Lecture

Relativistic kinematics

Elena Bratkovskaya

SS2020: ‚Dynamical models for relativistic heavy-ion collisions‘

Heavy-ion accelerators

1 event: Au+Au, 21.3 TeV

▪Relativistic-Heavy-Ion-Collider – RHIC -

(Brookhaven): Au+Au at 21.3 A TeV

▪Future facilities:

FAIR (GSI), NICA (Dubna)

▪Large Hadron Collider - LHC - (CERN):

Pb+Pb at 574 A TeV

L=3.8 km

STAR detector at RHIC

CBM, FAIR (GSI)

SPS

L=27 km

HIC experiments

Baryonic matter

||

Meson and baryon

spectroscopy

In-medium effects

EoS

0.1 1 10 100 1000 10000 100000

Low Intermediate High Ultra-High Ebeam [A GeV]

SIS FAIR NICA SPS RHIC LHC

‚Mixed‘ phase:

hadrons (baryons, mesons) +

quarks and gluons

||

In-medium effects

Chiral symmetry restoration

Phase transition to sQGP

Critical point in the QCD phase

diagram

QGP: quarks and gluons

||

Properties of sQGP

BM@N

Special relativity

Lorentz transformation:

x x‘

y y‘

z‘

z

u

S S‘

❑Consider the space-time point

▪ in a given frame S:

▪ and in a (moving) frame S‘:

1) S‘ moves with a constant velocity u along z-axis

)z,y,x,t()z,y,x,t(

)zt(t)zt(t

)tz(z)tz(z

yyyy

xxxx

SSSS

uu

uu

+=−=

+=−=

==

==

Lorentz factor: 21

1

u

−=

Space-time Lorentz transformation S→S‘:

Note: units c=1

❑ Consider the 4-momentum:

▪ in a given frame S:

▪ in the (moving) frame S‘:

)p,p,p,E()p,E(p zyx=

)p,p,p,E()p,E(p'

z

'

y

'

x=

)pE(E

)Ep(p

pp,pp

z

z

'

z

y

'

yx

'

x

u

u

−=

−=

==Lorentz transformation

for 4-momentum S→S‘:

uuu == ||

z

)zc

t(t

)zc

t(t

2

2

u

u

+=

−=

Special relativity

xx‘

y y‘

z‘

z

u

S S‘

2

z

||

yx

||

)p(pp

)0,p,p(p

ppp

u

uu

u

u

==

=

+=

⊥

⊥

Lorentz transformation for 4-momentum S→S‘:

2) S‘ moves with a constant velocity u in arbitary

direction relative to S

z

2

z

||

yx

||

)p(pp

)0,p,p(p

ppp

u

uu

u

u

=

=

=

+=

⊥

⊥

uuuu /)p(p/)p(p zz

==

⊥⊥ = pp

⊥+= ppp z

u

u

)p(EE

E1

)p(pp

u

uu

−=

−

++=

x

py

p

||p

⊥p

y

pz

px

cosppp

ppp

z||

2

y

2

x

==

+=⊥

Reference frames for collision processes

❑ Collision process a+b in the laboratory frame (LS):

4-momenta )0,(mp),p,(Ep bbaaa ==

bbb mE,0p ==

target b

ap

0pb =

❑ Collision process a+b in the center-of-mass frame (CMS):

4-momenta

*

ap *

bp

0pp*

b

*

a =+

)p,(Ep),p,(Ep*

b

*

bb

*

a

*

aa

==

Relative velocity of CMS and LS:ba

a

0pba

ba

mE

p

EE

pp

b

+=

+

+=

=

u

2

aa*

a2

b*

b

2

aa*

a2

b*

b

1

Epp

1

mp

1

)p(EE

1

mE

u

u

u

uu

u

u

−

−=

−−=

−

−=

−=

2

*

aaa

2

*

b

*

bb

2

*

a

*

aa

2

*

b

*

ab

1

Epp

1

Epp

1

)p(EE

1

)p(EE

u

u

u

uu

u

u

u

−

+=

−

+=

−

+=

−

+=

Lorentz transformation for 4-momentum CMS→LS:

