Behind and Beyond the Standard Model

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Behind and Beyond Standard Model the

Riccardo Rattazzi

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1. A critical view on the Standard Model

• Obvious limitations of the Standard Model

• Effective Quantum Field Theories: couplings, mass scales and accidental symmetries

• The Standard Model as an effective theory (baryon

& lepton number, flavor, precision EW tests)

• Naturally light particles & generation of mass hierarchies in field theory

2. Supersymmetry 3. Grand Unification

4. Composite Higgs sector: technicolor, etc..

• Precision Electroweak tests

2 Monday, September 28, 2009

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Standard Model: defined by gauge symmetry & multiplet content

gauge group SU(3) × SU(2) × U (1)_Y × gravity

bosons ⎨

⎧

⎧ ^G^A^µ ^W^µ^I ^B^µ ^g^µν

q_L, u_R, d_R, !_L, e_R H^α

fermions

L = − 1

4g₃² G²_µν − 1

4g₂² W_µν² − 1

4g_Y² B_µν² + |D_µH|² + V (H)

¯

q_L "Dq_L + ¯u_R "Du_R + ¯d_R "Dd_R + ¯!_L "D!_L + ¯e_R "De_R

+Y_uîjq¯_Lⁱ H^†u_R + Y_dîjq¯_Lⁱ Hd^j_R + Y_eîj!¯ⁱ_LHe_R + λîj

M (H!ⁱ)(H!^j) + · · · +√

gM_P² (R(g) − λ + · · · )

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Dark Matter is not made of any known particle

What could it be ? ^∼ ^{10 keV}

∼ 10⁻² meV

! 1 TeV

✦ WIMPS

✦ sterile neutrini

✦ axions

✦ Wimp-zillas

∼ 1 TeV

(5)

Dark Matter is not made of any known particle

What could it be ? ^∼ ^{10 keV}

∼ 10⁻² meV

! 1 TeV

✦ WIMPS

✦ sterile neutrini

✦ axions

✦ Wimp-zillas

∼ 1 TeV

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Gravity

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General Relativity at the quantum level only makes sense as an Effective Quantum Field Theory

There is an absolute upper bound on the energy scale at which General Relativity makes sense

Gravity couples to

all other particles absolute upper bound on energy scale up to which the SM can be valid

e⁻ e⁻

µ⁻ µ⁻

A^gravity = = s

M_P² 1

t

(8)

Mp is huge and thus gravity is not necessarily of urgent concern for the LHC

But previous argument only sets an upper bound

on relevant gravity scale. In the scenario of large extra dimensions gravity becomes indeed strong at around a TeV

The fate of gravity is of crucial importance to develop a theory of the very early universe

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The other 3 forces...

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Gauge Group

{

matter fermions

• why this apparently bizarre spectrum ?

• why is hypercharge quantized ?

non-abelian group

Ex: SU(2) [T₃,T_±]= ±T_±

[T₊,T₋]= 2T₃ T₃|ψ! = n

2 |ψ!

integer

abelian group: no quantization condition

Can one build new theory with non-abelian hypercharge ?

G = SU(3) × SU(2) × U(1)_Y

q_L = (3 , 2 , 1/3)

u_R = (3 , 1 , 4/3)

d_R = (3 , 1 , −2/3)

l_L = (1 , 2 , −1)

e_R = (1 , 1 , − 2)

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SU(3) SU(2) U(1)Y

g²₃ ! 1.5 g_W² ! 0.42

g_Y² ! 0.13

• they differ, but n o t w i l d l y

• strength of gravity at E M^≈ _Z

G_NM_Z² ≡ M_Z²

M² ∼ 10⁻³⁴

Strength of forces at E M^≈ _Z

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Matter

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Fermion masses Yukawa couplings

➤ ➤ ➤ ✕ ➤

Ex: up quarks

mass eigenvalues:

●

● fermion masses are inputs ... but the observed spectrum begs for an explanation

m_t = 175 m_b = 4.2 m_τ = 1.7 m_c = 1.2 m_s = 0.1 ^m^µ ⁼ ^0.1

ups downs leptons

3rd 2nd

1st family type

masses in GeV

H → υ_F

m_i = λ_i!H" ≡ λ_iυ_F = λ_i × (174GeV)

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ratios

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ratios

analogy with the spectrum of hydrogen lines before Bohr

●

1 2 3

Balmer fomula: ν =

! 1

n² − 1 m²

"

R

Rydberg const.

n,m= integers

explained by Bohr E_n = −2π²e⁴m_e h²n²

what is the analogue of Bohr atom in the case of fermion masses ?

