Hadronization via coalescence Hadronization via coalescence for light and heavy flavoursfor light and heavy flavours

IN COLLABORATION WITH:V. Minissale, F. Scardina, S. K. Das, V. Greco

S. Plumari

Università degli Studi di CataniaINFN-

Secondo incontro sulla fisica con ioni pesanti a LHCSecondo incontro sulla fisica con ioni pesanti a LHC9-10 October 20179-10 October 2017Universita' degli Studi di TorinoUniversita' degli Studi di Torino

OutlineOutline

Hadronization: FragmentationCoalescence modelp,π, k, Λ spectra and baryon/meson ratio

Heavy Quarks:Λc and D mesons spectra for RHIC and LHC energies Λc/D

0 ratio

Small systems: p-p @ 7 TeV (preliminary) p/π, Λ/k, Λc/D

0

Conclusions

Ultra-relativistic heavy ion collisionsUltra-relativistic heavy ion collisions

Initial Stage

Pre-equilibrium

stage

QGP

Hadronization

Expansion

Chemical and kinetic freeze-out

Rati

o

Meson

Baryon

dN h

d2 ph

=∑f∫ dzdN f

d2 pf

D f → h(z )

Hadronization: fragmentation and coalescenceHadronization: fragmentation and coalescence

Fragmentation function

We use the AKK fragmentation functionS. Albino, B.A. Kniehl, G. Kramer, NPB 803 (2008)

Proton to pion ratio Enhancement:

In vacuum from fragmentation functions the ratio is small

Dq→ p(z )

Dq→π (z)< 0.25

Elliptic flow splitting:

For pT>2 GeV Both hydro and fragmentation predicts similar v2 for pions and protons

R. J. Fries, V. Greco, P. Sorensen Ann.Rev.Nucl.Part.Sci. 58 (2008) 177

Another hadronization mechanism is by coalescence

Hadronization: CoalescenceHadronization: Coalescence

dN Hadron

d2 pT

=gH∫∏i=1

n

pi⋅dσ i

d3 p i

(2π)3f q (x i , pi) f W (x1 ,... , xn ; p1 , ... , pn) δ( pT−∑i

pi T )

Statistical factor colour-spin-isospin

Parton Distribution function

Hadron Wigner function

V. Greco, C.M. Ko, P. Levai PRC 68, 034904 (2003)V. Minissale, F. Scardina, V. Greco PRC 92, 054904 (2015)

f Meson=9π2Θ(σ x

2− xr1

2) Θ(σ p

2−pr1

2+ Δ m12

2)

f Baryon=9π2Θ(σ x

2−

12xr1

2 ) Θ (σp2−pr1

2+ Δ m12

2 )

×9π2Θ (σ x

2−xr2

2 ) Θ (σp2−pr2

2+ Δ m123

2 )

σ x=1/σ p Only one free parameter in fW

Thermal Distribution (pT< 2 GeV)

Collective flow

Fireball radius+radial flow constraints dNch/dy and dET/dy

Minijet Distribution (pT> 2 GeV)

dNq

d2 rT d2 pT

=gg τmT

(2π)3exp (− γT (mT−pT⋅βT )

T )

βT=β0rR

Constraints from experiments

Hadronization: CoalescenceHadronization: Coalescence

dN Hadron

d2 pT

=gH∫∏i=1

n

pi⋅dσ i

d3 p i

(2π)3f q (x i , pi) f W (x1 ,... , xn ; p1 , ... , pn) δ( pT−∑i

pi T )

Statistical factor colour-spin-isospin

Parton Distribution function

Hadron Wigner function

V. Greco, C.M. Ko, P. Levai PRC 68, 034904 (2003)V. Minissale, F. Scardina, V. Greco PRC 92, 054904 (2015)

f Meson=9π2Θ(σ x

2− xr1

2) Θ(σ p

2−pr1

2+ Δ m12

2)

f Baryon=9π2Θ(σ x

2−

12xr1

2 ) Θ (σp2−pr1

2+ Δ m12

2 )

×9π2Θ (σ x

2−xr2

2 ) Θ (σp2−pr2

2+ Δ m123

2 )

σ x=1/σ p Only one free parameter in fW

Constraints from experiments

Bulk matter consistent with viscous hydro, experiments, lQCD The parameter wave function width σp of baryons and mesons is the same at RHIC and LHC!

for (0-10)%Temperature T = 160 MeV

RHIC_________- tRHIC=4.5 fm/c- β0=0.37- R=8.7 fm- V~1000 fm3

LHC_________- tLHC=7.8 fm/c- β0=0.63- R=10.2 fm- V~2500 fm3

Coalescence codeCoalescence codeNumerically the multi-dimensional integral in the coalescence formula is evaluated by Monte-Carlo methodV. Greco, C.M. Ko, P. Levai PRC 68, 034904 (2003)V. Minissale, F. Scardina, V. Greco PRC 92, 054904 (2015)

