Femtoscopic study in high-energy heavy-ion
collisions
TCHoU Research Member Meeting(Date: 24/06/2022)

Ashutosh Kumar Pandey
Postdoctoral Research Associate,
Tomonaga Center for the History of the Universe,
University of Tsukuba

Introduction• Lattice QCD predicts QGP at ~150 MeV temperature• Big-Bang theory of the universe predicts that matter existed in QGP form after ~1µs of theuniverse formation

Ashutosh Kumar Pandey

2Relativistic Colliders• Compressed Baryonic Matter(CBM) at FAIR/GSI and JPARC heavy-ion experimental
project at KEK/JAEA

Ashutosh Kumar Pandey3Heavy-ion collisions• Heavy-ions like Gold (Au79+) and Lead (Pb82+)

are allowed to collide at ultra-relativistic speed inrelativistic colliders like the LHC and RHIC

• Very hot and dense QGP like medium formationin heavy-ion collisions due to inelastic collisions
between nuclei and conversion of kinetic energy
into heat• To obtain the equation of state, it is important toknow the dimensions of fireball which isimpossible to measure directly due to its verysmall size (size ~10-15 m, lifetime ~ 10-23 s)

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Introduction (contd.)• Femtoscopy (or HBT technique) provides a direct tool to measure the source size
• Probe space-time characteristics of the source using particle correlations inmomentum space• Main sources of correlations:➡ Quantum statistics (QS)HBT analysis• Identical bosons - Bose-Einstein quantum statistics• Identical fermions - Fermi-Dirac quantum statistics➡Final-state interactions (FSI)• Strong interaction• Coulomb interaction (repulsion/attraction)
QS + FSI→ Femtoscopic analysis
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Two-particle correlation function
paP2 (pa , pb )C2 =P1 (pa )P1 (pb )rarrb

q = pa - pb
pb

Koonin-Pratt Equation,
PLB70 (1977) 43,S. Pratt et al., PRC42 (1990) 2646

C(q) =Z

Source function

dr| (q, r)|2 S(r)Experimental correlation function

Pair wave function➡ P2(pa,pb) – probability of detection of particles with momenta pa and pb➡ P1(pi) – probability of detection of particle with momentum pi
Ashutosh Kumar Pandey

6Bertsch–Pratt ‘Out-Side-Long’ coordinate system• side is interpreted as geometric size
• out gives info on emission process• long is used for emission time approx.
Longitudinal co-moving system (LCMS):
A rest frame moving along the beam direction such that Pz = 0

Pair rest frame (PRF):

Where MT 2= P02 – Pz2 , PT2 = Px2 + Py2 , Minv2 = P2 and kT = (pT1 + pT2)/2
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Experimental Correlation function

S(q)C(q) =B(q)• S(q) - distribution of q of pairs from same events (signal)• B(q) - distribution of q of pairs from different events (background)
where q : relative momentum of the pair particles

Fit function for identical charged pions and kaons:C(q) = N [(1G(q) = exp() + K(qinv )(1 + G(q))]
22Rout qout

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Rside qside

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Rlong qlong )- fraction of the pairs originate from the spatio-temporal region relevant for correlations,★ K(qinv) - squared Coulomb wave function averaged over a spherical source★

Ashutosh Kumar Pandey

8Identical pion-pion femtoscopy• Source radii decreases with increasing average transverse momentum
Ashutosh Kumar Pandey

9Identical pion-pion femtoscopy (cont.)

