Lecture Relativistic kinematics

Elena Bratkovskaya

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

▪Relativistic-Heavy-Ion-Collider – RHIC - STAR detector at RHIC (Brookhaven): Au+Au at 21.3 A TeV 1 event: Au+Au, 21.3 TeV L=3.8 km

▪Large Hadron Collider - LHC - (CERN): Pb+Pb at 574 A TeV ▪Future facilities: L=27 km FAIR (GSI), NICA (Dubna)

CBM, FAIR (GSI)

SPS HIC experiments

Low Intermediate High Ultra-High Ebeam [A GeV]

0.1 1 10 100 1000 10000 100000

SIS BM@N FAIR NICA SPS RHIC LHC

‚Mixed‘ phase: Baryonic matter hadrons (baryons, mesons) + QGP: quarks and gluons || quarks and gluons || Meson and baryon || Properties of sQGP spectroscopy In-medium effects In-medium effects Chiral symmetry restoration EoS Phase transition to sQGP Critical point in the QCD phase diagram Special relativity

Lorentz transformation: ❑Consider the space-time point ▪ in a given frame S: ( t , x, y,z ) ▪ and in a (moving) frame S‘: ( t, x, y,z ) x x‘ u 1) S‘ moves with a constant velocity u along z-axis

S S‘ z‘ Space-time Lorentz transformation S→S‘: S  S S  S z x = x x = x y y‘ y = y y = y 1  = z =  ( z −ut ) z =  ( z +ut ) Lorentz factor: 2 1 −u t =  ( t −uz ) t =  ( t +uz ) ❑ Consider the 4-momentum: ▪ in a given frame S: p  ( E , p ) = ( E , px , py , pz )  Note: units c=1 ▪ in the (moving) frame S‘: ' ' ' p   ( E, p ) = ( E, px , py , pz ) u t =  ( t − z ) c 2 p' = p , p' = p u Lorentz transformation x x y y t =  ( t + z ) c 2 for 4-momentum S→S‘: ' pz =  ( pz −uE )  uz =|u |=u E =  ( E −upz ) Special relativity

x‘ x 2) S‘ moves with a constant velocity u in arbitary direction relative to S z‘ u     p = ( pu ) / u p = ( pu ) / u S S‘ z z

z Lorentz transformation for 4-momentum S→S‘: y y‘       p = p|| + p⊥ p = p|| + p⊥      z p⊥ = ( px , py ,0 ) p⊥ = ( px , py ,0 )            pu ( pu )u  p u ( pu )u  z z pz p|| = = 2 p|| = = 2   u u u u p p   ||    u  p = p p = pz + p⊥ py y ⊥ ⊥  u px  p⊥ x      ( pu )  p = p + u  − E 2 2   + 1  p⊥ = px + py   E =  E − ( pu ) p|| = pz = pcos Reference frames for collision processes

❑ Collision process a+b in the laboratory frame (LS):    pa pb = 0 pb = 0, Eb = mb  target b 4-momenta p a = (Ea ,pa ), p b = (mb ,0 )

❑ Collision process a+b in the center-of-mass frame (CMS):     * * * * pa pb pa + pb = 0

* * * * 4-momenta p a = (E a ,pa ), p b = (E b ,pb )     p + p p Relative velocity of CMS and LS: u = a b = a Ea + Eb  Ea + mb pb =0 Lorentz transformation for 4-momentum CMS→LS:

CMS:   LS: * *  * *  m E − ( p u ) Ea + ( pbu ) Ea + ( pau ) * b * a a E = E = Eb = Ea = b 2 a 2 1 −u 2 1 −u 2 1 −u 1 −u    * *   *  * mbu * pa − Eau  pb + Ebu  pa + Eau pb = − pa = pb = pa = 1 −u 2 1 −u 2 1 −u 2 1 −u 2 * * pa = − pb Kinematical variables

1 1 +u z  Rapidity y = ln  2 1 −u z    p 1  E + p  1  E + p  Since u = y = ln z = ln || E     2  E − pz  2  E − p|| 

Rapidity is additive under Lorentz transformation: y = y* + y !

