BEYOND THE STANDARD MODEL

1

BEYOND THESTANDARD MODEL

G.F. Giudice

(I) Supersymmetry (general structure)

(II) Supersymmetry (phenomenology)

(III) Extra Dimensions

(IV) Strong dynamics, Little Higgs & more

CERN-Fermilab HCP Summer School CERN June 8-17, 2009

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

http://www.cern.ch/

2

Supersymmetry• Evades Coleman-Mandula (Poincaré internal group is the largest symmetry of the S-matrix)

• Relates particles with different spin involves space-time transformations

New concept of space

Local susy contains gravity

• Special ultraviolet finiteness properties

Just a mathematical curiosity? Too beautiful to be ignored by nature? A solution in search of a problem



x y

z

4-d space

Quantum dimensions (not described by ordinary

numbers)

3-d space

translations/rotations

4-d space-time

Poincaré

superspace

supersymmetry

3

Virtual particles are like ordinary particles, but have unusual mass-energy relations

The Higgs field propagating in vacuum “feel” them with strength E mH ≈ Emax (maximum energy of virtual particles)

temperature T

In quantum theory, the vacuum is a busy place Particle-antiparticle pairs can be produced out of nothing, borrowing an energy E for a time t E t ≤ h

If interacts with , after a while, we expect E ≈ T

4

mH ≈ Emax What is the maximum energy? MGUT = 1016 GeV? MPl = 1019 GeV?

Having MW << MPl requires tuning up to 34th digit !

temperature T E = 10-17 T

The “stability” of the hierarchy MW / MPl requires an explanation

Higgs mass is “screened” at energies above mH new forces and new particles within LHC energy range

What is the new phenomenon?

5

A problem relevant for low-energy supersymmetry: hierarchy/naturalness

€

mH2 =

3GF

4 2π 22mW

2 + mZ2 + mH

2 − 4mt2

( ) Λ2 ≈ − 0.2 Λ( )2

< TeV

€

mH <182 GeV (95% CL limit on SM Higgs)

If susy is effective at the Fermi scale:

stoptop + = 0Higgs

€

chiral symmetry⇒ ̃ m H = 0

supersymmetry ⇒ ˜ m H = mH

⎫ ⎬ ⎭⇒ mH = 0

The Fermi scale (mH) is induced only by susy breaking

6

Dynamical supersymmetry breaking

• If susy unbroken at tree-level, it remains unbroken to all orders in perturbation theory

• Non-perturbative effects can break susy with mS ~ e-1/ MP

• stable against quantum corrections (because of symmetry)

• naturally much smaller than MP

(because of dynamics)

Weak scale

Supersymmetric SM with mS < TeV solves the Higgs naturalness problem

Can we construct a realistic theory?

7

Supersymmetry transformation corresponds to group element

€

G x,θ,θ ( ) = ei −xP +θQ +θ Q ( )

€

G a,0,0( ) : x,θ,θ ( ) a x + a,θ,θ ( )

G 0,ξ ,ξ ( ) : x,θ,θ ( ) a x + iθσξ − iξσθ ,θ + ξ ,θ + ξ ( )

Translation

Susy

In superspace differential operators represent action of generators

€

Pμ → − i∂μ

Qα →∂

∂θ α− iσ α ˙ α

μ θ ˙ α ∂μ

Q ˙ α →∂

∂θ ̇ α

− iθ ασα ˙ β μ ε

˙ β ˙ α ∂μ

4-d space

superspacesusy

translation

8

Superfields

Function of superspace Power series in

€

Φ x,θ,θ ( )

€

Φ x,θ,θ ( ) = A(x) + θϕ (x) + θ χ (x) + θθB(x) + θ θ C(x)

+ θσ μθ V μ (x) + θθθ λ (x) + θ θ θψ (x) + θθθ θ D(x)

• finite number of component fields

• contains fields with different spin

• compact description of susy multiplets

• easy to write susy Lagrangians

9

CHIRAL SUPERFIELD General Φ is reducible. Additional constraints:

€

D ̇ α Φ = 0

€

Φ=A(x) + 2θψ (x) + iθσ μθ ∂μ A(x) +

θθF(x) −i

2θθ∂μψ (x)σ μθ +

1

4θθθ θ ∂ μ∂μ A(x)

components:

€

A(x) complex scalar field, ψ (x) Weyl spinor, F(x) auxiliary field

SUSY ACTION FOR A CHIRAL SUPERFIELD

€

d4 x∫ d4θ Φ+Φ

d4 x∫ d2θ W Φ( )

Kinetic term for A and F*F eliminated via e.o.m.

