Introduction to Particle Physics I

Risto Orava Spring 2017

relativistic kinematics

Lecture III_1

relativistic kinematics

outline •  Lecture I: Introduction, the Standard Model •  Lecture II: Particle detection •  Lecture III_1: Relativistic kinematics •  Lecture III_2: Non-relativistic Quantum

Mechanics •  Lectures IV: Decay rates and cross sections •  Lecture V: The Dirac equation •  Lecture VI: Particle exchange •  Lecture VII: Electron-positron annihilation

outline continued... •  Lecture VIII: Electron-proton elastic

scattering •  Lecture IX: Deeply inelastic scattering •  Lecture X: Symmetries and the quark

model

•  Particle decay

•  Two-particle scattering •  Scattering angle •  Elastic scattering •  Angular distribution •  Relative velocity •  Center of mass and laboratory systems

•  Crossing symmetry

•  Interpretation of antiparticle-states

Lecture III; Relativistic kinematics

relativistic kinematics E 2 =m2c4 + p2c2

E 2 =m2 + p2

v = βc = β

γ = (1−β 2 )−1/2

E = γm

p = γβm = (γ 2 −1)m

β = γ 2 −1 /γ

particle energy E, momentum p, rest mass m in natural units (c = 1) particle velocity Lorentz factor

relativistic kinematics references: •  Nachtmann [I.4], Hagedorn [II.1], Byckling & Kajantie [II.2]

notations:

eproper tim 1

invariant Lorentz

tensormetric

1000010000100001

g

vector-fourcovariant ),(x

vector-fourant contravari ),(),,,(

2

2222

3210

γ

τ µµνµ

µν

µνµν

µ

µ

dtdtxd-dtdτ

xxxxxgxt

g

xt

xtxxxtxx

=⎟⎠

⎞⎜⎝

⎛=

===−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−

−

−==

−=

===

!

!

!

!

notations

mechanics.Newtonian of 1for expression erecover th wei.e.

...21...

211

and , particle icrelativist-non aFor . if icrelativist be tosaid is particleA

relation momentumenergy thefind we

,

invariant, Lorentz ingcorrespond thegcalculatinBy

),(),1(

:as defined is momentum-four The

vector.like- timea isu ,01)1( Since

),1( :velocity-four The

2

2

222

2222

22

222222

0

222

<<

++=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛++=+=

<<>>

+=

−

−===

====

>=−=

===

v

mpm

mpmpmE

mpmp

pmE

pEmump

pEpvmmup

vu

vddt

dtdx

ddxu

!

!!!

!!

!

!

!!

!

!

γ

γ

γττ

µµ

µµµ

particle decay ).0,0,0,(by given is - framerest itsin - particle decaying a of momentum-four The Mp =

p

1p

2p.

:frame laboratoryin lifetime theis where

)1(

:is - lifetime - decay time The

2

222

ττγ

τ

dddt

dt

vdtd

>=

−=!

m

EGeVE

s

ππ

νµπ

γ

τµ

==

⋅= −→ ++

,20

106.2

:allyExperiment

8

.143'

9999.0143 ,20

106.2 :allyExperiment 8

=⇒

=⇔===

⋅= −→ ++

π

π

π

ππ

νµπ

γ

τµ

tt

vmEGeVE

s

constraints

Constraints: (i)  energy-momentum conservation and (ii) mass-shell condition

21 ppp +=

22ii mp =

),( ),( )0,(

222111

22

22

21

21

22

pEppEpMp

mpmpMp

!!!===

===

constraints...

[ ] [ ]

.calculateddirectly becan momenta theof values absolute theand energies the whileunknown,remain and of directions only the i.e.

,))()(2(4

1 :get we

)(2

1

)(21)(1

get we, 21)(

21 :usingBy

)(11

:Therefore

21

22

22

21

22

21

242

21

21

21

22

21

22

22

21

221

211

22

21

222

21

22121

21

pp

pmmmmMMM

mEp

mmMM

E

mmMM

pppM

E

mmMpppppp

ppppM

ppM

EMEpp iiiiii

!!

!!=−++−=−=

+−=

−+=⋅+=

−−=−−+=⋅

⋅+⋅=⋅=⇒=⋅

Mandelstam variables: s, t, u

•  two body scattering. A + B è C + D

pA+pBèpC+pD •  scalar products of 4-vectors are invariants •  possible combinations:

pA�pB pA�pC pA�pD

•  total 4-momentum is conserved => there are

only two independent Lorentz-invariant kinematic variables on which the reaction cross section can depend

Mandelstam variables: s, t, u Three convenient variables: for which: The Mandelstam variables nicely relate to the propagator masses in the leading order diagrams.