Ashutosh Kumar Pandey

10Femtoscopic studies w.r.t. Ѱ1
ALICE Collaboration, Phys. Rev. Lett. 111 (2013) 232302

STAR Collaboration, Phys. Rev. C 73 (2006) 34903

ηAu + Au 62.4 GeV, charged particle v1• Directed flow v1 ( = 〈cos(ϕ - Ѱ1)〉 ) is produced due to interaction between spectator andparticipant particles• v1(η) is zero three times at around mid, forward and backward rapidities : possible
signature of phase transition (J. Brachmann et al. Phys. Rev. C 61 (2000) 024909)• v1 can’t be explained by hydrodynamical models unlike v2 or v3
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Femtoscopic studies w.r.t. Ѱ1• v1 signal can be generated from assuming the “tilted source” initial conditions• Femtoscopic measurements w.r.t. Ѱ1 can give the information about tilt angle θs

M. A. Lisa et al. New J. Phys. 13 (2011) 065006

Ashutosh Kumar Pandey

12Femtoscopic studies w.r.t. Ѱ1• Fit function for identical particles with cross term:C(q) = N [(1) + K(qinv )(1 + G(q))]

2 222222G(q) = exp(−Routqout − Rsideqside− Rlongqlong− Rosqout qside − Rol2 qout qlong − Rsl2 qsideqlong)

Fit function{

222Rμ,0+ 2Rμ,1cos(ϕ − Ψ1) + 2Rμ,2cos(2(ϕ − Ψ1)), (μ = o, s, l, ol)
222Rμ,0+ 2Rμ,1sin(ϕ − Ψ1) + 2Rμ,2sin(2(ϕ − Ψ1)), (μ = os, sl)

2−4R1sl,1θs = tan−12( R2l,0 − R2s,0 + 2R2s,2 )Ashutosh Kumar Pandey

13Event plane reconstruction:• wi = η for TPC• wi (For EPD) = pm*wgr[ringID]*aa• ringID = Tile/2 ,Q∑∑sin(ϕ)/ww
11x,niii• double wgr[16] = { 6.5, 4.0, 2.5, 1.0,
Ψn = tan−1=n( Qy,n ) n ( ∑ wi cos(ϕi)/ ∑ wi ) 0.0, 0.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0,-1.0, -1.0, -1.0, -1.0};• Float_t pm =1; if (Id<0) pm=-1;• aa= NMip ;if (NMip<0.3) aa=0.; else if(NMip>3.0) aa=3.0;

Recentering:

Q′x,n￼ =Q′y,n￼ =Qx,n − ⟨Qx,n⟩σQx,nQy,n − ⟨Qy,n⟩σQy,nFlattening:2−⟨sin(knΨ′n￼)⟩cos(knΨ′n￼) + ⟨cos(knΨ′n￼)⟩sin(knΨ′n￼)}
Ψ′n′￼ ￼ = Ψ′n￼ +∑k{k=1➡ k is taken upto 8

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Event plane resolution:
Res{Ψn,a}Res{Ψn,b} = ⟨cos(n[Ψn,a − Ψn,b]⟩Res{Ψn,b}Res{Ψn,c} = ⟨cos(n[Ψn,b − Ψn,c]⟩Res{Ψn,c}Res{Ψn,a} = ⟨cos(n[Ψn,c − Ψn,a]⟩Res{Ψn,a} =⟨cos(n[Ψn,a − Ψn,b]⟩⟨cos(n[Ψn,c − Ψn,a]⟩⟨cos(n[Ψn,b − Ψn,c]⟩➡ TPC - detector a, EPD East - detector b, EPD West - detector cRes{Ψn,EPD} = ⟨cos(n[Ψn,EPD − Ψn,TPC ]⟩/Res{Ψn,TPC}

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Non-identical particles femtoscopy• In a hydrodynamical induced system :particle =f +tcomponent of mean emission point ofa single particle parallel to the velocity

Adam Kisiel, Phy.Rev.C 81, 064906 (2010)

emission asymmetry• Lighter particles emitted closer to the centre/later than heavier particles• Emission asymmetry only arises in a system where both random (thermal) andcorrelated (flow) velocities exist and are comparable in magnitude
Ashutosh Kumar Pandey (IIT BOMBAY)16

Extracting the source size and emission asymmetryC(k⇤ ) =known from experimentZ

dr0 | (k⇤ , r0 )|2 S(r0 )unknown(rout µout )2
S(r) = exp
2
RoutPair wave function: (treated as weight)ψpair = ψmotion + ψFSI2
rside
2
Rside

