Introduction to Particle Physics I.III

Risto Orava Spring 2015

cross sections

Lecture III Lorentz invariant scattering

cross section and phase space

Outline

• Lecture I: Units, Elementary Interactions

• Lecture II: Relativistic kinematics

• Lectures III: Lorentz invariant scattering

cross section

• Lecture IV: Accelerators and collider

experiments

• Lecture V: Elements of Quantum

ElectroDynamics

Outline continued...

• Lecture VI: Testing QED

• Lecture VII: Unitary symmetries and QCD

as a gauge theory

• Lectures VIII: QCD in e+e- annihilation

• S-operator

• Fermi’s golden rule

• Total decay rate

• Scattering cross section

• Invariant phase space for nf –particles

• Differential cross section

• 2 → 2 scattering cross section

• Phase space

• Differential cross section

• Unitarity of the S-operator

Lecture III; Lorentz invariant scattering cross section and phase space

observables in particle physics

decay rates

scattering cross

sections

Feynman diagrams

• schematic repesentation of scattering amplitudes • aid in calculating the interaction cross sections

• strictly defined in the momentum space

• time adn space axes defined for the initial and final state

• initial state/final state • ‘on mass shell’ -> length of a 4-vector = rest mass

• diagrams with the identical initial and final states interfere

• intermediate particles • virtual ‘propagators’; may have ‘masses’ ≠ rest mass (may have

m2 < 0); the cost to pay for involving propagators ‘virtuality’

µ1

q2 - m2

S-operator

In both cases denotes the initial state, denotes a multiparticle

final state in a Fock space, and the box represents the dynamics/

interactions and is called the S-operator – predicted by the theory of

the interaction in question.

Example: In QM I & II, it is shown that S H’(t) V(t) in the 1st order

perturbation theory of the Scrödinger equation. S is often a complicated

entity: it describes all the dynamics of the transitions .

The experiments are typically carried out for a specific intial state and

for a specific final state by choosing the beam particle (proton, electron,..)

and by choosing a specific final state (leptons only, two-jet events etc.).

The S-operator is represented by considering its matrix elements:

i f

fi

iSfSSfiSff ifif

ff

' where''' ''

''

S-operator

iTS

TppiS

SfiP

fi

SSffSff

f

amplitudescattering

fi

nservationmomentumco

iffifi

fi

fiif

f

if

fff

1 :notation shorthand a usingby or

)()2(

can write, wegeneralIn

is, place take to process for they probabilit theand

''

get we, state final specific aFor

4

)4(4

ninteractio no

2

'

'

'

''

S-operator By using Feynman diagrams:

In particle physics, one uses transition probability per unit time:

T

Sw

fi

fi

2

golden rule of Fermi

In the previous discussion, the squared Dirac -function was left

open. Enrico Fermi’s approach was the following:

2)4(4

2

)3(3)3(3)(32)3(3

0000)(200

)()2(

)()2()()2()()2(

)(2)(2)(200

fiif

fi

fi

ifif

xppi

if

ifif

tppi

if

TppVT

Sw

ppVppxedpp

ppTppdtepp

if

if

An example: A cubic box of volume V = L3 with infinitely high potential

wells, the allowed momenta are discretely distributed.

golden rule of Fermi...

fff

f

n

f

fiiffi

n...nn

pdV

TppVdw

pdL

nddpL

dnnL

p

f

21 state final particle for the stands where

)2()()2(

hence and

,22

2

3

13

2)4(4

33

3

the total decay width

aa

fiifn

n

nan

naa MppE

pd

E

pd

Ef

f

f

f

f

1

lifetime theand

)(2

...2)2(

1

2

1 2)4(

3

1

13

43

Note: Since Ea is not a Lorentz invariant quantity, a also depends

on the reference frame. The ”lifetime” listed in Particle Data Booklet is

always the lifetime measured in the rest frame of the particle, and

therefore the shortest version of it.

example: cosmic muons

Classically expect cosmic ray muons to travel a distance of

cm 600 m before they decay, i.e. they would be practically impossible

see on the earth. However, when accounting for the special relativity –

time dilatation – the distance of cosmic muons becomes cm 10 km,

in accordance with the observed rate.

scattering cross section The scattering cross section for the case ni=2, i.e. for for the process

2)(()((

2

1)(

:factorflux Moller invariant Lorentz the-again once - find ein which w

2)2(

1

4

1

get, weabove, equations theusingBy

velocity)(relativedensity)(number flux incoming as, stated be alsocan r denominato The

flux incoming timeand surfaceunit per particles incoming of no.

unit timeper ...21 ns transitioof no.)...21(

...21

22 ,22222

2

1

)4(

1

3

43

smmsmmsmmppvEEF

MpppE

pd

F

V

v

wnbanba

nba

ba

ff

ff

mms

babababaabba

fi

n

f

baf

n

f f

f

nni

ab

fif

f

f

invariant phase space for nf-

particles The scattering angle is related to the 4-momentum transfer t

(see previous considerations). Concerning multiparticle final states,

we define the nf-particle phase space as

. 0..