Condider i test particles

Give a probability P(i) from partonic distribution

Compute the coalescence integral for mesons and baryons

dNMeson

d2 pT

=gM∑i , j

Pq(i)Pq( j)δ(2)( pT−p i T−p j T ) f M (x i , x j; p i , p j)

dN Baryon

d 2 pT

=gB∑i , j , k

Pq (i)Pq( j)Pq(k)δ(2)( pT−p i T−p j T−pk T ) f B(x i , x j , xk ; p i , p j , pk )

π (I = 1, J = 0)Resonances:_______________

k∗ (I = 1, J = 1/2) → kπ

ρ (I = 1, J = 1) → ππ

∆ (I = 3/2, J = 3/2) → Nπ

p (I = 1/2, J = 1/2)Resonances:_______________

∆ (I = 3/2, J = 3/2) → Nπ

k± (I = 0, J = 1/2)Resonances:_______________

k∗ (I = 1, J = 1/2) → kπ

Λ (1116) (I = 0, J = 1/2)Resonances:___________________________________

Σ0(1193) (I = 1, J = 1/2) → Λγ

Λ (1405) (I = 0, J = 1/2) → Σπ

Σ0(1385) (I = 1, J = 3/2) → Λπ with B. R. = 88%

→ Σπ with B. R. = 11,7%

Resonance decayResonance decay

[(2J+ 1)(2I+ 1)]H∗[(2J+ 1)(2I+ 1) ]H (mH∗

mH)

3/2

e−(EH∗−EH )/T

Statistical factor

mesons

baryons

Main hadronic channels including the ground states and the first excited states have been taken into account

RHIC: spectra and baryon/mesonRHIC: spectra and baryon/meson

Resonances improve the description at low pT

Height and pT position of the peak well described

Lack of fragmentation at pT ≈ 5 GeV (seen in pp with AKK)

Without radial flow … (similar to pp collisions but not exactly)V. Minissale, F. Scardina, V. Greco PRC 92, 054904 (2015)

LHC: spectra LHC: spectra ππ, p, , p, kk, , ΛΛwave function widths σp of baryon and mesons kept the same at RHIC and LHC!

LHC: baryon/mesonLHC: baryon/meson

V. Minissale, F. Scardina, V. Greco PRC 92, 054904 (2015)

wave function widths σp of baryon and mesons kept the same at RHIC and LHC!

LHC: LHC: φ mesonφ meson

Discussed question for long time:

φ meson behavior: meson-like or mass effectCoalescence gives a similar slope for φ and p.

V. Minissale, F. Scardina, V. Greco PRC 92, 054904 (2015)

Proton is composed by 3 quarks flowing each with a mass of about 330 MeV

φ is composed by 2 quarks flowing each with a mass of about 550 MeV

Heavy flavourHeavy flavour Hadronization: Fragmentation Hadronization: Fragmentation

dN h

d2 ph

=∑f∫ dzdN f

d2 pf

D f → h(z )

D f → h(z)∝1

z [1−1z−

ϵ

1− z ]2

The distribution function is evaluated at the Fixed-Order plus Next-to-Leading-Log (FONLL)M. Cacciari, P. Nason, R. Vogt, PRL 95 (2005) 122001

We use the Peterson fragmentation functionC. Peterson, D. Schalatter, I. Schmitt, P.M. Zerwas PRD 27 (1983) 105

The parameter ε for meson hadronization fixed by pp collisions data for D mesonsFor baryons we fix it in accordance with e++e- collisions as done in S.K. Das et al, PRD94 (2016) no.11, 114039

Data from STAR Coll., PRD 86, 072013 (2012)

M. Lisovyi, et al. EPJ C76 (2016) no.7, 397

measurement in e± p, pp and e+ e- are in agreement within uncertainties:fragmentation at most independent of the specific production process

dN h

d2 ph

=∑f∫ dzdN f

d2 pf

D f → h(z )

D f → h(z)∝1

z [1− 1z−

ϵ

1− z ]2

The distribution function is evaluated at the Fixed-Order plus Next-to-Leading-Log (FONLL)M. Cacciari, P. Nason, R. Vogt, PRL 95 (2005) 122001

We use the Peterson fragmentation functionC. Peterson, D. Schalatter, I. Schmitt, P.M. Zerwas PRD 27 (1983) 105

(Λc+

D 0 )e+ e−p p

≃0.1 (D s+

D0 )e+ e−p p

≃0.13

Thermal models predicted ratios of about 3 and 2 times larger.A. Andronic et al., Phys. Lett. B571, 36 (2003)I. Kuznetsova, J. Rafelski, EPJ C51, 113 (2007)