Superpotential: holomorphic function that defines interactionsE.g.:

=h2 required for cancellation of 2

€

W = mΦ2 ⇒ L = −m

2ψψ +ψ ψ ( ) − m2A+A

W = λ Φ3 ⇒ L = −λ ψψA + h.c.( ) − λ2 A+A( )2

In general: no quadratic divergences in susy theory

10

VECTOR SUPERFIELD

Global symmetry of superpotential can be made local (

chiral)

€

Φ→ e iα Φ

Φ → e iΛΦ

if we introduce a vector superfield V=V+ such that

€

Φ+Φ→ Φ+e i(Λ−Λ+ )Φ is made invariant

€

L = Φ+eV Φ V →V + i Λ+ − Λ( )

When expanded in components, V contains

€

λ(x) Weyl spinor, Vμ (x) vector field, D(x) auxiliary field

+ gauge degrees of freedom

11

4-d space

superspace

boson (integer spin) fermion

(half-integer spin)

superparticle

Q

Chiral multiplet:

A

Vector multiplet:

λ V

Gravity multiplet:

G g

12

Supersymmetric Standard Modelparticle

sSparticl

esquark

ssquar

ksslepto

nslepto

ns

Higgsdouble

ts

Higgsinos

bino

winos

gluinos

13

Ex: SU(3) color

interactions

all vertices controlled by the SU(3) coupling

there is a quartic scalar vertex

sparticles enter interactions in pairs:

sparticle parity = (-1) is conserved

(number of sparticles)

New particles, new interactions, but no new free parameters

14

1st problem: indirect new-physics effectsAny FT can be viewed as an effective theory below a UV cutoff

€

Leff = Ld = 4 g,λ( ) +1

ΛLd = 5 +

1

Λ2Ld = 6 + ...

€

g gauge

λ Yukawa

has physical meaning: maximum energy at which the theory is valid. Beyond , new degrees of freedom

€

B number⇒1

Λ2qqql p - decay ⇒ Λ ≥1015 GeV

L number ⇒1

Λl lHH ν mass ⇒ Λ ≥1013 GeV

individual L ⇒1

Λ2e σ μν μHFμν μ → eγ ⇒ Λ ≥10 8 GeV

quark flavour ⇒1

Λ2s γ μ d s γ μ d ΔmK ⇒ Λ ≥10 6 GeV

LEP1, 2 ⇒ 1

Λ2H +Dμ H

2,

1

Λ2e γ μe l γ μ l ⇒ Λ ≥10 4 GeV

Higgs naturalness gives an upper bound on . However,

15

New theories at TeV are highly constrained

€

f = QLDRc H1 + QLUR

c H2 + LL ER H1 +

URc DR

c DRc + QLDR

c LL + LLLL ERc + H2LL

Violate B or L

€

τ p =1

λ4

mS

TeV

⎛

⎝ ⎜

⎞

⎠ ⎟4

10 -10 sec

A first problem:

R-parity = + for SM particles, R-parity = for susy particles

• no tree-level virtual effects from susy

• susy particles only pair produced

• LSP stable (missing energy + dark matter)

Important for phenomenology

Usually one invokes R-parity (it could follow from gauge symmetry of underlying theory)

Flavor gives powerful constraints on the theory

16

2nd problem: supersymmetry breaking

Break susy, but keep UV behavior soft breaking

17

€

mSλλ gaugino mass

mS2 ϕ +ϕ scalar mass

mS ϕ 3 A - term

mS

• Soft susy breaking introduces a dimensionful parameter mS

• Susy particles get masses of order mS

• Susy mass terms are gauge invariant

• Treat soft terms as independent; later derive them from theory

• Different schemes make predictions for patterns of soft terms

18

ELECTROWEAK SYMMETRY BREAKING

Higgs potential

• m1,2,32 = O(mS

2) determined by soft terms

• quartic fixed by supersymmetry

• Stability along H1 = H2 m12 + m2

2 > 2 |m32|

• EW breaking, origin unstable m12 m2

2 < m34

Two robust features of low-energy susy: EW breaking & gauge coupling unification

€

V = m12 H1

0 2+ m2

2 H20 2

− m32 H1

0H20 + h.c.( ) +

g2 + ′ g 2

8H1

0 2− H2

0 2

( )2

19

RG running:

gauge effects

Yukawa effects

• If λt large enough SU(2)U(1) spontaneously broken

• If s large enough SU(3) unbroken

• Mass spectrum separation m22 < weak susy < strong susy

EW breaking induced by quantum corrections

20

HIGGS SECTOR8 degrees of freedom 3 Goldstones = 5 degrees of freedom

2 scalars (h0,H0), 1CP-odd scalar (A0), 1 charged (H)

3 parameters (m1,2,32 ) MZ = 2 free param. (often mA and tan)

€

mh ≤ mZ cos2β , mh < mA < mH , mH ±2 = mA

2 + mW2

mh,H2 =

1

2mA

2 + mZ2 m mA

2 − mZ2

( )2

+ 4 sin2 2β mA2 mZ

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥

mH±

mH

MZ

MZ

mA

m

mh

Large tan mh

mH

decoupling region

21

mS IS THE SEED OF EW BREAKING

EW breaking is related to susy breaking, mS mZ

• mS plays the role of 2 cutoff

• The quantum correction is negative and drives EW breaking

Minimum of the potential

€

mZ2 =

2 m12 − m2

2 tan2 β( )

tan2 β −1≈ −2m2

2

€

2δm22 <

mZ2

Δ⇒ ˜ m t < 300 GeV

10%

Δ

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

€

m22 = −

3λ t2

8π 2

k 2dk 2

k 2 + mt2

Λ2

∫ +3λ t

2

8π 2

k 2dk 2

k 2 + mt2 + mS

2

Λ2

∫ = −3λ t

2

4π 2mS

2 lnΛ

mS

€

mh2 ≈ mZ

2 +3

2π 2λ t

4v 2 ln˜ m tmt

>114 GeV ⇒ ˜ m t >1 TeV

Tension with data

stoptop

22

“Natural” supersymmetry has already been ruled out

23

Connection susy breaking EW breaking at the basis of low-energy supersymmetry

• Susy particle content dynamically determines EW breaking pattern

• Higgs interpreted as fundamental state, like Q and L

• Higgs mass determined by susy properties and spectrum

After LEP, “natural” susy is ruled out

• Source of “mild” tuning (is it observable at LHC?)