*

b

*

a pp

−=

CMS: LS:

Kinematical variables

E

p

=u

Rapidity

−

+=

z

z

1

1ln

2

1y

u

u

Since

−

+=

−

+=

||

||

z

z

pE

pEln

2

1

pE

pEln

2

1y

Transverse mass: 22mpM += ⊥⊥

)ycosh(ME

)ysinh(Mp||

⊥

⊥

=

=

yyy* +=

Rapidity is additive under Lorentz transformation:

y rapidity in Lab frame = y* in cms + y relative rapidity of cms vs. Lab

!

Mandelstam variables for 2→2 scattering

Definitions of the Lorentz invariants (s,t,u) for a+b→1+2:

bp

ap 1p

2p

s

u

t

ab

2

b

2

aframe.Lab

2*

b

*

a

2

0

*

b

*

a

2*

b

*

acms

21

2

2

2

1

2

21

ba

2

b

2

a

2

ba

Em2mm

)EE()pp()EE(

)pp(2mm)pp(

)pp(2mm)pp(s

++=⎯⎯⎯ →⎯

+=+−+=⎯→⎯

++=+=

++=+=

=

2b

2

2

2

bframe.Lab

1a1a1a

2

1

2

acms

2

2b

2

1a

Em2mm

cospp2EE2mm

)pp()pp(t

−+=⎯⎯⎯ →⎯

+−+=⎯⎯ →⎯

−=−=

1b

2

1

2

bframe.Lab

2a2a2a

2

2

2

acms

2

1b

2

2a

Em2mm

cospp2EE2mm

)pp()pp(u

−+=⎯⎯⎯ →⎯

+−+=⎯⎯ →⎯

−=−=

Invariant variables for 2→2 scattering

There are two independent variables and s,t,u are related by:

2

2

2

1

2

b

2

a

2

p

1ba

2

1

2

b

2

a

2

1b

2

1a

2

ba

mmmmuts

)ppp(ppp

)pp()pp()pp(uts

2

+++=++

−++++=

−+−++=++

Kinematical limits:

( )2

21

2

ba )mm(,)mm(maxs ++

1a1a

2

1

2

a1a1a

2

1

2

a pp2EE2mmtpp2EE2mm +−+−−+

2a2a

2

2

2

a2a2a

2

2

2

a pp2EE2mmupp2EE2mm +−+−−+

( )1cos 1a −= ( )1cos 1a =

‚Thereshold energy‘

2→2 cross section

bp

ap 1p

2p

2

2

if

4

dΦMF

π)2(dσ =

ab

*

a

2

b

2

a

2

ba

pmF:frame.Lab

spF:CMS

mm)pp(4F:fluxF

=

=

−=−

( )( )( )23

21ba

4

2

2

3

1

1

3

2

2

1pppp

E2

pd

E2

pdd

−−+=

dEp

Edppdp2EdE2)p(d)E(dmpE

dcosdd,EpdEdΩdpdΩppdpd

22222

23

=→=→=+=

===

|p|p

)p,E(pp

=

Differential cross section a+b→1+2:

|Mif|2 – squared matrix element

Two-body phase space:

Two-body phase space

( )

)m)ppp((E2

pd

)2(

1

)2(

1)pppp()E(mppd

E2

pdd

2

2

2

1ba

1

1

3

6

621ba

4

2

2

2

2

22

4

1

1

3

2

−−+=

−−+−=

−

)E()mp(pd)mpp(pddEE2

pd 22423

0

3

−

−=−=

=

0E,0

0E,1)E(

ddE2

pddEEp

E2

1

E2

pd1

1111

11

1

3

==

substitute in two-body phase space:

(Invariant under Lorentz transformation)

! Further considerations require to choose a reference frame!

