

−

∂

= ⁺ h.c.

ˆ 2 ˆ

W χ µ

R L

χ e u d ν

f χ

nµ µ iL

iL µ iL

µ iL

c

ν e u d j f T s Q f

j

i i i i

) (

, ^{3 2}

, , , ,

−

= +

=

∑

= = γ

γ γ

µ ν ν

µ µν

νc µb a abc

µ a ν

ν a µ

µν

B B

B

W W

ε g W

W W

∂

−

∂

=

−

∂

−

∂

=

µ+

W

ν−

W

Z , γ

µ+

W

ν−

W

µ µ µ

µ Z Z

W ⁺, , ,γ

ν ν ν ν Z

W ⁻, ,γ , γ a

µν µνa

W

W

− 1

W +

W −

e+

e−

γ

W −

e+

W +

W −

e+

e−

νe

(a)

e− (b)

(a) (a+b)

(a+b+c)

• Higgs gauge interactions

gauge unitarity

2 1

2 2 ²₂ ^{2 2} 



 



 +











 +

M

W

W

M

Z

Z

v

h v

h

The flavour sector

•

• The Higgs coupling to fermions is proportional to their masses

– The coupling to light (« 174 GeV) fermions is small

– The coupling to the top leads to non-decoupling effects – The Higgs is mainly produced by gauge interactions

• Despite the rich flavour structure:

– No flavour or CP violation with leptons (neglecting neutrino masses)

v m

H Q u

H Q d

H L e

L

−

2 h.c.

2

2 + + +

+

+ +

=

h u

v u h m

d v d

h m e

v e m

u u m

d d m

e e m

Lj Ri ijU

Lj Ri

ijD Lj

Ri ijE

Lj U Ri

Lj ij D Ri

Lj ij E Ri

ij

unitarity gauge

The lepton flavour sector (neglecting ν masses)

• Mass eigestates:

Then

• Extend to gauge multiplets (commutes with G):

• Then

No LFV nor CPV,

R E EL

† diag

E

U

m U

m

L

R E e

E diag

E

U m m

U

i e i

Li e Ri

Lj E Ri

ij

e e m e e m e e

m







′ =

′ =

Lj E ij

Li

ij Rj Ri E

e U

e

e U

e

L R ) (

) (

Ri i Ri

Ri i i Ri

i

i D L e i D e L i D L e i D e L







′ =

′ =

j E ij

i

Rj E ij

Ri

L U

L

e U

e

L R ) (

) (

i † e Ri

j † E Ri

ij

e L H

e

L

H

10 11

2 . 1 )

(

µ

e γ

BR

,

• Mass eigestates:

Then

• Cannot extend both to gauge multiplets

H Q d V H

Q u

H Q d H

Q d

Q U

Q

d U

d

u U

u

i j u ij

Ri j Uij

i † d i

j † D Ri

ij ij j

i D

Rj ij

D Ri

Rj ij

U Ri

i i

L R R

′

= ′

′

= ′

 →







′ =

′ =

′ =

λ λ

λ

λ but

) (

) (

) (

e.g.







′ =

′ =







′ =

′ =

ij Lj U

Li

ij Rj U

Ri ij Lj

D Li

Rj D ij

Ri

u U

u

u U

u d

U d

d U

d

L R L

R ( )

) , (

) (

)

( _{d i i d i i}

Li d Ri

Li d Ri

Lj U Ri

ij Lj

D Ri ij

u u m d

d m

u u m d

d m u

u m

d d

m

i i

i i

′ + ′

′

= ′

′ + + ′

′

= ′ +

+ h.c. h.c.