▲

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Neutrino masses

∆m²_atm ! 2 × 10⁻³ eV² ∆m²_sol ! 0.8 × 10⁻⁴ eV²

sin² 2θ_atm = 0.9 − 1.0 tan² θ_sol = 0.3 − 0.6 we were hoping to get illuminated on the structure of

quarks and charged lepton spectrum, but we weren’t

the smallness of neutrino masses can be viewed as yet another success of SM

overall neutrino mass scale points to existence of new dynamics at a scale around 10¹⁴ GeV

Is this simple picture correct? Are neutrini Majorana particles?

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The fifth force

or

how weak interactions became weak

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Fermi scale Higgs field H vacuum expectation value

V (H) = m² H² + λ H⁴

Higgs potential:

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V (H) = m² H² + λ H⁴

Higgs potential:

m² > 0

H

< H >= 0

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V (H) = m² H² + λ H⁴

Higgs potential:

m² > 0

H

< H >= 0

m² < 0

H

●

< H >≡ υF = |m²|

2λ

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V (H) = m² H² + λ H⁴

Higgs potential:

m² < 0

H

●

< H >≡ υF = |m²|

2λ

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V (H) = m² H² + λ H⁴

Higgs potential:

m² < 0

H

●

< H >≡ υF = |m²|

2λ

< H > = υ_F

gives rise to all other masses in our world:

m² < 0

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V (H) = m² H² + λ H⁴

Higgs potential:

m² < 0

H

●

< H >≡ υF = |m²|

2λ

< H >

< H > < H >

lepton lepton vector

boson

vector boson

m! = λ! < H > ^m_W² ⁼ ^g² ^< ^H ^>²

< H > = υ_F

gives rise to all other masses in our world:

m² < 0

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V (H) = m² H² + λ H⁴ perturbativity

picks up all sorts of additive quantum corrections

m²

|m²| ∼ O(M_Planck² )

if SM valid up to Planck scale then it is natural to expect

λ <∼ 16π² < H > =

! −m²

2λ >∼ O( |m| 4π )

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m²

|m²| ∼ O(M_Planck² )

either

< H >= 0

or

< H >= O(M_Planck)

λ <∼ 16π² < H > =

! −m²

2λ >∼ O( |m| 4π )

(27)

m²

|m²| ∼ O(M_Planck² )

either

H = 0

or

H = O(M )

< H >= εM_Planck

but we need

λ <∼ 16π² < H > =

! −m²

2λ >∼ O( |m| 4π )

(28)

➤

Graphical picture of hierarchy puzzle

phase diagram

SM lives extremely close to the critical line is

λ₁ λ₂

< H >= 0

< H >!= 0

●

L^{f und} = L(g₁, g₂, Λ, . . . ; H, W_µ^I , q, !, . . . )

m_W , m_q, m_! = 0

m_h, m_W , m_q, m_! ∼ Λ m_h ∼ Λ

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Why ^v^F ^! ^M^{P lanck}

?

( G_F ! G_N )

In order to better appreciate this question, we must understand why is not considered to be a problem^mproton ! M_{P lanck}

One way to phrase the hierarchy problem is more simply

As we shall see, the problem is in a sense deeper than stated above:

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We need to better understand what is the Standard Model

In order to infere where* do we expect new physics to show up Not enough to generically ask why

* at what energy scale

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Effective field theory approach to particle physics

working at tree level first

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Physical scales & couplings

Ex: most general Lagrangian

for scalar field

+

!4

"₄

λ_i, η_i = dimensionless

(assume ) O(1)

λ₄

= ⁺ ^η4 ^E−→^→⁰ λ₄

E² Λ²

L^=∂µφ∂^µφ − m²φ² + λ₄φ⁴ + λ₆

Λ² φ⁶ + λ₈

Λ⁴φ⁸ + ···

+ η₄

Λ²φ²∂_µφ∂^µφ + η₆

Λ⁴φ⁴∂_µφ∂^µφ···

[∂_µ]= 1

Length = E

dimensions

[L^]=_(Length)^Energy₃ ⁼ ^E⁴

⎨⎧

⎧ ^{[φ] =} ^E

A(2 → 2) =

m <∼ E " Λ

assuming

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!₄

"₄

!₄

!₆

+ + + . . .