22

..

4322

0

3

1 1

)4(

3

2

23

1

13

)()()(2

:quantityinvariant Lorentz a is shown that becan It

2...

22

is L.IE

i

IL

ii

IL

iiiiii

i

n

n

f

n

i

ifn

n

nn

i

f i

f

f

ff

EmppdpdmpdEE

pd

R

ppE

pd

E

pd

E

pddRR

differential cross section

For obtaining the differential cross section, we define

))(()2(

1

4

1

writeand

)),,(()(:

22

43

2

kjjkfinnjk

kjkjjk

pptMdRFdt

d

ppfppt

ff

By using this expression, differential distributions for other kinematical

variables can be deduced.

22 scattering cross section

Next we will consider the special case:

a + b 1 + 2

))(2(2

))(()(

)()())(()(

)()()()()()(22

functions.- theof ridget tois goal The

.parameters free 2only have wei.e. s,constraint 4 and sn variableintegratio 6 contains space phase The

22

211

)(

21

21

0

1

22

21

21

21

211

0

2

1

0

1

1122

21

21

211

4

21)4(

2122

222

421

211

421

)4(

2

23

1

13

2

2

21

21

211

0

2

1

mmpppsmE

ddE

mpppmpEdpdpdE

EEEEmpppmppd

ppppEEmppdmppdppppE

pd

E

pdR

R

ba

mpEpdp

EE

ba

EE

baba

baba

ba

ba

22 scattering cross section...

0since0

21

211

21

211

1

21

21

21

21)4(

24

21

2111

21

21

211

1

1

2

1)(

),(1

~ ),()(~

2

1

used wewhere

p ,

ba

mEpmEpp

mpE

pppppd

pmEEmpEdEE

22 scattering cross section...

*22

212

*1*2

221

*1

*1*

0

*12

),,(8

1

4)2(

2

obtain weand

0

:frame mass-of-centre In the

dmmss

dR

s

pdmmEss

pddER

sEEpp

s

baba

22 scattering cross section...

),,(2

obtain wesymmetric,y sphericall is integrand thecaseIn

))(2

1(

2

1)2(

fact that theandearlier derivedrelation a used westeps,last For the

22

2122

22

21

*1

22

21

*1

mmss

dRR

mmss

Es

mmEss

differential cross section

2

2

,2

22

2

2*

**1

*

2

2

*1

**

16

1

),,(16

1

following get the we21 ofsection crossinvariant for the Finally,

64

1

obtain we, scattering elastic of case special For the

. since

,64

get we2 case for the equations previous usingBy

22

21

fi

mms

fi

ba

fi

el

a

a

fi

f

Ms

Mmmsdt

d

ba

Msd

d

pp

psF

MsF

p

d

dt

dt

d

d

d

,n

unitarity of the S-operator

operator.unitary a is S s,other word i.e. ,i 1

obtain we

, relation, sscompletene theusingby andfurther thisdevelopingBy

)(2

V

for stands where

,1 ,

n transitiofor the elementsmatrix thefromy probabilitn transitio thecomputecan We

**

numbers quantum types,particle spins, f

3

3

f

222

1

1

SSiSSiiSffSi

ff

pd

SiSfS

fi

f

f

f

f

fifi

unitarity of the S-operator

oncontributi absorbtive ReIm

oncontributi virtual ImRe

thatus tells of partsimaginary and real theof comparison obviousAn

0Im

theory ginterestinan for hand,other On the . hence and 0 theory,free aFor

)()2(

,expression theconcernsn implicatioother The

j and i states orthogonal for two First, ns.implicatio profound hasfact important This

)4(4

fifi

fifi

fi

fi

fififi

fiiffifi

ij

ST

ST

S

.S

δS T

TppiS

ijiSSj

unitarity of the S-operator

:diagramsFeynman theusingby ly symbolicalit rewritecan We

all.at occur n to transitiofor thisy probabilit for the measure a

is which in transitiofor thesection cross total with theS ofelement specific very a relating

),,(Im

: theoremopticalimportant theyields ),0*,f( scattering forward elastic of case special The

state. teintermediaan denotesn where,)()2(Im2

find, weon,contributi absorbtive at thelook closer a Taking

fi

22

*)4(4*

f

mmsM

i

TTppiTTTi

totbaii

n

infniffififi

We will return to

matrix element Mfi

calculation in

lecture V.

• Particle accelerators: motivations

• Center of mass energy

• Acceleration methods

• Cyclotron

• Synchrotron

• Particle physics experiments

• Cross section

• Luminosity

• Particle detectors

• Kinematics and data analysis methods

• Pseudorapidity and transverse momentum

• Momentum conservation in particle jets

• Missing mass method

• Invariant mass method

NEXT: Lecture IV; Accelerators and

collider experiments