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

Transverse mass: 2 2 M ⊥ = p⊥ + m

p|| = M ⊥ sinh( y )

E = M ⊥ cosh( y ) Mandelstam variables for 2→2 scattering t  Definitions of the Lorentz invariants (s,t,u) for a+b→1+2:  p1 pa 2 2 2 s = ( p a + p b ) = ma + mb + 2( p a  p b ) s 2 2 2 = ( p 1 + p 2 ) = m1 + m2 + 2( p 1  p 2 )   ⎯ ⎯ → = ( E* + E* )2 − ( p* + p* )2 = ( E* + E* )2  u  cms a b a b a b pb p2 =0 2 2 ⎯ Lab⎯ .⎯frame⎯ → = ma + mb + 2mb Ea

2 2 t = ( p a − p 1 ) = ( p b − p 2 ) 2 2 ⎯ cms⎯⎯ → = ma + m1 − 2Ea E1 + 2 pa p1 cosa1

2 2 ⎯ Lab⎯ .⎯frame⎯ → = mb + m2 − 2mb E2

2 2 u = ( p a − p 2 ) = ( p b − p 1 ) 2 2 ⎯ cms⎯⎯ → = ma + m2 − 2Ea E2 + 2 pa p2 cosa 2

2 2 ⎯ Lab⎯ .⎯frame⎯ → = mb + m1 − 2mb E1 Invariant variables for 2→2 scattering

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

2 2 2 s + t + u = ( p a + p b ) + ( p a − p 1 ) + ( p b − p 1 ) = p2 + p2 + p2 + ( p + p − p )2   a  b  1  a b 1

p 2

2 2 2 2 s + t + u = ma + mb + m1 + m2

Kinematical limits:

2 2 ‚Thereshold energy‘ s  max(( ma + mb ) ,( m1 + m2 ) )

2 2 2 2 ma + m1 − 2Ea E1 − 2 pa p1  t  ma + m1 − 2Ea E1 + 2 pa p1

(cosa1 = −1) (cosa1 = 1)

2 2 2 2 ma + m2 − 2Ea E2 − 2 pa p2  u  ma + m2 − 2Ea E2 + 2 pa p2 2→2 cross section

Differential cross section a+b→1+2:   p 4 1 ( 2π) 2 pa dσ = M dΦ F if 2

2 |Mif| – squared matrix element   pb p2 2 2 2 F − flux : F = 4 ( p a p b ) − ma mb

* CMS : F = pa s  p  p = ( E , p ) Lab. frame : F = mb pa  Two-body phase space: p | p |

3 3 d p1 d p2 4 1 d 2 =  (p a + p b − p 1 − p 2 ) 2E 2E 3 2 1 2 ((2 ) )

 d 3 p  dp = p2dpdΩ = EpdEdΩ, d = d cosd E E 2 = p2 + m 2  d( E 2 ) = d( p2 ) → 2EdE = 2 pdp → dp = dE p Two-body phase space

d 3 p   1,E  0 = dEd 3 p  ( p p − m 2 ) = d 4 p ( p2 − m 2 ) ( E )  ( E ) =       0,E  0 2E 0 − 1  =  dE ( E 2 ) ➔ Invariant under Lorentz transformation! 2E 0 substitute in two-body phase space: d 3 p  1 d = 1 d 4 p  p2 − m 2  ( E ) 4 ( p + p − p − p ) 2   2 ( 2 2 ) 2  a  b  1  2 6 2E1 − ( 2 ) 3 1 d p1 2 2 = 6  (( p a + p b − p 1 ) − m2 ) ( 2 ) 2E1

3 d p1 1 p1 = p1 E1 dE1d = dE1d 2E1 2E1 2 (Invariant under Lorentz transformation)

! Further considerations require to choose a reference frame!