Heavy flavourHeavy flavour Hadronization: Fragmentation Hadronization: Fragmentation

~60%

~8% ~6%

Heavy flavourHeavy flavour Hadronization: Coalescence Hadronization: Coalescence

dN Hadron

d2 pT

=gH∫∏i=1

n

pi⋅dσ i

d3 p i

(2π)3f q(x i , pi) f W (x1 ,... , xn; p1 , ... , pn) δ( pT−∑i

pi T )

Statistical factor colour-spin-isospin

Parton Distribution function

Hadron Wigner function

● The width parameters σ in fW(...) fixed by the root-mean-square charge radius as predicted by quark models

C.-W. Hwang, EPJ C23, 585 (2002).C. Albertus et al., NPA 740, 333 (2004)

● Normalization in fW(...) fixed by requiring that Pcoal=1 for p=0

⟨ r2 ⟩D+=0.184 fm2 ⟨ r2 ⟩ Ds+=0.124 fm2 ⟨ r2 ⟩Λc

+=0.152 fm2

charm distribution function at mid-rapidity from parton simulations solving Boltzmann transport eq. that give good description of both RAA and v2(pT) from RHIC to LHC energies.

Heavy flavour:Heavy flavour: Resonance decay Resonance decayIn our calculations we take into account main hadronic channels, including the ground states and the first excited states for D and Λc

___MESONS____________

D+ (I=1/2,J=0)

D0 (I=1/2,J=0)

Ds+ (I=0,J=0)

Resonances______________________

D*+ (I=1/2,J=1) → D0 π+ B.R. 68% → D+ X B.R. 32%

D*0 (I=1/2,J=1) → D0 π0 B.R. 62% → D0 γ B.R. 38%

Ds*+ (I=0,J=1) → Ds

+ X B.R. 100%

Ds0*+ (I=0,J=0) → Ds

+ X B.R. 100%

___BARYONS___________

Λc+ (I=0, J=1/2)

Resonances__________________________

Λc+(2595) (I=0,J=1/2) → Λc

+ B.R. 100%

Λc+(2625) (I=0,J=3/2) → Λc

+ B.R. 100%

Σc+(2455) (I=1,J=1/2) → Λc

+π B.R. 100%

Σc+(2520) (I=1,J=3/2) → Λc

+π B.R. 100%

[(2J+ 1)(2I+ 1)]H∗[(2J+ 1)(2I+ 1) ]H (mH∗

mH)

3/2

e−(E H∗−EH )/T

Statistical factor

Data from STAR Coll. PRL 113 (2014) no.14, 142301

RHIC: resultsRHIC: results

Data from STAR Coll., arXiv:1704.04364 [nucl-ex].

Data from STAR Coll. PRL 113 (2014) no.14, 142301

RHIC: resultsRHIC: results

RHIC: Baryon/mesonRHIC: Baryon/meson

● Compared to light baryon/meson ratio the Λc/D

0 ratio has a larger width (flatter)

● Similar to the one predicted in Y. Oh, C.M. Ko, S.H. Lee, S. Yasui PRC 79, 044905 (2009)

Some more calculations on Λc/D can be found in:S. Ghosh, S. K. Das, V. Greco, S. Sarkar, J. Alam, PRD90 (2014) no.5, 054018.S. K. Das, J. M. Torres-Rincon, L. Tolos, V. Minissale, F. Scardina, V. Greco, PRD94 (2016) no.11,114039.

Data from STAR Coll., arXiv:1704.04364 [nucl-ex].

RHIC: Baryon/mesonRHIC: Baryon/meson

Data from STAR Coll., arXiv:1704.04364 [nucl-ex].

Coal+fragm with wave function width σp of D0 and Λc changed to

have Λc/D0=thermal ratio at pT→0

Blast Wave model:

Λc+

D 0=gΛgD

mTΛ

mTD

K 1(mTΛ/T C)

K1(mTD/TC)

≈gΛgD (

mTΛ

mTD )

1/2

e−(mΛ−mD

)/TC≈0.17for pT << m

Following: L.W.Chen, C.M. Ko, W. Liu, M. Nielsen, PRC 76, 014906 (2007). K.-J. Sun, L.-W. Chen, PRC 95, 044905 (2017).For hypersurface of proper time τ and non relativistic limit:

Coalescence

Λc+

D0∝

gΛgD (

mTΛ

mTD ) e−(m

Λ−mD

)/T C τμ2

Is the reduced mass of the baryon

for pT << m

μ2=m3(m1+ m2)

m1+ m2+ m3

Data from ALICE Coll. JHEP 1209 (2012) 112

LHC: resultsLHC: resultswave function widths σp of baryon and mesons kept the same at RHIC and LHC!