=0

2 )E(dEE2

1

pddEE2E2

pddEpdEdpd

323

34 ===

➔ Invariant under Lorentz transformation!

Lorentz invariants

dyddppE2

pdTT

3

=

(Invariant under Lorentz transformation)

Show that

ddpdpp2ddpdpdcosdpdppd ||TT||

2

T

23 ===

)ycosh(ME

)ysinh(Mp||

⊥

⊥

=

=2) Use that Edydy)ycosh(Mdp|| == ⊥

1)

Edyddpp2pd TT

3 =3) thus,

dyddppE2

pdTT

3

=

dyddpp

d

pd

dE

TT

3

3

3

=

Invariant cross section:

Two-body phase space

Let‘s choose the center-of-mass system (cms): 0pppp*

2

*

1

*

b

*

a =+=+

*

ap *

bp

*

1p

*

2p

*

2*

b

*

a

2

cmsin0

*

b

*

a

2*

b

*

a

2

21

2

ba

)EE()pp()EE(

)pp()pp(s

+=+−+=

+=+=

=

Consider

)mmsE2s(

)mm)EE(E2s(

)mm)pp(p2)pp((

)m)ppp((

2

2

2

1

*

1

2

2

2

1

*

b

*

a

*

1

2

2

2

1ba1

2

ba

2

2

2

1ba

−+−=

−++−=

−++−+=

−−+

in cms:

s2

)m,m,s(mEp,

s2

mmsE

2

2

2

12

1

2*

1

*

1

2

2

2

1*

1

=−=

−+=

))zy(x)()zy(x(

zy4)zyx()z,y,x(

222

222222222

−−+−=

−−−=Kinematical function:

)sE2mms(ddE2

p

)2(

1

)m)ppp((E2

pd

)2(

1d

*

1

2

2

2

1

**

1

*

1

6

2

2

2

1ba

1

1

3

62

−−+=

−−+=

Two-body phase space

=|a|

1dx)ax(

s2

1d

2

p

)2(

1d

**

1

62

=

s2

1)sE2mms(dE

*

1

2

2

2

1

*

1 =−−+

0)x(f),xx(|)x(f|

1dx))x(f( 00

0

=−

= Use that

Thus,

Two-body cross section in CMS

**

162 dps4

1

)2(

1d

=

2d

**

16

2

if

F

*

a

4

2

2

if

4

dps4

1

)2(

1M

sp

π)2(dΦM

F

π)2(d

==

*2

if*

a

*

1

2dM

sp4

p

)2(

1d

=

Differential cross section:

in the cms:

s2

)m,m,s(p

2

2

2

1*

1

=

s2

)m,m,s(p

2

b

2

a*

a

=

*2

if*

a

*

1

2dM

sp4

p

)2(

1

=

Cross section reads:

Two-body cross section in terms of invariants

Let‘s express the cross section in terms of Lorentz invariants (s,t), where

)pp(2mm)pp(t 21

2

1

2

a

2

1a −+=−=

**

1

*

a

*

1

*

a

*

1

*

a

*

1

*

a21 cosppEEppEE)pp( −=−=

**

1

*

a

*

1

*

a

2

1

2

a cospp2EE2mmt +−+=

in the cms:

*

1

*

a*pp2

cosd

dt=

*

1

*

a

*

pp2

dtcosd

=

*

*

1

*

a

***d

pp2

dtdcosdd

==

*

*

1

*

a

2

if*

a

*

1

2

*2

if*

a

*

1

2d

pp2

dtM

sp4

p

)2(

1dM

sp4

p

)2(

1d

==

2d

2

0

* =

2

if2*

a

Mp

1

s64

1

dt

d

=

s2

)m,m,s(p

2

b

2

a*

a

=

If the matrix element doesn‘t depend of :