2
rlong
2
Rlong

Interaction term
Motion of pair as whole (plane wave: exponential function, only additional phase)
Combination of the regular and singular s-wave Coulomb functions

Ψ(k*, r′)￼ =AC(η) e[Gamow factor⃗−i k *.r*G̃(ρ, η)⃗F(−iη,1,iζ) + fC( k *)r* ]Confluent hypergeometric function
Strong scattering amplitude, modified by the Coulomb interaction•

Adam Kisiel, Phy.Rev.C 81, 064906 (2010)
Ashutosh Kumar Pandey (IIT BOMBAY)17!

Extracting the source size and emission asymmetry1
η=k*aC

Bohr radius of the pair (248.52 fm for pion-kaon)

ζ = k*r*(1 + cos θ*)
Angle between k* and pair relative position r* in PRF⃗ =fC( k *)21 12⃗ ) − i k *A⃗⃗⃗
+ d0 k* − h( k *aCC( k *)
aC[ f0 2]−1

where f0 = 0.137 fm for like-sign pair of pion-kaon, -0.071 fm for unlike-sign pair of pion-kaon,
d0 is the effective radius (taken to 0 for small k* where 1/f0 term dominates).

F(α,1,z) = 1 + αz + α(α + 1)z 2 /2!2 + . . .
Ashutosh Kumar Pandey (IIT BOMBAY)18

Non-identical particles femtoscopy• 3D correlation function converted into infinite set ofm
1D functions in terms
Z of spherical harmonics (Yl )1Clm (k~⇤ ) = pd'd(cos✓)C(k ⇤ , ✓, ')Ylm (✓, ')4⇡
R out (fm)10

ALICE Pb<Pb s NN = 2.76 TeV
0.19 < p T < 1.5 GeV/ c , |d | < 0.8
THERMINATOR-2
default
6 o = 1.0 fm/ c , mt = 2.0 fm/ c
6 o = 2.1 fm/c, mt = 2.0 fm/ c

5± 15% corr. unc.

6 o = 3.2 fm/ c , mt = 2.0 fm/ cµ out (fm)0<5± 20% corr. unc.

6ALI-PUB-490519

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dN ch/dd1/319

Summary• Average source radii decreases with increasing pair transverse momentum• Average source radii increase with increasing system size/multiplicity• Lifetime of the fireball increases from periferal to central collisions• Finite emission asymmetry observed between pions and kaons which shows pions
are emitted later than kaons• It is expected in a system with strong collectivity which includes flow of resonances(consistent with model predictions, e.g. Therminator2 coupled with viscoushydrodynamics)• Source size and emission asymmetry increase from peripheral to central collisions• Results may suggest a 2.1 fm/c delay in emission time which means different particle
species freeze-out at different times

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THANK YOU

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Correlation fromZ strong interaction
S(r)| (q, r)|2 d4 r

C(q) =emission function(source size/shape)

measured correlation

q = 2k*pair wave function(includes cross section)exp(ik ⇤ r)
= exp( ik ⇤ r) + fr
111 ⇤f (k ) =+ d0 k ⇤2 ik ⇤f0
2

s-wave scattering
approximation
effective range
approximation

For only Strong Final State Interaction:C(k ⇤ ) = 1 +X
S

⇢S"S⇤

1 f (k )2
Rspin fractions

2

1dS0p

2 ⇡R!

S+⇤

2<f (k )p
F1 (2k ⇤ R)
⇡R

⇤

S

=f (k )F2 (2k ⇤ R)
R

#Lednicky, Lyuboshitz, Sov. J. Nucl. Phys., 35, 770 (1982)

Z• The correlation function is finally characterised by three parameters:

z2z2e/zdz• radius R, scattering length f0 and effective radius d0 F1 (z) =0F2 (z) = (1 e z )/z• Cross-section 𝞂 (at low k*) is simply: 𝞂 = 4π| f |2Ashutosh Kumar Pandey

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