a. bay beijng october 20051 summary standard model of particles (sm) symmetries, gauge theories,...

A. Bay Beijng October 2005 1

Summary

Standard Model of Particles (SM) Symmetries, Gauge theories, Higgs, ...


syn-: together

metron : measure

Symmetries


What does it mean being "symmetric"

… 6 equivalentpositions for theobserver


What does it mean being "symmetric" .2

the number of possibilities is


Emmy Noether http://www.emmynoether.com

Today theories are based on the work ofE. Noether. She studies the dynamicconsequences of symmetries of a system.

In 1915-1917 she shows that every symmetryof nature yields a conservation law, andreciprocally.

The Noether theorem:

SYMMETRIES CONSERVATION LAW


SYMMETRIES CONSERVATION LAW

Examples of continuous symmetries:

Symmetry Conservation law

Translation in time EnergyTranslation in space MomentumRotation Angular momentumGauge transformation Charge

Ex.: translation in space r r + dif the observer cannot do any measurement on a systemwhich can detect the "absolute position" then p is conserved.

d is a displacement


Symmetries in particle physicsNon-observables symmetry transformations conservation law

/ selection rulesdifference between permutation B.E. / F.D. statis. identical particlesabsolute position r r + p conservedabsolute time t r + E conservedabsolute spatial direction rotation r r' J conservedabsolute velocity Lorentz transf. generators L. groupabsolute right (or left) r r Paritysign of electric charge q q Charge conjugationrelative phase between states with different charge q eiq charge conserved different baryon nbr B eiB B conserved different lepton nbr L eiL L conserveddifference between coherent mixture of (p,n) isospin

€

p

n

⎛

⎝ ⎜

⎞

⎠ ⎟→ U

p

n

⎛

⎝ ⎜

⎞

⎠ ⎟


An introduction to gauge theoriesSome history.

We observe that the total electric charge of a system is conserved.

Wigner demonstrated that if one assumes1) conservation of Energy2) the "gauge" invariance of the electric potential V

=> than the electric charge must be conserved

Point 2) means that the absolute value of V is not important,any system is invariant under the "gauge" change V V+v(in other terms only differences of potential matter)


Wigner conservation of e.m. chargeSuppose that we can build a machine to create and destroy charges.Let's operate that machine in a region with an electric field:

V1

V2

V1

V2

creation of qneeds work W

V1

V2

move charge to V2

V1

V2

destroy q, regain W

regaining W cannotdepend on theparticular valueof V (inv. gauge)

here we gain q(V2-V1)

1 2 3

4

E conservationis violated !


Maxwell and the local charge conservation

Differential equations in 1868:

€

∇E = ρ ∇ × E = −∂B

∂t∇B = 0 ∇ × B = j

€

0 =∇(∇ × B) =∇jTaking the divergence of the last equation:

if the charge density is not constant in time in the element ofvolume considered, this violates the continuity equation:

€

∇j = −∂ρ

∂t

To restore local charge conservation Maxwell introduces in theequation a link to the field E:

€

∇×B = j +∂E

∂t

The concept of global charge conservation has been transformedinto a local one. We had to introduce a link between the two fields.


Gauge in Maxwell theory

€

B =∇ × A E = −∇V −∂A/∂t

Introduce scalar and potential vectors: V, and A

We have the freedom to change the "gauge":for instance we can do

€

V → V + ∂χ /∂twhere is an arbitrary function.To leave E (and B) unchanged, we need to change also A:

€

A → A +∇χ

In conclusion: E and B still satisfy Maxwell eqs, hencecharge conservation, but we had to act simultaneously on V and A.

* Note that we can rebuild Maxwell eqs, starting from A,V,requiring gauge invariance, and adding some relativity:

A,V add gauge invariance Maxwell eqs


Gauge in QM

In QM a particle are described by wave function. Take r,tsolution of the Schreodinger eq. for a free particleWe have the freedom to change the global phase :

€

(r,t) → e iαψ (r,t)still satisfy to the Schroedinger equation for the free particle.

We can rewrite the phase introducing the charge q of the particle

€

(r,t) → e iqθψ (r,t)

We cannot measure the absolute global phase: this is a symmetryof the system. One can show that this brings to the conservationof the charge q: it is an instance of the Noether theorem.

assume global gauge invariance charge conservation

independenton r and t


Gauge in QM .2

If now we try a local phase change:

€

(r,t) → e iqθ r,t( )ψ (r,t)

we obtain a which does not satisfy the free Schroedinger eq.