• Missing principle?

24

GRAND UNIFICATION

susySM GUT ?

MGUTmZmS

• Fundamental symmetry principle to embed all gauge forces in a simple group

• Partial unification of matter and understanding of hypercharge quantization and anomaly cancellation

To allow for unification, we need to unify g,g’,gS from effects of low-energy degrees of freedom (depends on the GUT structure only through threshold corrections)

b3=7, b2=19/6, b1=41/6

b3=3, b2=1, b1=11

€

dgi−2

d lnQ=

bi

4πsusy

SM

25

QuickTime™ and aTIFF (LZW) decompressor


26

3 equations, 2 unknowns (GUT, MGUT): predict S in

terms of and sin2W

• success of susy

• does not strongly depend on details of soft terms

• remarkable that MGUT is predicted below MP and above p-decay limit

sexp =0.11760.0020

27

THEORY OF SOFT TERMS• Explain origin of supersymmetry breaking

• Compute soft terms

Similar to EW breaking problem

• Origin of EW breaking

• Compute EW breaking effects

€

V H( ) = −mH2 H

2+ λ H

4

€

L = Dμ H +Dμ H − λHψ ψ

W,Z

q,l

EW

gauge

Yukawa

Gauge boson mass

Fermion mass

28

Invent a new sector which breaks supersymmetry

Couple the breaking sector to the SM superfields

But

€

STr M 2 = −1( )2J

2J +1( )MJ2 = 0

J

∑ at tree level, with canonical kinetic terms

Squarks, sleptons, gauginos, higgsinos

SUSY ???

sparticle < particle

What force mediates susy-breaking effects?

29

GRAVITY AS MEDIATORGravity couples to all forms of energy

Assume no force stronger than gravity couples the two sectors

Susy breaking in hidden sectormS = FX / MP mS = TeV FX

1/2 = 1011 GeV

ATTRACTIVE SCENARIO• Gravity a feature of local supersymmetry

• Gravity plays a role in EW physics

• No need to introduce ad hoc interactions

BUT• Lack of predictivity (102 parameters)

• Flavour problem

For simplicity, most analyses take universal m, M and A

30

Searching for supersymmetry at the LHC

• At a hadron collider, the total energy of the parton system is not known

• The initial momentum of the parton system in the transverse direction is zero

ET is a characteristic signal of supersymmetry

Background: • (mostly produced by W/Z or heavy quarks)

• incomplete solid angle coverage

• finite energy resolution of the detectors

• mismeasurement of jet energies

31

Colored particles have large cross sections at the LHC

Already with 10 fb-1, parameter space is

explored up to 1-2 TeV in gluino and squark masses

€

σ TeV ˜ g ( ) ≈ pb

However, determining parameters and masses is a much more

complicated issue

If MC tools for SM background are fully validated, If detector response is properly understood, then TeV susy particles can be discovered with low integrated luminosity

32



• Many new particles in final states

• Kinematics of the event cannot be fully reconstructed: unknown CM frame and pairs of particles carrying missing energy

Precise determination of masses and couplings is essential

• Confirm supersymmetric relations

• Understand pattern of supersymmetry breaking

• Identify “unification” relations

• Determine the DM mass

• Reconstruct relic abundance

33

Susy mass (differences) from edges in invariant mass distributionsConsider the decay chain

Repeating this technique along complicated chains and combining different channels, one can solve for (most) masses

Consider two-body decay

€

˜ q → q ˜ χ 20 ˜ χ 2

0 → ˜ l +l −

→ l + ˜ χ 10

max m2 l +l −( ) is obtained for l + and l − back to back in ˜ χ 2

0 ref frame

⇒ max m l +l −( )[ ] = m ˜ χ 2

1−m˜ l

2

m ˜ χ 2

2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ 1−

m ˜ χ 1

2

m˜ l 2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

˜ q → q ˜ χ 20 ˜ χ 2

0 → ˜ χ 10l +l − through Z 0 or ˜ l exchange

max m2 l +l −( ) is obtained for ˜ χ 1

0 and l +l −( ) at rest in ˜ χ 2

0 ref frame

⇒ max m l +l −( )[ ] = m ˜ χ 2

− m ˜ χ 1

34

This technique doesn’t exploit the kinematic constraints on ET

€

pp → ˜ g ̃ g → qqqqχ 10χ 1

0

→ q ˜ q

→ qχ 10

New techniques to derive all masses from kinematic distributions

Example:W “transverse mass” from Wl

€

mT2 = ml

2 + mν2 + 2 ET

l ETν −

r p T

l ⋅r p T

ν( ) ≤ mW

2

mW obtained from end-point of mT

The end-point of the “gluino stranverse mass” has a kink structure when plotted as a function of the test LSP mass

The location of the kink corresponds to the physical mg and m ISR, finite resolution, background and finite width can smear end-points

35

ET may not be the discovery signature(even in gravity mediation)

Long-lived charged particle at the LHC (ττG)~ ~If the gravitino is the LSP:

Distinctive ToF and energy loss signatures

“Stoppers” in ATLAS/CMS caverns:

• Measure position and time of stopped τ time and energy of τ Reconstruct susy scale and gravitational coupling

~

36

GAUGE MEDIATION

Soft terms are generated by quantum effects at a scale M << MP

• If M << F, Yukawa is the only effective source of flavour breaking (MFV); flavour physics is decoupled (unlike sugra or technicolour)

• Soft terms are computable and theory is highly predictive

• Free from unknowns related to quantum gravity

MPFMmZ

37

BUILDING BLOCKS OF GAUGE MEDIATION

SUSY SMSUSY Messengers

gauge loop

SUSY SM: observable sector with SM supermultiplets

SUSY: “hidden” sector with <X> = M + 2 F

Messengers: gauge charged, heavy (real rep), preserve gauge unification (complete GUT multiplet)

Ex.:

€

Φ+Φ =5 + 5 of SU(5) with f = XΦΦ , V = M 2 ϕ2

+ ϕ 2

( ) + F ϕϕ + h.c.( )

Parameters: M, F, N (twice Dynkin index; N=1 for 5+5)

38

Gaugino mass at one loop, scalar masses at two loops:

€

mS ≈g2

16π 2

F

M

F/M ~ 10-100 TeV, but M arbitrary

To dominate gravity and have no flavour problem

€

F

MP

<10−2 g2

16π 2

F

M⇒ M <1015 GeV

From stability:

From perturbativity up to the GUT scale:

€

N <150 /lnMGUT

M

€

F < M ⇒ M >10 −100 TeV

€

M ˜ g Q( ) =g2(Q)

16π 2N

F

M

€

˜ m Q2 (M) = 2c

g4

16π 2( )

2 NF 2

M 2

39

• Theory is very predictive

• Gaugino masses are “GUT-related”, although they are not extrapolated to MGUT

• Gaugino/scalar mass scales like N1/2

• Large squark/slepton mass ratio and small A do not help with tuning

40

Higgs mass is the strongest constraint: stop masses at several TeV

41

Crucial difference between gauge and gravity mediation

€

m3 / 2 =F

3MP

⇒ in gravity m3 / 2 ≈ mS, in gauge m3 / 2 ≈F

100 TeV

⎛

⎝ ⎜

⎞

⎠ ⎟

2

2 eV

In gauge mediation, the gravitino is always the LSP

q

q~~G

€

Δ ˜ m 2

FGoldberger-Treimanino relation

€

L = −1

FJQ

μ∂μ˜ G = −

1

F˜ m ϕ

2ψ Lϕ +M ˜ g

4 2λ aσ μν Fμν

a ⎛

⎝ ⎜

⎞

⎠ ⎟ ˜ G + h.c.

NLSP decays travelling an average distance

€

l ≈100 GeV

mNLSP

⎛

⎝ ⎜

⎞

⎠ ⎟

5F

100 TeV

⎛

⎝ ⎜

⎞

⎠ ⎟

4E 2

mNLSP2

−1 0.1 mm

From microscopic to astronomical distances

on mass shell

42

Intermediate region very interesting

(vertex displacement; direct measurement of F)

Susy particles

NLSP

0 τR

E E ττE Stable charged particle

€

F ≤106 GeV

€

F ≥106 GeV

€

F ≥106 GeV

€

F ≤106 GeV

~

0 or τR are the NLSP (NLSP can be charged)

In gravity-mediation, “missing energy” is the signature

~

43

DARK MATTER

• rotational curves of galaxies• weak gravitational lensing of distant galaxies• velocity dispersion of galaxy satellites• structure formation in N-body simulations

Indirect evidence for DM is solid



• Opportunity for particle physics

• Intriguing connection weak-scale physics dark matter



44

T >> M T << MT ≈ M

45

Relic abundance

€

Ω =mn∞

ρ c

=4π( )

2

3

π

45

x f gS γ( )

g*1/ 2

Tγ3

H02MP

3σ

If σ =k

128π m2⇒ Ωχ =

0.22

k

m

TeV

⎛

⎝ ⎜

⎞

⎠ ⎟2

Weak-scale particle candidate for DM

No parametric connection to the weak scale

Observation provides a link MDM <H>

Many BSM theories have a DM candidate

Susy has one of the most appealing

46

Supersymmetric Dark Matter

R-parity LSP stable

RG effects colour and electric neutral massive particle is LSP

Heavy isotopes exclude gluino, direct searches exclude sneutrino

Neutralino or gravitino are the best candidates

NEUTRALINO

Because of strong exp limits on supersymmetry, current eigenstates are nearly mass eigenstates:

Bino, Wino, Higgsino

47

BINO

f

ff~

B~B~

HIGGSINO

WINO

W,Z

W,ZH~

H~

W,Z

W,ZW~

W~

48

Bino

€

1.1<˜ m e R

M1

< 3

Higgsino

€

1.5 <˜ m tμ

< ∞

Wino

€

1.5 <˜ m l L

M2

< ∞

€

ΩDM h2 = 0.105 ± 0.008

Neutralino: natural DM candidate for light supersymmetry

Quantitative difference after LEP & WMAP

Both MZ and ΩDM can be reproduced by low-energy supersymmetry, but at the price of some tuning.

Unlucky circumstances or wrong track?

49

TO OBTAIN CORRECT RELIC ABUNDANCE

• Heavy susy spectrum: Higgsino (1 TeV) or Wino (2.5 TeV)

• Coannihilation Bino-stau (or light stop?)