Decay rate A→1+2

1p

2p

Ap

Decay rate A→1+2

12

2

if

4

dΦMF

π)2(d =

Am2F:fluxF =−

( )( )( )23

21A

4

2

2

3

1

1

3

12

2

1ppp

E2

pd

E2

pdd

−−=

)E()mp(pd)mpp(pddEE2

pd 22423

0

3

−

−=−=

( )

)m)pp(2mm()2(

1

E2

pd)m)pp((

)2(

1

E2

pd

)2(

1ppp)E()mp(pd

E2

pdd

2

21A

2

1

2

A6

1

1

32

2

2

1A6

1

1

3

621A

4

2

2

2

2

22

4

1

1

3

12

−−+=−−=

−−−=

|Mif|2 – squared matrix element

Futher considerations require to choose a reference frame!

Decay rate A→1+2

*

1p Rest frame of A = center-of-mass system 1+2 :

*

2p

0p*

A =

0ppp*

2

*

1

*

A =+=

*

1A

*

1

*

A1A Em2EE2)pp( ==

)0,m()p,E(p A

21cmsin

AAA

+

==

*

A

*

1

6

A

**

1

6

2

2

*

1A

2

1

2

A

**

1

*

1

612

dm2

p

)2(

1

m2

1d

2

p

)2(

1

)mEm2mm(ddE2

p

)2(

1d

==

−−+=

6

2

if

A

4

12

2

if

4

)2(

1M

m2

)2(dΦM

F

π)2(d

==

*

2

A

*

12

if2d

m

pM

32

1d

=

Decay rate:

Dalitz decay A→1+2+3

Decay rate for Dalitz decay A→1+2+3

13

2

if

4

dΦMF

π)2(d =

Am2F:fluxF =−

( )( )( )33

321A

4

3

3

3

2

2

3

1

1

3

13

2

1pppp

E2

pd

E2

pd

E2

pdd

−−−=

|Mif|2 – squared matrix element

1p

3p

Ap

2p

Introduce a new variable

by -function:

21 ppp +=

1p

3p

Ap

2p

p

Thus, A→1+2+3 is treated as A→R12+3

))pp(p(pd 21

44

+−

22

12

2pEsp

),p,E(p

−=

Dalitz decay A→1+2+3

( )

))pp(p(pd

ppppE2

pd

E2

pd

E2

pd

)2(

1d

21

44

321A

4

3

3

3

2

2

3

1

1

3

913

+−

−−−=

1p

3p

Ap

2p

p

➔ 3-body phase space is replaced by the product of 2-body phase space factors

by introducing an ‚intermediate state‘: A → 1+2+3 → R12+3

( )

( )

)p,p,p(d)p,p,p(dds)2()p,p,p,p(d

pppE2

pd

E2

pd

)2(

1

pppE2

pd

E2

pd

)2(

1ds)2()p,p,p,p(d

21123A1212

3

321A13

21

4

2

2

3

1

1

3

6

3A

4

3

3

33

612

3

321A13

=

−−

−−=

22

2112

3

12

4p)pp(s,

E2

pddspd =+=

Inclusive reactions

a

b

1

2

3

.

.

.

n

a

b

1

X

Multiparticle production a+b → 1+2+3+…+n

exclusive reaction inclusive reaction

n

2

if

4

dΦMF

π)2(dσ =

n-body phase space:

Cross section a+b→1+2+…+n (1+X):

aa

frame.Lab*

a

cms2

b

2

a

2

ba pmspmm)pp(4F:flux ⎯⎯⎯ →⎯⎯→⎯−=

( )( )n3

n

1i

iba

4n

1i i

i

3

n

2

1ppp

E2

pdd

−+

=

==

22

References

Ref.: E. Byckling, K. Kajantie, „Particle kinematics“,

ISBN : 9780471128854

PDG: Kinematics:

http://pdg.lbl.gov/2019/reviews/rpp2019-rev-kinematics.pdf