d 3 p d 4 p = dEd 3 p = dE 2E = dE 2d 3 p 2E Lorentz invariants

d 3 p Show that = p dp dyd 2E T T

3 2 2 1) d p = p dpd cosd = dpT dp||d = 2 pT dpT dp||d

p = M sinh( y ) 2) Use that || ⊥ dp|| = M ⊥ cosh( y )dy = Edy E = M ⊥ cosh( y )

3 3) thus, d p = 2 pT dpT Edyd

d 3 p = p dp dyd (Invariant under Lorentz transformation) 2E T T

Invariant cross section: d 3 d 3 E 3 = d p pT dpT dyd Two-body phase space

* * * * Let‘s choose the center-of-mass system (cms): pa + pb = p1 + p2 = 0

* 2 2 s = ( p a + p b ) = ( p 1 + p 2 ) p1   = ( E* + E* )2 − ( p* + p* )2 = ( E* + E* )2 a b a b a b * * =0 in cms p * p a  b Consider

2 2  (( p a + p b − p 1 ) − m2 ) 2 2 2 =  (( p a + p b ) − 2 p 1 ( p a + p b )+ m1 − m2 ) * in cms: =  ( s − 2E* ( E* + E* )+ m 2 − m 2 ) p2 1 a b 1 2 * 2 2 =  ( s − 2E1 s + m1 − m2 )

2 2 2 2 s + m − m 2 ( s,m ,m ) E* = 1 2 , p* = E* − m 2 = 1 2 1 2 s 1 1 1 2 s

Kinematical function: ( x 2 , y 2 ,z 2 ) = ( x 2 − y 2 − z 2 )2 − 4 y 2 z 2 = ( x 2 − ( y + z )2 )( x − ( y − z )2 ) Two-body phase space

3 1 d p1 2 2 d 2 = 6  (( p a + p b − p 1 ) − m2 ) ( 2 ) 2E1 1 p* = 1 dE* d *  ( s + m 2 − m 2 − 2E* s ) ( 2 )6 2 1 1 2 1

Use that 1  ( f ( x ))dx =  ( x − x ), f ( x ) = 0  0 0 | f ( x0 )| 1  ( ax )dx =  | a |

Thus, * 2 2 * 1 dE  ( s + m − m − 2E s ) =  1 1 2 1 2 s

1 p* 1 d = 1 d * 2 ( 2 )6 2 2 s Two-body cross section in CMS

1 1 d = p*d * 2 ( 2 )6 4 s 1 Differential cross section:

4 4 ( 2π) 2 ( 2π) 2 1 1 d = M dΦ = M p*d * F if 2 p* s if ( 2 )6 4 s 1 a   F d 2

* 1 p 2 in the cms: d = 1 M d * ( 2 )2 4 p* s if a ( s,m 2 ,m 2 ) p* = a b a 2 s

Cross section reads: 2 2 * ( s,m1 ,m2 ) * p = 1 p 2 1  = 1 M d * 2 s 2  * if ( 2 ) 4 pa s Two-body cross section in terms of invariants

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

2 2 2 t = ( p a − p 1 ) = ma + m1 − 2( p 1  p 2 )

* * * * * * * * * in the cms: ( p 1  p 2 ) = Ea E1 − pa  p1 = Ea E1 − pa  p1 cos

2 2 * * * * * t = ma + m1 − 2Ea E1 + 2 pa  p1 cos

dt * * * dt * = 2 pa  p1 d cos = * * d cos 2 pa  p1 * * * dt * d = d cos d = * * d 2 pa  p1

* * 2 2 1 p1 * 1 p1 dt * d = 2 * M if d = 2 * M if * * d ( 2 ) 4 pa s ( 2 ) 4 pa s 2 pa  p1 2 If the matrix element doesn‘t depend of  :  d * = 2 0

d 1 1 2 2 2 * ( s,ma ,mb ) = 2 M if dt 64s * pa = pa 2 s Decay rate A→1+2

 p Decay rate A→1+2  1 4 pA ( 2π) 2 d = M dΦ F if 12  F − flux : F = 2m p2 A 2 |Mif| – squared matrix element