• Nearly degenerate Bino-Higgsino or Bino-Wino

• S-channel resonance (heavy Higgs with mass 2m)

• TRH close to Tf

All these possibilities have a very critical behavior with underlying parameters

• Decay into a lighter particle (e.g. gravitino)

50

How can we identify DM at the LHC?

Establishing the DM nature of new LHC discoveries will not be easy. We can rely on various hints

• If excess of missing energy is found, DM is the prime suspect

• Reconstructing the relic abundance (possible only for thermal relics and requires high precision; LHC + ILC?)

• Identify model-dependent features (heavy neutralinos, degenerate stau-neutralino, mixed states, mA = 2 m)

• Compare with underground DM searches

51

SPACE DIMENSIONS AND UNIFICATION

Minkowski recognized special relativistic invariance of Maxwell’s eqs connection between unification of forces and number of dimensions

Electric & magnetic forces unified in 4D space time

€

r∇ ⋅

rE = ρ

r ∇ ×

r E = −

∂r B

∂tr

∇ ⋅r B = 0

r ∇ ×

r B =

∂r E

∂t+

r J

⎧

⎨ ⎪ ⎪

⎩ ⎪ ⎪

€

space - time t,r x → x μ = (t,

r x )

EM potentials r E = −

r ∇φ −

∂r A

∂t,

r B =

r ∇ ×

r A → Aμ = (φ,

r A )

EM fields r E ,

r B → Fμν = ∂μ Aν −∂ν Aμ =

0 −Ex −Ey −E z

Ex 0 Bz −By

Ey −Bz 0 Bx

E z By −Bx 0

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

current ρ,r J → Jμ = ( ρ,

r J )

Maxwell's eqs → ∂μ F μν = Jν



52

UNIFICATION OF EM & GRAVITYNext step:

New dimensions?

1912: Gunnar Nordström proposes gravity theory with scalar field coupled to T

1914: he introduces a 5-dim A to describe both EM & gravity

1919: mathematician Theodor Kaluza writes a 5-dim theory for EM & gravity. Sends it to Einstein who suggests publication 2 years later

1926: Oskar Klein rediscovers the theory, gives a geometrical interpretation and finds charge quantization

In the ‘80s the theory, known as Kaluza-Klein becomes popular with supergravity and strings







53

GRAVITY

In General Relativity, metric (4X4 symmetric tensor) dynamical variable describing space geometry (graviton)

€

ds2 = gμν dx μ dxν

€

gμν

Dynamics described by Einstein action

€

SG =1

16π GN

d4∫ x −g R(g)

• GN Newton’s constant

• R curvature (function of the metric)





54

Consider GR in 5-dim

€

ˆ S G =1

16π ˆ G Nd5∫ x −ˆ g R( ˆ g )

Choose

€

ˆ g MN ( ˆ x ) =gμν + κ 2φ Aμ Aν κ φ Aμ

κ φ Aν φ

⎛

⎝ ⎜

⎞

⎠ ⎟( ˆ x )

€

ˆ g MN ⇔ gμν , Aμ , φDynamical fields

Assume space is M4S1

• First considered as a mathematical trick

• It may have physical meaning

(t,x)

x5

R

55

Extra dim is periodic or “compactified”

€

x5 + 2π R = x5

All fields can be expanded in Fourier modes

€

ϕ ( ˆ x ) =ϕ (n )(x)

2π Rn=−∞

+∞

∑ exp in x5

R

⎛

⎝ ⎜

⎞

⎠ ⎟

5-dim field set of 4-dim fields: Kaluza-Klein modes

€

ϕ (n )(x)

Each has a fixed momentum p5=n/R along 5th dim

€

ϕ (n )

4-d space

extra dimensions

mass

D-dim particle

E2 = p 2 + p2extra + m2

KK mass

From KK mass spectrum we can measure the geometry of extra dimensions

56

R

r << R r >> R

2-d plane1-d line

Suppose typical energy << 1/R only zero-modes can be excited

Expand SG keeping only zero-modes and setting ϕ=1

€

ˆ S G ( ˆ g MN ) = SG (g(0)μν ) + SEM (A(0)

μ )

€

SG (g) =1

16π GN

d4 x −g R(g)∫

SEM (A) = −1

4d4 x Fμν F μν∫

⎧

⎨ ⎪

⎩ ⎪

To obtain correct normalization:

€

SG →1

GN

=dx5∫ˆ G N

=2π R

ˆ G N

SEM → κ = 16π GN

Gravity & EM unified in higher-dim space: MIRACLE?

57

Gauge transformation has a geometrical meaning

€

dˆ s 2 = ˆ g MN ( ˆ x ) dˆ x M dˆ x N

€

ˆ g MN ( ˆ x ) =gμν + κ 2φ Aμ Aν κ φ Aμ

κ φ Aν φ

⎛

⎝ ⎜

⎞

⎠ ⎟( ˆ x )

Keep only zero-modes:

€

dˆ s 2 = g(0)μν dx μ dxν + φ(0) dx 5 + κ A(0)

μ dx μ( )

2

Invariant under local

€

x 5 → x 5 −κ Λ

A(0)μ → A(0)

μ + ∂μ Λ(where g and ϕ

do not transform)