R L

R L U U

† diag U U

D D

† diag

D

U

m U m U m U

m

The quark flavour sector

) ,

(both

u

d

Q

• In terms of mass eigenstates:

• All flavour and CP violation effects originate from the quark charged current → calculable FCNC and CPV processes

(CP: g → g* in an arbitrary phase convention for the mass

eigenstate fields ⇒ CP is conserved iff all the couplings are real for at least one phase choice)

µ jL ij iL

µ iL µ iL

had

c u d V u d

j _, = γ = ^′ γ ^′ )

( _,

, = _n^µ_had ′

µhad

n

j

j

) ( _,

, = _em^µ _had ′

µ had

em

j

j

Parameters: summary (no ν masses)

) CC (

18 : Tot 3

(l) sector Flavour

10 ,

, ,

(q) sector

Flavour

2 ,

sector Higgs

3 sector

Gauge

3 , 2 , 1

+ +

e u d

h s

θ

m

δ θ

m m

m v

g,g',g

i i i

Tests of the electroweak sector

• The electroweak sector (fermions + gauge interactions + masses from the Higgs sector) is the best tested

part of the SM

– Wide range of predictions:

g, g’, v ↔ ↔ QED, W&Z masses, their

self-interactions and all fermion gauge interactions (tree level)

– Measurements at the ‰ level

• Excellent agreement with the experiment (even too good)

v s

, ),

(α

Low energy tests

• β, lepton and pion decays (CC): GF, universality

• Atomic parity violation:

• Parity violation in Møller scattering

• (Parity violation in ep scattering)

• Neutrino-nucleous DIS:

θ

sin

θ

sin

θ

sin

θ

sin

(NC)

An example: Atomic Parity Violation

e e

Z

(Z,A) (Z,A)

A Z

s Q

r δ σ

m Q

H G

e

PV = F ⋅∇ ( ), = (2 − 4 ) −

∆ 2 ²

55133

Ce in crossing -

level p

- s

) 3 ( 19 . 73

|

| ), 48 ( 69 . 72

|

|

Q

Q

[E158]

(The 3σ NuTeV anomaly has probably more to do with QCD (parton distributions) than

High energy tests

• At LEP II, LEP I, SLC, Tevatron

• M

,Γ

Z resonance in e

e

→ff

N

=2.9841±0.0083: 3 light neutrinos + anomaly cancellation = 3 families

• M

,Γ

e

e

→W

W

at LEP II

•

•

•

• A

...

l

σ

gc

e WWZ

WW γ , couplings ∝ ,

) (

Γ )

( Γ

) (

Γ )

( Γ

L R R

L

L R R

f L

LR

Z f f Z f f

f f Z

f f A Z

→ +

→

→

−

= →

• Accuracy in most cases is at the ‰ level → sensitivity to 1-loop

corrections, which involve

– g, g’, v

– m_t, α_s(M_Z), ∆α_had(M_Z) – m_h

and bring together

– the gauge sector:

– the flavour sector:

– the EW-breaking sector:

) 4

2 /( π λt

) 4 /(

), 4

/( ²

2 π g π

g ′

) /

log(

) 4

2 /(

h MW

m π

g

Bounds on NP









⋅

>

⇒

<











= 

+ +

=

<

∑

1

~ loop, one

ve, perturbati is

NP if TeV 0.5

1

~ level, tree

ve, perturbati is

NP if TeV 5

g interactin strongly

is NP if TeV 50

~ TeV 5

Λ

TeV) 5

(

1 : Λ

EWPT

16

) Λ Λ

(

2 2

2 2 2

2

λ λ c

c π λ λ

c

c O L

E L

i SM

SM i

n i

i i

SM SM i

eff SM …

Accidental symmetries

• Global symmetries of :

– Must commute with the gauge symmetry

– Must be U(1) symmetry (non degenerate fermion spectrum)

• 5 (independent) global U(1) symmetries of :

• Symmetries of could be violated by (accidental symmetries)

GeV j

i L H L

L H

L

GeV L B

Q Q Q

k L j

i

B B k l j i

15 15

10

~ Λ masses

neutrino )

( and

violates )