λ²₄ + +

1 E²

!

only a finite number of terms in the lagrangian are important the infinite set of couplings with negative mass dimensions is

L⁼∂_µφ∂^µφ − m²φ² + λ₄φ⁴ + λ₆

Λ²φ⁶ + λ₈

Λ⁴φ⁸ + ···

+η₄

Λ²φ²∂µφ∂^µφ + η₆

Λ⁴φ⁴∂µφ∂^µφ ···

λ₄η₄ E²

Λ² ^λ⁶

E² Λ²

E ! Λ

A(2 → 4) =

(34)

dimensionless quantity controlling strength of interaction

★ d > 0 : relevant at small E

• Ex: can treat mass as perturbation at E>> m

★ d = 0 : relevant at all energies (marginal)

★ d < 0 : irrelevant al small E

• perturbative expansion breaks down at high enough E

weak coupling

g g¯ ≡ gE ⁻^d

coupling with dimension

¯

g ! 1

!m¯ ² = m² E²

"

gauge and Yukawa couplings

[g] = d

(35)

Ex. : Fermi Lagrangian L_Fermi = G_F (p¯γ_µn)(e¯γ^µν)

G¯ _F ≡ G_FE²

G¯_F

M_W

➤ E

●

● ●

➤

g_W² 8

G⁻_F^1/2

Standard Model

Fermi Model

(36)

• fully describe an elementary (pointlike) particle

• correspond to inner structure

• to probe structure, E ≈ Λ is needed wavelength ≈

Imagine all couplings with d < 0 scale like inverse powers of a single scaleⁱ

dynamics at E << couplings with d ≥ 0_i

● ● ●

(m², λ₃, λ₄ ) (λ5, λ6, . . . )

−Λ1

Λ

(37)

Now at the quantum level...

( a more physical picture of renormalizability )

(38)

Problem: internal momentum of loops is not fixed by external momentum contributions enhanced by powers of cut-off

A(2 → 2) =

L = 1

2 ∂_µφ∂^µφ − 1

2 m²φ² − 1

4!λ₄φ⁴ − 1 6!

λ₆

Λ² φ⁶ − λ₈

Λ⁴ φ⁸ + · · ·

λ₄

!

1 + λ₄

32π² ln(− s

Λ² ) + . . .

"

+

does not vanish when Λ → ∞

λ₆

λ₄ λ₄ λ₄

1 Λ²

λ₆ 32π²

! _∼Λ² 0

p²dp²

p² + m² → λ₆ 32π²

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tree level ^λ⁶^E^external²

Λ²

λ₆E_virtual² Λ²

loop level

not small

in principle ^E^virtual _∼ ^Λ small

Apparently operators of arbitrarily high dimension matter!

But notice that UV enhanced contribution is local

+ λ^!₄ ≡ λ₄ + λ₆

32π²

UV enhanced contribution is just a renormalization of quartic term

Result generalizes to all orders

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Power divergent effects can be reabsorbed by renormalization of coefficient of lower dimension operators

must exist a scheme where these effects are absent ab initio

Dimensional Regularization

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, neglecting effects , L^{ef f} is equivalent to

L^!_{ef f} = L(g^!, λ^!)^d^≤⁴ + 0

virtual effects of accounted just by renormalization

g, λ −→ g^!, λ^!

L^d=5, L^d=6, . . .

physics is described by renormalizable Lagrangian _L^d^≤⁴ L^{ef f} = L(g, λ)^d^≤⁴ + 1

Λ L^d=5 + 1

Λ² L^d=6 + . . .

E ! Λ !E

Λ

"#

E ! Λ

at

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the ‘renormalizable’ terms (dimension 4 or less) fully describe elementary (pointlike) particles

‘non-renormalizable’ terms (dimension 5 or more) describe inner structure of particles

needed to directly probe structure wawelengths

E ∼ Λ

∼ 1 Λ

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➤

ρ(x)

R

▲ Analogy with multipole expansion in electrodynamics

= charge density

= electric potential

E_int = ^Z Φ(x)ρ(x)d³x = Φ(0)^Z ρ(x) + ∂iΦ(0)^Z xⁱρ(x) + 1

2∂i∂jΦ(0)^Z xⁱx^jρ(x) + . . .

ρ(x)

Φ(x)

= Φ(0)Q₀ + ∂iΦ(0)Qⁱ₁ + ¹₂∂i∂jΦ(0)Q^{i j}₂ + . . .