2

2

2

2 2 2 2

( )

( )

( )

A B

A C

A D

A B C D

s p pt p pu p p

s t u M M M M

= +

= −

= −

+ + = + + +

two particle scattering

3p1p

2p 4p

31

2 4 The Mandelstam s = (p1 + p2 )2

variables t = (p1 − p3)2

u = (p1 − p4 )2

s+ t +u = mi2

i=1

4∑

24

23

22

21 0

:by defined is frame (c.m.s.) mass ofcenter The

pppp!!!!

+==+

4321

22 )4,...,1(

pppp

imp ii

+=+

==

!!!!!!!!! "!!!!!!!!! #$dependentlinearly 4 t,independenlinearly 2 ,invariants 6

43423241312122

4231

, , , , , and

:invariants Lorentz heconsider tNext . and scattering elasticFor

ppppppppppppmp

mmmm

ii ⋅⋅⋅⋅⋅⋅=

==

cross sections and luminosity

Cross section σ can be defined by:

number of events = σ � L or equivalently

number of events per unit time = σ � dL/dt where an ”event” is an interaction such as pp scattering, is the luminosity, i.e. ”number of chances of an event per unit area”. For a fixed target within the beam of incident particles dL/dt =NJ, where N is the number of target particles and J is the flux per unit area of particles in the incident beam.

cross sections and luminosity

L = f n1n24πσ xσ y

σ =σ referenceN

Nreference

frames of reference

)'*,( )'*,(

),*(

),*(

and '** ,** :frame c.m.s. In the

as labelled

momenta particle and used is ,0 system,Breit the(DIS), processes inelastic deepIn

:Lan with labelled are variables the), target"fixed(" 0 frame, laboratory In the

* :asterixan by denotedoften are variables theframe, c.m.s. In the

0

:by defined is frame mass ofcenter The

44

33

22

212

21

211

4321

31

2

24

23

22

21

pEppEp

pmpEp

pmpEp

pppppp

pp

pp

ppp

pp

pppp

Bii

Li

labi

icmsi

!

!

!!

!!

!!!!!!

!!

!

!!!!

−=

=

−+==

+==

=−==−=

=

=+

==

=

+==+

1 2

3

4

p! p

!−

'p!

'p!

−

*Θ

two particle scattering

[ ] [ ] 2222222

22

21

21

21

224,2

23,13,1

221

2212121

)()(2)()( 222

:function (tringle)Källén theuse we where),,,(41*)( ),(

21*

:1.1) no. excercises (see s of in terms ' and , express nowcan We

one. is whereasinvariant, Lorentz no is )0,**(

cbcbaacbacbabcac-ab--cbaλ(a,b,c)

mmss

mEpmmss

E

pp*,E

*)E*(E)p(psEEpp

i

s

−++−=−−⋅+−=++=

=−=−+=

+=+=+=+

λ!

!!

!"#"$%

two particle scattering

{ }

2'****

:obtains one , limit,energy high At the channel.-s in the process theof threshold theis

0)(,)(max

:followsit 0', From .determined be toprocess scattering theof properties some allows This

),,( :behaviour asymptotic *

and cbaunder symmetric *

:properties following thehasfunction Källén The

4321

2

243

221min

22

2

sppEEEE

ms

mmmms

pp

acbab,c:a

i

======

>>

≥++=

>

→>>

↔↔

!!

!!

λ

scattering angle

tand or and

:st variableindependen by two described is scattering 22 above, theof basis On the

),,(),,(

))(()(*cos

),,(*cos derive we

)(2)(

*cos****

usingBy

*cos''

by defined is * angle scattering theframe, c.m.s. In the

24

23

22

21

24

23

22

21

2

24231

23

21

231

313131

sΘ*s

mmsmms

mmmmuts

mtsfunction

ppppmmpp t

ppEEpp

pppp

i

→

−−+−=Θ

=Θ

−=⋅−+=−=

Θ⋅⋅−=⋅

Θ⋅⋅=⋅

Θ

λλ

!!

!!!!

elastic scattering

( ) ( )

⎪⎩

⎪⎨⎧

⇔⎪⎭

⎪⎬⎫

≥

≤Θ≤

+=Θ⇒

Θ−−=−−=−=

−−⋅+−==

==

→==

≤≤−

+≥

04)(2

2

2231

231

221

221

22

4231

4231

2

2210

1*cos1- :yieldsregion allowed physically theoRelation t

21*cos

*)cos1(2)()(

:scattering elasticin angle scattering for the giving

)()(41'

** *,*

and ), (e.g. and ,scattering elasticIn

tpmmsp

p

t

pppppt

mmsmmss

pp

EEEE

epepmmmm

!

!

!

!!!