• Gauge transformation is balanced by a shift in 5th dimension

• EM Lagrangian uniquely determined by gauge invariance

58

CHARGE QUANTIZATION

Matter EM couplings fixed by 5-dim GR

Consider scalar field

€

S = d5 ˆ x −ˆ g ˆ g MN∂Mϕ∫ ∂Nϕ

Expand in 4-D KK modes:

€

S = dx5 d4 x −g(0) ∂ μ − inκ

RA(0)μ ⎛

⎝ ⎜

⎞

⎠ ⎟ϕ (n )

2

−n2

R2

ϕ (n )2

φ

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

∫n

∑∫

€

2π R

Each KK mode n has: mass n/R charge n/R

• charge quantization

• determination of fine-structure constant

• new dynamics open up at Planckian distances

€

= 2

4π R2=

4GN

R2⇒ R =

4GN

α≈ 4 ×10−31 m = 5 ×1017 GeV( )

−1

59

Not a theory of the real world

ϕ=1 not consistent (ϕ dynamical field leads to inconsistencies: e.g. F(0)

F(0)=0 from eqs of motion)

• Charged states have masses of order MPl

• Gauge group must be non-abelian (more dimensions?)

Nevertheless

• Interesting attempt to unify gravity and gauge interactions

• Geometrical meaning of gauge interactions

• Useful in the context of modern superstring theory

• Relevant for the hierarchy problem?

60

61

Usual approach: fundamental theory at MPl, while W is a derived quantity

Alternative: W is fundamental scale, while MPl is a derived effect

New approach requires• extra spatial dimensions

• confinement of matter on subspaces

Natural setting in string theory Localization of gauge theories on defects (D-branes: end points

of open strings)

We are confined in a 4-dim world, which is embedded in a higher-dim space where gravity can propagate

62

COMPUTE NEWTON CONSTANT

Einstein action in D dimensions

€

SED =

1

16π ˆ G NdD x −ˆ g R( ˆ g )∫

Assume space R4SD-4: g doesn’t depend on extra coordinates

Effective action for g

€

SE =VD−4

16π ˆ G Nd4 x −g R(g)∫

⇒1

GN

=VD−4

ˆ G N

€

MPl = MD RMD( )D−4

2

€

ˆ G N =1

MDD−2

VD−4 = RD−4

63

Suppose fundamental mass scale MD ~ TeV

€

MPl = MD RMD( )D−4

2 very large if R is large (in units of MD-1)

Arkani-Hamed, Dimopoulos, Dvali

€

5 ×10−4 eV( )−1

≈ 0.4 mm D − 4 = 2

R = 20 keV( )−1

≈10−5 μ m D − 4 = 4

7 MeV( )−1

≈ 30 fm D − 4 = 6

Radius of compactified space

• Smallness of GN/GF related to largeness of RMD

• Gravity is weak because it is diluted in a large space (small overlap with branes)

• Need dynamical explanation for RMD>>1

64



€

V (r) = −GN

m1m2

r1+ α exp −r λ( )[ ]

λ

Gravitational interactions modified at small distances

€

FN (r) = GN

m1m2

r2 at r > R

At r < R, space is (3+)-dimensional (=D-4)

€

FN (r) = ˆ G N(4 +δ ) m1m2

r2+δ=

= GN Rδ m1m2

r2+δ

From SN emission and neutron-star heating:

MD>750 (35) TeV for =2(3)

65

Probing gravity at the LHC?Probing gravity at the LHC?


are needed to see this picture.QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.



Gravitational wave jet +

Gravitational deflection dijet

Black hole multiparticle eventET

graviton

gluon

Gravitational phenomena into collider arena

66

Probability of producing a KK graviton

€

≈E 2

MPl2

€

σ pp → G(n ) jet( ) =α s

πGN =10−28 fb 1 event ⇒ run LHC for 1016 tU

Number of KK modes with mass less than E (use m=n/R)

€

∝ nD−4 ≈ ER( )D−4

≈E D−4 MPl

2

MDD−2

Inclusive cross section

€

σ pp → G(n ) jet( ) ≈α sE

D−4

π MDD−2

n

∑

graviton

gluon

It does not depend on VD (i.e. on the Planck mass)

Missing energy and jet with characteristic spectrum

67

68

Contact interactions from graviton exchange

€

L = ±4π

ΛT4

T

T =1

2Tμν T μν −

1

D − 2Tμ

μTνν ⎛

⎝ ⎜

⎞

⎠ ⎟

• Sensitive to UV physics

• d-wave contribution to scattering processes

• predictions for related processes

• Limits from Bhabha/di- at LEP and Drell-Yan/ di- at Tevatron: T > 1.2 - 1.4 TeV

• Loop effect, but dim-6 vs. dim-8

• only dim-6 generated by pure gravity

• > 15 - 17 TeV from LEP

€

L = ±4π

ΛΥ2

Υ

Υ =1

2f γ μγ 5 f

f = q,l

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

69

TRANSPLANCKIAN REGIME2

1

3

+⎟⎠

⎞⎜⎝

⎛=

λc

GDP

h

1

1

3

1

1

2

3

2

81 ++

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ +Γ

+=

π c

sGR D

S

Planck length quantum-gravity scale

€

classical limit h → 0( ) : RS >> λ P

transplanckian limit s >> MD( ) : RS >> λ P

Schwarzschild radius

classical gravity

same regime

G-emission is based on linearized gravity, valid at s << MD2

The transplanckian regime is described by classical physics (general relativity) independent test, crucial to verify

gravitational nature of new physics

70

b > RS

Non-perturbative, but calculable for b>>RS (weak gravitational field)

