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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
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