!!

angular distribution

'),,(),,(

4*

cos2*

2 i.e. vector,- by the defined axis therespect towith invariant ly rotational ison distributiangular The

24

23

22

21

ppmmsmms

sdt

d

Θ*πd d

πdφp

!!

!

⋅==

Ω

=Ω

=∫

π

λλ

π

relative velocity The relative velocity will be of relevance in defining the particle flux,

v12 =!v1 −!v2 =

!p1

E1

−!p2

E2

=!p1 *E1 *

−!p2 *E2 *

=!p1 *

E1 *E2 *(E1 *+E2*)

s" #$ %$

from which we get,

v12E1 *E2*= s !p1 * = s E1 *2 −m12 = s 1

4s(s+m1

2 −m22 )2 −m1

2

= (p1 ⋅ p2 )2 −m12m2

2The Moller flux factor.

a frame independent quantity!

Note: The Moller flux factor is needed for normalizing the cross sections, since the classical volume element is not Lorentz invariant.

CMS and LAB systems

!L

mmE

Em

EEs

L

12

,L

122

22

1

2221

2E2mmms lab

)energy total(*)*( c.m.s.

:systems laboratory and c.m.s. For the

211 >>

→++=

=+=

An example: Fixed target and colliding beams mode at the Fermilab Tevatron, Ebeam=980 GeV.

CMS and LAB systems

fixed target: m2=mp - secondary beams!

p

p

p

WettfixedpN

Wcolliderpp

mGeVs

mGeVs

<=

∗>=

7.42

21960

arg

s-channel

crossing symmetry

t-channel

the 2→2 scattering process exhibits underlying symmetries

4231

3 2

0,0,0

4321

43

)()(

naively,get then we, and exchange wesuppose instance,For reactions. crosseddifferent describecan then it

region, on the Depending . range whole thely toanalytical extended becan then

),,(),,(

SUSY,...), EW, QCD, (QCD,ally theoreticpredicted is and variablesMandelstam threetheon dependsIt later. more discussed be willand process theof dynamics scattering thedescribes

channel.-snotation thehence s, isreaction for this variableMandelstam positiveonly The

:conserved is momentum-4 thefor which 4,321 :page) (previousreaction channel-s Examineroles. their einterchangu andbut t affectednot is s ,p and p exchange When we:Example

pppp

pp

s,t,uT

utsTutsT

T

pppp

utss

s

+−=−+

ℜ∈

=

+=+

+→+

≤≤>

crossing symmetry

crossing symmetry

),,(),,(

:have Weprocess. channel"-" theofspeak weparticles, incoming theare 3 and 1 Since

:expression the toleading , particle theof leantipartic for the stands in which

:tioninterpreta themake now We

0,0,0

4231

≤>≤=

+=+

=−

utst

nn

utsTutsT

t

pppp

nn

pp

anti-particle states

!

""""" #""""" $%""" #""" $%&

density charge

densityy probabilitchargeelectron *)*(

current,-4 at thelook weclear when becomesfor that reason The. momentum-4 with lesantipartic as dinterprete are momentum-4 with particles The

ϕϕϕϕρ µµµ ∂−∂−=⎟⎟⎠

⎞⎜⎜⎝

⎛= iej

j

p-p

QMED

µµµ

µ

µ

µµ

µµ

ϕ

ppejej

pE

NepNe)(ej-pe

pNepNe)(ejp e

pE

NepNe)(ejpe

Ne

-µµ-

µµ

-µ-

xip

−→=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−−=−−=

−−=+=+

⎟⎟⎠

⎞⎜⎜⎝

⎛

+

+−=−=+

=

−+

++

⋅−

on substituti e with th)()(

:rule thehence And

2)(2: momentum-4 with

)(22: momentum-4with

22: momentum-4 with

get wecurrent,-4 theof definition in the , electron, free theoffunction wave theInserting

22

22

22

!

!

Note: The whole 4-vector pµ takes a minus sign, not only the spatial part.

Note: In the phase, the signs of both pµ and xµ can be flipped without changing the wave function – no place here for particle travelling backwards in time!

anti-particle states

- A particle with 4-momentum –pµ is a representation for the corresponding antiparticle with 4-momentum pµ. - Alternatively: Emission of a positron with energy +E corresponds to the absorption of an electron with energy –E (figure above).

anti-particle states In the Dalitz plot, the three reactions (s-,t- and u-channel ones) are described by a single diagrammatic representation. Function T(s,t,u), evaluated in the relevant kinematical region, describes all three.

example: Moller & Bhabha scattering

Moller: e-e-→e-e- -crossing symmetry- Bhabha: e+e-→e+e-

decay & production

NEXT: Lecture III_2: Non-relativistic Quantum Mechanics