€

≈∂b

∂L≈

bcδ

mvbδ +1 rel . ⏐ → ⏐ GD s

bδ +1θE =

4GD s

b

D-dim gravitational potential:

€

V (r) =GDmM

rδ +1D = 4 + δ

Quantum-mechanical scattering phase of wave with angular momentum mvb

€

b = −bc

b

⎛

⎝ ⎜

⎞

⎠ ⎟

δ

bc ≈GDmM

vh

⎛

⎝ ⎜

⎞

⎠ ⎟

1

δ

Gravitational scattering

bvm

71

Diffractive pattern characterized by

€

bc ≈GDs

h

⎛

⎝ ⎜

⎞

⎠ ⎟

1

δ

Gravitational scattering in extra dimensions: two-jet signal at the LHC

72

b < RS At b<RS, no longer calculable

Strong indications for black-hole formation

Characteristic events with large multiplicity (<N> ~ MBH / <E> ~ (MBH / MD2)/(+1)) and typical energy <E> ~ TH

BH with angular momentum, gauge quantum numbers, hairs (multiple moments of the asymmetric distribution of gauge charges and energy-momentum)

σ ~ πRS2 10 pb (for MBH=6 TeV and MD=1.5 TeV)

Gravitational and gauge radiation during collapse spinning Kerr BH

Hawking radiation until Planck phase is reached TH ~ RS

-1 ~ MD (MD / MBH)1/1)

Evaporation with τ ~ MBH(+3)/(+1) / MD

2(+2)/(+1) (10-26 s for MD=1 TeV)

Transplanckian condition MBH >> MD ?

73

WARPED GRAVITY





A classical mechanism to make quanta softer

For time-indep. metrics with g0=0 E |g00|1/2 conserved . (proper time dτ2 = g00 dt2)

€

Schwarzschild metric g00 =1−2GN M

r⇒

Eobs − Eem

Eem

= g00 −1= −GN M

rem

On non-trivial metrics, we see far-away objects as red-shifted

74

GRAVITATIONAL RED-SHIFT

€

ds2 = e−2K |y|η μν dx μ dxν + dy 2

Masses on two branes related by

€

mπR

m0

= e−πRK

Same result can be obtained by integrating SE over y

€

R ≈10 K−1 ⇒mπR

m0

≈MZ

MGUT

y=0 g00=1

y=πR g00=e-2πRK

75

PHYSICAL INTERPRETATION• Gravitational field configuration is non-trivial

• Gravity concentrated at y=0, while our world confined at y=πR

• Small overlap weakness of gravity

WARPED GRAVITY AT COLLIDERS• KK masses mn = Kxne-πRK [xn roots of J1(x)] not equally spaced

• Characteristic mass Ke-πRK ~ TeV

• KK couplings

• KK gravitons have large mass gap and are “strongly” coupled

• Clean signal at the LHC from G l+l- €

L = −T μν Gμν(0)

MPl

+Gμν

(n )

Λπn=1

∞

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟ Λπ ≡ e−πRK MPl ≈ TeV

76

Spin 2

Spin 1

77

A SURPRISING TWISTAdS/CFT correspondence relates 5-d gravity with

negative cosmological constant to strongly-coupled 4-d conformal field theory

Theoretical developments in extra dimensions have much contributed to model building of 4-dim theories

of electroweak breaking: susy anomaly mediation, susy gaugino mediation, Little Higgs, Higgs-gauge

unification, composite Higgs, Higgsless, …

Warped gravity with SM fermions and

gauge bosons in bulk and Higgs on brane

Technicolor-like theory with slowly-running couplings in 4 dim

78

DUALITY

AdS/CFT Composite Higgs

5-D warped gravity

large-N technicolor

SM in warped extra dims strongly-int’ing 4-d theory

KK excitations “hadrons” of new strong force

Technicolor strikes back?TeV brane Planck

brane

5th dim

IR UV

RG flow

5-D gravity 4-D gauge theory

Motion in 5th dim RG flow

UV brane Planck cutoff

IR brane breaking of conformal inv.

Bulk local symmetries global symmetries

79

What screens the Higgs mass?What screens the Higgs mass?

€

→ + a

no m2φ2

boson

Spont. broken global symm.

€

→ e iaγ 5ψ

no mψ ψ

fermion

Chiral symmetry

€

Aμ → Aμ + ∂μ a

no m2Aμ Aμ

vector

Gauge symmetry

mH

Dynamical EW breaking

Delayed unitarity violat.

Fundamental scale at TeV

• Very fertile field of research• Different proposals not mutually excluded

LITTLE HIGGS SUPERSYMMETRY HIGGS-GAUGE UNIF.

TECHNICOLOR HIGGSLESS EXTRA DIMENSIONS

Symmetry

Dynamics

80

Cancellation of Existence of

positron

charmtop

10-3 eV??CAVEAT EMPTOR

electron self-energyπ+-π0 mass differenceKL-KS

mass differencegauge anomaly

cosmological constant

€

Necessary tuning MZ

2

Λ2→

MZ2

MGUT2

≈10−28

Qu

ickTim

e™

and

aTIF

F (U

nco

mp

resse

d) d

eco

mp

resso

rare

nee

de

d to

see th

is pic

ture

.

n

It is a problem of naturalness, not of consistency!