If we insist on this local gauge, the only way out is to introduce a new field ("gauge field") to compensate the bad behaviour. This compensating field corresponds to an interaction => the Schrödinger eq. is no more free !

add local gauge invariance interaction field

This is a powerful program to determine the dynamics of a systemof particles starting from some hypothesis on its symmetries.


The electron of charge q is represented by the wavefunction satisfying the free Schroedinger eq. (or Dirac, or...)

The symmetry is U(1) : multiplication of by a phase eiq

* Requiring global gauge symmetry we get conservation of charge: we recover a continuity equation

* Requiring local gauge symmetry we have to introduce the massless field (the photon), i.e. the potentials (A,V), and the way it couples with the electron: the Schroedinger eq. with e.m. interaction

QED from the gauge invariance

€

∇j = −∂ρ

∂t

€

1

2m(−ih∇ + qA)2 + qV

⎡ ⎣ ⎢

⎤ ⎦ ⎥ψ (r,t) = ih

∂

∂tψ (r,t)

! Adding artificially a mass to the photon destroys the procedure !


Particles: the set of leptons and quarks of the SM.

The symmetry is SU(2)U(1) U(1) multiplication by a phase eiq

SU(2) similar: multiplication by exp(igT) but T are three 22 matrices and is a vector with three components

This is an instance of a Yang and Mills theory.

Applying gauge invariance brings to a dynamics with 4 massless fields (called "gauge" fields).

Fine for the photon, but how to explain that W+ W- and Z have a mass ~ 100 GeV ? Need to introduce the Higgs mechanism.

EW theory from gauge invariance


Higgs mechanismAnalogy: interaction of the e.m. field with the Cooper pairsin a superconductor. For a T below some critical value Tcthe material becomes superconductor and "slow down" the penetrationof the e.m. field. This looks like if the photon has acquired a mass.

Suppose that an e.m. wave A induces a current J close to the surfaceof the material, J A. Let's write J = M2A.In the Lorentz gauge: A = JReplacing: A = M2A or

A + M2A = 0

This is a massive wave equation:the photon, interacting with the(bosonic) Cooper pairs field has acquired a "mass" M

A


Higgs mechanism in EW

W

We apply the same principle to the gauge fields of the EW theory. We have to postulated the existence of a new field, the Higgs field, which is present everywhere (or at least in the proximity of particles).

The Higgs generates the mass of the W and Z. The algebra of the theory allows to keep the photon mass-less, and we obtain the correct relations between couplings and masses:

On the other hand, the model does not predict the values of themasses and couplings: only the relations between them.


Higgs mechanism in EW .2A new boson is created by quantum fluctuation of vacuum: the Higgs.Consider a complex field and its potential

normalvacuum

V is minimal on the circle of radius

while = 0 is a local max !

Any point on the circle is equivalent...

€

φvide =1

2(v,0)

v

Let's choose an easy one: A fluctuation around this

point is given by:

€

φ=1

2(v + H,0) H is the bosonic

field

Nature hasalso to choose


Spontaneous Symmetry Breaking

Nature has to choose the phase of All the choices are equivalent.Continue analogy with superconductor: superconductivity appearswhen T becomes lower than Tc. It is a phase transition.Assume that the Higgs potential V( ) at high temperature (earlyBigBang) is more parabolic. The phase transition appears whenthe Universe has a temperature corresponding to E ~ 0.5-1 TeV

High T Low T

Nature has to makea choice for Maybe different choicesin different parts of theUniverse.Are there "domains"with different phases ?


Summary of EW with Higgs mechanism

The search for the Higgs particle is one of the most importantof today research projects, at the LHC in particular.Because its mass is not known, it is a difficult search.Moreover there are alternative theories with more than 1 Higgs,or even with no Higgs at all !

I'll give a short description of past, present and future searchesfor the Standard Model Higgs.

The gauge symmetry allows to build the dynamics of the EW theory.In order to give masse to W and Z we use the Higgsmechanism, obtaining as a by-product a new neutral boson: the Higgs.

Bounds on its mass: 60 < MH < 700 GeV


Higgs, Peter W.

P.W. Higgs,Phys. Lett. 12 (1964) 132


Higgs searches. The possible decays

* For M~1 4 GeV: H gg

* For M 2mb: H and cc

--

* For M 2mb up to 1000 GeV/c2:

then gluons hadronize to KK,...