3 3 d p1 d p2 4 1 d12 =  ( p A − p 1 − p 2 ) 2E 2E 3 2 1 2 ((2 ) )

d 3 p   = dEd 3 p  ( p p − m 2 ) = d 4 p ( p2 − m 2 ) ( E )      2E 0 − 3 d p1 4 2 2 4 1 d 12 = d p2 ( p 2 − m2 ) ( E2 ) ( p A − p 1 − p 2 ) 6 2E1 ( 2 ) 3 3 d p1 1 2 2 d p1 1 2 2 2 = 6  (( p A − p 1 ) − m2 ) = 6  ( m A + m1 − 2( p A  p 1 )− m2 ) 2E1 ( 2 ) 2E1 ( 2 )

Futher considerations require to choose a reference frame! Decay rate A→1+2

  Rest frame of A = center-of-mass system 1+2 : * *    p1 p2 * * * pA = p1 + p2 = 0 *  pA = 0 p A = ( E A , pA ) = ( mA ,0 ) in cms 1+2

* * * ( p A  p 1 ) = 2E A E1 = 2m A E1

1 p* d = 1 dE*d * ( m 2 + m 2 − 2m E* − m 2 ) 12 ( 2 )6 2 1 A 1 A 1 2 * * 1 p1 * 1 1 p1 * = 6 d = 6 d ( 2 ) 2 2m A ( 2 ) 2m A

4 4 ( 2π) 2 ( 2 ) 2 1 d = M if dΦ12 = M if 6 F 2m A ( 2 )

Decay rate: * 2 1 p1 * d = 2 M if 2 d 32 m A Dalitz decay A→1+2+3

 p Decay rate for Dalitz decay A→1+2+3  1 p 4 A  ( 2π) 2 d = M if dΦ13 p2 F  F − flux : F = 2m p3 A 2 |Mif| – squared matrix element

3 3 3 d p1 d p2 d p3 4 1 d13 =  (p A − p 1 − p 2 − p 3 ) 2E 2E 2E 3 3 1 2 3 ((2 ) )  p1   Introduce a new variable p = p 1 + p 2  p p2 pA d 4 p 4 ( p − ( p + p )) by -function:     1  2  p3 Thus, A→1+2+3 is treated as A→R +3  12 p  ( E , p ), 2 2  2 p  s12 = E − p Dalitz decay A→1+2+3

 p 1 1 d 3 p d 3 p d 3 p d = 1 2 3  4 (p − p − p − p )   13 9  A  1  2  3  p ( 2 ) 2E1 2E2 2E3 p p2 A 4 4  d p  ( p − ( p 1 + p 2 ))

 3 p3 4 d p 2 2 d p = ds12 , s12  ( p 1 + p 2 ) = p 2E

3 3 3  1 d p d p3 4  d 13( p A , p 1 , p 2 , p 3 ) = ( 2 ) ds12  6  ( p A − p − p 3 ) ( 2 ) 2E 2E3  3 3  1 d p1 d p2 4    6  ( p − p 1 − p 2 ) ( 2 ) 2E1 2E2 

3 d 13( p A , p 1 , p 2 , p 3 ) = ( 2 ) ds12  d 12( p A , p , p 3 )d 12( p , p 1 , p 2 )

➔ 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 Inclusive reactions

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

1 1 2 a a 3 . . b X b . n exclusive reaction inclusive reaction

4 ( 2π) 2 Cross section a+b→1+2+…+n (1+X): dσ = M dΦ F if n

2 2 2 cms * Lab . frame flux : F = 4 ( p a p b ) − ma mb ⎯ ⎯ → pa s ⎯ ⎯ ⎯⎯ → ma pa

n-body phase space: n 3 n  d pi  4   1 d n =    p a + p b − p i  n  2E    3  i=1 i   i=1  ((2 ) ) 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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