81

HIGGS AS PSEUDOGOLDSTONE BOSON

€

Φ= + f

2e iθ / f Φ = f Φ → e iaΦ :

ρ → ρ

θ →θ + a

⎧ ⎨ ⎩

Non - linearly realized symmetry h → h + a forbids m2h2

Gauge, Yukawa and self-interaction are non-derivative couplings Violate global symmetry and introduce quadratic divergences

Top sector ●●

➤

➤

No fine-tuning

If the scale of New Physics is so low, why do LEP data work so well?

82A less ambitious programme: solving the little hierarchy



strong dynamics

new physics

energy 1 TeV 10 TeV

Little Higgs Composite Higgs

Higgsless

LEP

€

H +τ aHWμνa Bμν 10 9.7

H +Dμ H2

5.6 4.6

iH +Dμ H L γ μ L 9.2 7.3

e γ μe l γ μl 6.1 4.5

e γ μγ 5eb γ μγ 5b 4.3 3.2

1

2q Lλ uλ u

+γ μq( )2 6.4 5.0

H +d R λ d λ uλ u+σ μν qLF μν 9.3 12.4

LEP1

LEP2

MFV

-- +

€

L = ±1

Λ2O

Bounds on [TeV]

83

Explain only little hierarchy

At SM new physics cancels one-loop power divergences

LITTLE HIGGS

LH

224

4

22

222

2

TeV10 loops Two

TeV loop One

≈≈≈⇒=

≈<⇒=

SMFSM

FH

FSMSMSM

FH

mGm

Gm

Gm

Gm

ππ

ππ

“Collective breaking”: many (approximate) global symmetries preserve massless Goldstone bosonℒ1ℒ

2

H2

222

44=

ππ Hm

ℒ1 ℒ2

84

Realistic models are rather elaborate

Effectively, new particles at the scale f cancel (same-spin) SM one-loop divergences with couplings related by symmetry

Typical spectrum:

Vectorlike charge 2/3 quark

Gauge bosons EW triplet + singlet

Scalars (triplets ?)

85

New states have naturally mass

New states cut-off quadratically divergent contributions to mH

Ex.: littlest Higgs model

Log term: analogous to effect of stop loops in supersymmetry

Severe bounds from LEP data

86

TESTING LITTLE HIGGS AT THE LHC

• Discover new states (T, W’, Z’, …)

• Verify cancellation of quadratic divergences

€

mT

f=

λ t2 + λT

2

2λT

f from heavy gauge-boson masses

mT from T pair-production

λT : we cannot measure TThh vertex (only model-dependent tests possible)

87

MT from T production can be measured up to 2.5 TeV

f and gH from DY of new gauge

bosons

Production rate and BR into leptons in region favoured by LEP (gH>>gW)

➤

➤

Can be seen up to ZH mass of 3 TeV

88

Possible to test cancellation with 10% accuracy for mT < 2.5 TeV and mZ < 3 TeV

Cleanest peak from

In order to precisely extract λT from measured cross section, we must control b-quark partonic density

€

Γ T → bW( ) = 2Γ T → tZ( ) = 2Γ T → th( ) ∝ λT2

Measure T width?

89

Concept of symmetry central in modern physicsinvariance of physics laws under

transformation of dynamical variables

Now fundamental and familiar concept, but hard to accept in the beginning

Ex.: Earth’s motion does not affect c

Lorentz tried to derive it from EM

Einstein postulates c is constant (invariance under velocity changes of observer)

dynamics determine symmetries

symmetries determine dynamics



Einstein simply postulates what we have deduced, with

some difficulty and not always satisfactorily, from the

fundamental equations of the electromagnetic field

90

General relativity deeply rooted in symmetry

SM: great success of symmetry principle

Impose SU(3)SU(2)U(1) determine particle dynamics of strong, weak and EM forces

Will symmetries completely determine the properties of the “final theory”?

Or new principles are needed to go beyond our present understanding?

91

life biochemistry atomic physics SM “final theory”

Microscopic probes

Complexity

Breaking of naturalness would require new principles

• the “final theory” is a complex phenomenon with IR/UV interplay

• some of the particle-physics parameters are “environmental”

92

A different point of view



Vacuum structure of string theory

~ 10500 vacua

(N d.o.f in M config. make MN)

Expansion faster than bubble propagation

Big bang universe expanding like an inflating balloon

Unfolding picture of a fractal universe multiverse

93

In which vacuum do we live?

• Large and positive blows structures apart

• Large and negative crunches the Universe too soon Weinberg

Is the weak scale determined by “selection”? Are fermion masses

determined by “selection”? Will these ideas impact our approach to the final theory?The LHC will address this question!

SPLIT SUPERSYMMETRY abandons the hierarchy problem, but uses unification &

DM

Not a unique “final” theory with parameters = O(1) allowed by symmetry

but a statistical distribution

Determined by “environmental

selection”

94

CONCLUSIONSLHC will soon begin operation:

Unveiling the mechanism of EW breakingHiggs?Unconventional Higgs?Alternative dynamics?

If Higgs is found,

New physics at EW scale curing the UV sensitivity? (many theoretical options, none of which is free from tuning) New principle in particle physics?

Date post:	05-Jan-2016
Category:	Documents
Upload:	cece
View:	32 times
Download:	0 times

BEYOND THE STANDARD MODEL

Documents