Decay channels depends on MBR

discoverychannels

* Low mass: H , ee,


TEVATRON/LEP/SLD: indirect bounds

Tevatron measurementof the top mass (LP 2005):m(top) =174.3 ± 3.4 GeV

with this constraint:

MH = 98 +52 -36 GeV

or MH < 208 GeV at 95%CL


Example of Higgs searches at LEP .3

muon

muon

jet 1

jet 2

Simulated Higgs event in the DELPHI detector

Z*

Z

H

e+

e

b

b


Example of Higgs searches at LEP .4A closer look to the interaction region. The initial b quarks are found in b hadrons, a B0 for instance.A B0 has an average lifetime of 1.536 ps.Its velocity is not far from c, with a Lorentz boost ~5

€

= 1− (v /c)2[ ]

1/ 2

e+ e

the B0 travels an averagedistance c ~ 2 mm beforedecaying.We can tag such eventsby verifying that some trackspoint at displaced vertices.


b tagging with vertex detector

Solid state DELPHI vertex detector

vertices

example of event with displaced vertices


Higgs searches at LEP

A few events at MH ~ 115 GeV significance 1.7

~ 6 events


The Large Hadron Collider

The LHC is a pp collider built in the LEP tunnel.Ebeam = 7 GeV.

Because the p is a composite particle the total beam E cannotbe completely exploited. The elementary collisionsare between quarks or gluons which pick up only a fractionx of the momentum:

proton

proton

quarksspectators

quarksspectators

p2

p1

x1p1

x2p2

momentum availableis only x1p1+ x2p2


LHC physicsLHC is a factory for W, Z, top, Higgs,...Even running at L~1033 cm-2s-1, during 1 year (107s), integratedluminosity of 10fb-1, the following yields are expected:

Process Events/s Events World statistics (2007)

W e 30 108 104 LEP / 107 Tevatron

Z ee 3 107 106 LEP

Top 2 107 104 Tevatron

Beauty 106 1012 – 1013 109 Belle/BaBar

H (130 GeV) 0.04 105

In one year an LHC experiment can get 10 times the numberof Z produced at LEP in 10 years.


LHC environmentWe have to cope with a huge number of particles

you wish to extract this Higgs 4


LHC experiments

ATLAS CMS

LHCbALICE


SM Higgs production at LHC

(pb)


--

BR

discoverychannels

Higgs searches. The possible decays


Higgs discovery

MH> 130 GeV

gold-plated H ZZ 4

MH< 130 GeVH

ttH ttbb

B

S

LEP


Example: H

Measure the 2 photons 4-momenta (E,p)Combine them and compute the invariant mass of the parent * need to identify the photons * detectors must have the best resolution both in E and position

e.m. calorimeters E resolution:

CMS crystals:

ATLAS liquid Ar Pb sampling

E

5%-3

E

(E) ≈

E

10%

E

(E) ≈


Example: H the background

From photons qq and gg

Also from many 0 random combinations will producea large "combinatorial" background.

1

2

3

4

5

6

In the figure, we must take all the possiblecombinations: (1,2), (1,3),..., (5,6).Some of these combinations can mimicthe H decay.

Because 0 are mostly found in jets, apowerful selection strategy is to require thatthe photons are far from the jets:they must be isolated.


Example: H discovery~ 1000 events in the peak

ATLAS100 fb-1

CMS100 fb-1

K=1.6


More complex: ttH production, H bb

Final state with 4 jets with b hadrons, plus the decay productsof the two W: W2 jets or Wlepton and neutrino

b

b

b

q, lq,

W

Backgrounds: combinatorial from signal itself : with 4 b jets => 6 combinations W+jets, WWbbjj, etc. t t j j ~ 60% of the total

gluons frombeam protons


More complex: ttH production, H bb .2

ATLAS100 fb-1

mH=120 GeV


Higgs in LHCbp

p

beam jet 1

beam jet 2

q

q'

W

H

b

b

lept

on

neutrino

jet b

jet b

•Process is

•b-quarks will hadronize jets of particles

€

HW ±(Z 0) → bb l ±ν l (l +l −)

b jets

lepton

beam jets

a. bay beijng october 20051 summary standard model of particles (sm) symmetries, gauge theories,...

Documents

bay beijng

space r r dif

electric potential v

total electric charge

charge density

symmetries conservation

gauge change v v vin

wigner conservation