)(

(

10 4

Λ decay

proton violates

Λ

≠ ⇒

⋅

>

⇒ ) 2 / 1 1

2 / 1 3

/ 1 3

/ 2 6

/ 1 (

0 0

0 0

0 0

0 3

/ 1 3

/ 1 3

/ 1

−

−

−

−

− Y

δ δ

L B

H e

L u

d Q

ik ik

k

R i i i

i R i R

SMren

L

SMren

L

SMren

L L

Smallness of neutrino masses

Natural scale of fermion masses:

v = 174 GeV

Why

(must have a different origin than

? 10

/v < ⁻¹² m_ν

) 10 3 . 0

/v ≈ ⁻⁵ m_e

GeV) 10

2 (

eV 05

. GeV 0

10 5

. 0

~ Λ

Λ

) )(

Λ (

GUT 16 15 SMren SMeff

⋅

≈

 

 



⋅ 

⋅

=

+ +

=

M

h m hv v

m

L H L

h H L

L

ν ν

…

The SM as an effective theory

Majorana

In neutrino physics we are still (and we will remain…) at the level of the Fermi theory

c c

c c

e e

d d

ν ν

u u

Right-handed neutrinos



 





Y c xSU(2)LxU(1) SU(3)



 







 





c c

c c

e e

d d

ν ν

u u

Right-handed neutrinos



 





B-L R

c xSU(2)LxSU(2) xU(1) SU(3)



 



 



 





c c

c c

e e

d d

ν ν

u u

Right-handed neutrinos



 



 SO(10)



 





c c

c c

e e

d d

ν ν

u u

Right-handed neutrinos



 





Y c xSU(2)LxU(1) SU(3)

heavy be

therefore can

and singlet

SM a

c

is v

v λ m

LH v

λ

→

=

(unlike the other fermions) c

c

v M v

L

⊃ − 2

T D ν D

T

M m m

m

M λ h λ

L H L

h H

1 1 Λ

) )(

Λ (

−

=

−

→

X

H

L

H

L νc ν_c

M

See-saw

: out

Integrate ν

Majorana

Beyond the Standard Model

• Experimental “problems” of the SM:

– Gravity

– Dark matter

– Baryon asymmetry

• Experimental hints of physics beyond the SM

– Neutrino masses

– Quantum number unification

• Theoretical problems of the SM:

– Naturalness problem

– Cosmological constant problem – Strong CP problem

• Theoretical puzzles of the SM:

– Hierarchy

– Family replication

– Small Yukawa couplings, pattern of masses and mixings

– Gauge group, anomaly cancellation, charge quantization, quantum numbers

E

ΛSM

− MPl

−

SM NP

The SM is an effective theory valid below Λ_SM

• Where is ?

• What type of physics above ? ΛSM

ΛSM

The naturalness problem as a guideline for NP

GeV H =174

−

2 2

2 2

2 2

2

2 (4 2 ) Λ

2 4 ) 3 Λ

( _SM ^F _{t W Z h SM}

h

h m M M m

π m G

m = + − − −

Where is ? ^Q

The main upper limit follows from solving the hierarchy problem











 =





 + 

 =





 + 

=

GeV 250

if TeV

2 ) Λ

Λ (

GeV 115

TeV if 5

. 0 ) Λ

Λ (

2 2 2

2 2 2

SM h SM h

h

SM h SM h

h

m m

m

m m

m

• is the scale of the degrees of freedom cutting off the Higgs mass quadratic divergence

• barring accidental cancellations ΛSM

TeV Λ_SM <~

Unification

3 / 2 1

3

3 / 1 1

3

6 / 1 2

3

1 1

1

2 / 1 2

1

U(1) SU(2)

SU(3)

−

−

R i R i i R i

i

u d

Q e

L

10 ,

,

5 ,

SU(5)

R i i i

R

R i i

u Q e

d L

16 ,

, , ,

,

SO(10)

R i R i

i i i R

R

i ν e Q u d

L

(SU(5))

heavy

5V τ = ig G t +igW T +ig'B Y + ig _µ^{α α} _{s µ}^{A A} _µ^{a a} _µ

s g g g MU

g ' @

3

5 = 5

=

=

With supersymmetry