∼ QR²

∼ QR

∼ Q

At wavelengths > R light emission is dominated by dipole term

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Accidental symmetries

dynamics determined by a few `renormalizable’ couplings

extra (accidental) symmetries

E ! Λ

(45)

Example: parity in QED is respected by `renormalizable’ interactions

L^QED = −1

4 F_µνF ^µν + ¯ψiγ^µD_µψ + ¯ψ(m₁ + iγ₅m₂)ψ + a

4 F_µνF˜^µν

(m₁ + iγ₅m₂) → m = !

m²₁ + m²₂ ψ → e^iβγ⁵ψ

F_µνF˜^µν = total derivative

by chiral rotation

(46)

Example: parity in QED is respected by `renormalizable’ interactions

L^QED = −1

4 F_µνF ^µν + ¯ψiγ^µD_µψ + ¯ψ(m₁ + iγ₅m₂)ψ + a

4 F_µνF˜^µν

(m₁ + iγ₅m₂) → m = !

m²₁ + m²₂ ψ → e^iβγ⁵ψ

F_µνF˜^µν = total derivative

by chiral rotation

O_!_P = 1

Λ² ( ¯ψγ_µψ)( ¯ψγ_µγ₅ψ)

dim 6 operator violates parity

generated in SM by Z-exchange ¹

Λ² ∼ G_F = 1 v²

(47)

●

➤ ● ➤ ➤ ● ➤ ●

Standard Model interactions

gauge Yukawa self-Higgs Higgs mass

λ_{i j} λ g_a

i j

µ²

dim = 0 dim = 2

(48)

●

➤ ● ➤ ➤ ● ➤ ●

Standard Model interactions

gauge Yukawa self-Higgs Higgs mass

λ_{i j} λ g_a

i j

µ²

dim = 0 dim = 2

★ By allowing dim < 0 we would also have:

➤

➤ ➤

➤ ➤ ➤

1 Λ²_!_B

!u_αCγµu_β"

(eCγ^µd_δ)ε^αβγ Baryon number violation

1 Λ_!L

!!^aC !^b"

H_aH_b

Lepton number violation Flavor violation

●

➤ ➤

● ●

m_µ

Λ²_!_F (e¯γ_µγ_ν µ)F^µ^ν

(49)

➤

➤ ➤

proton ^● ➤ _➤ p → e⁺ π⁰

➤

1) B+L violation: proton decay

Superkamiokande: τ > 8.2 x 10 years _p ³³ Λ_!_B ≥ 10¹⁵GeV

(50)

➤

➤ ➤

proton ^● ➤ _➤ p → e⁺ π⁰

➤

1) B+L violation: proton decay

Superkamiokande: τ > 8.2 x 10 years _p ³³ Λ_!_B ≥ 10¹⁵GeV

2) L violation: neutrino masses

➤ ➤

H → υF ●

ν ν

υ_F ₁ υ_F

Λ_!L

× ×

m_ν ∼ υ²_F

Λ_"_L

observed neutrino oscillations: m_ν ∼ 0.1eV Λ_!_L ∼ 10¹⁴ GeV

(51)

3) Flavor violation

L = ¯q_LYˆ_dH^†d_R + ¯q_LVCKMYˆ_uHu_R + ¯!Yˆ_!H^†e_R

Yˆu =



 λu

λc

λ_t



 Yˆ! =



 λe

λµ

λ_τ

 ˆ 

Yd =



 λd

λs

λ_b





(52)

3) Flavor violation

Yˆu =



 λu

λc

λ_t



 Yˆ! =



 λe

λµ

λ_τ

 ˆ 

Yd =



 λd

λs

λ_b





absence of ν_R L_e, L_µ, L_τ are conserved

(53)

3) Flavor violation

Yˆu =



 λu

λc

λ_t



 Yˆ! =



 λe

λµ

λ_τ

 ˆ 

Yd =



 λd

λs

λ_b





V_CKM

very special quark Flavor violation all due to Glashow-Iliopoulos-Maiani

(GIM)

suppression mechanism

K − K¯ mixing

∼ G_Fα_W

4π (sin θ_C cos θ_C)² ! m_c M_W

"2 #

d¯_Lγ^µs_L$2

d¯ s

¯ s d

(54)

3) Flavor violation

Yˆu =



 λu

λc

λ_t



 Yˆ! =



 λe

λµ

λ_τ

 ˆ 

Yd =



 λd

λs

λ_b





non-renormalizable

contribution ^Λ¹²_!_F

!d¯_Lγ^µs_L"2 ∆m_K

m_K

!!

!exp −→ Λ_!_F > 10⁶GeV

V_CKM

very special quark Flavor violation all due to Glashow-Iliopoulos-Maiani

(GIM)

suppression mechanism

K − K¯ mixing

∼ G_Fα_W

4π (sin θ_C cos θ_C)² ! m_c M_W

"2 #

d¯_Lγ^µs_L$2

d¯ s

¯ s d