The Λc/D0 ratio is smaller at LHC energies:

fragmentation play a role at intermediate pT

Small systemsSmall systems

R.D.Weller, P. Romatschke arXiv:1701.07145 [nucl-th]

Objections to applying hydro in pp

• Too few particles, cannot be collective

• System not in equilibrium

Traditional view:• QGP in Pb+Pb• no QGP in p+p (“baseline”)

ALICE Coll., PRL 111 (2013) 222301ALICE Coll., J. Phys.: Conf. Ser. 509 (2014) 012091ALICE Coll. (L. Bianchi),NPA 956 (2016) 777-780.

Small systemsSmall systems

R.D.Weller, P. Romatschke arXiv:1701.07145 [nucl-th]

Objections to applying hydro in pp

• Too few particles, cannot be collective

• System not in equilibrium

Traditional view:• QGP in Pb+Pb• no QGP in p+p (“baseline”)

ALICE Coll., PRL 111 (2013) 222301ALICE Coll., J. Phys.: Conf. Ser. 509 (2014) 012091ALICE Coll. (L. Bianchi),NPA 956 (2016) 777-780.

Small systems: baryon/mesonSmall systems: baryon/meson

If we assume in p+p @ 7 TeV a medium similar to the one simulated in hydro:

wave function widths σp of baryon and mesons kept the same at RHIC and LHC!

PRELIMIN

ARY

p+p @ 7 TeV - tpp=1.7 fm/c- β0=0.4- R=2.5 fm- V~30 fm3

Thermal Distribution (pT< 2 GeV)

Collective flow

Fireball radius+radial flow constraints dNch/dy and dET/dy

Minijet Distribution (pT> 2 GeV)NO QUENCHING

dNq

d2 rT d2 pT

=gg τmT

(2π)3exp (− γT(mT−pT⋅βT )

T )βT=β0

rR

Data taken from:ALICE Coll., PRL 111 (2013) 222301ALICE Coll., EPJ C75 (2015) no.5, 226ALICE Coll., talk by A. Grelli SQM2017

ConclusionsConclusions

Good agreement with RHIC and LHC data: p, π, k, Λ spectrabaryon/meson ratio

Heavy Quarks:Λc production at intermediate pT dominant role of coalescence mechanism Λc/D

0 ~1.5 for pT ~3 GeV with Coal.+fragm. model

In p+p at √s=7 TeV assuming a medium like in hydro:Coal.+fragm. qualitative description from light to heavy baryon/meson ratio. coalescence shows a mass ordering at low pT similar to the data

Extension to study Λb and B0 spectra and ratio in p+p, p+Pb, Pb+Pb

Data from STAR Coll. , PRL 113, 142301 (2014)

Data from STAR Coll. PRL 118, 212301 (2017)

● In (0-10)% coalescence implies an increase of the RAA for pT > 1 GeV.

● The impact of coalescence decreases with pT and fragmentation is dominant at high pT.

● In (0-80)% the v2(pT) due to only coalescence increase a factor 2 compared to the v2(pT) charm.

● In (0-80)% coalescence+fragmentation give a good description of exp. data.

F. Scardina, S. K. Das, V. Minissale, S. Plumari, V. Greco, arXiv:1707.05452

Electromagnetic field: time evolutionElectromagnetic field: time evolution

like in:K. Tuchin, PRC 88, 024911 (2013).K. Tuchin, Adv. High Energy Phys. 2013, 1 (2013).U. Gürsoy, D. Kharzeev, K. Rajagopal PRC 89, 054905 (2014).

Assumptions:Electric conductivity σel const. in timeModification in the bulk due to currents is negligibleNo event-by-event fluctuations

∇⋅E=eδ( z−β t)δ(x−xT)

∇⋅B=0 ∇×E=−∂B∂ t

∇×B=∂ E∂ t+ σ el E+ eβδ (z−β t)δ( x−xT )

Solve the Maxwell eq.s by starting with a point-like charge at the xT in the transverse plane and moving in the +z direction with velocity β.

Fold them with the nuclear transverse density profile of the spectator nuclei and sum forward (+) and backward (-) Standard

hydro τ0

Charmform. time

vacuum

S. K. Das, S. Plumari, S. Chatterjee, J. Alam, F. Scardina, V. Greco, PLB768 (2017) 260-264.

Direct Flow vDirect Flow v11 of charm quarks of charm quarks

F ext=qE+qE p

( p×B )

For light quarks was predicted v1≈ 10-3 -10-4

U. Gürsoy, D. Kharzeev, K. Rajagopal PRC 89, 054905 (2014).

For charm quarks due to early form. time we find a sizeable v1 with the same E-B evolutionS. K. Das, S. Plumari, S. Chatterjee, J. Alam, F. Scardina, V. Greco, PLB768 (2017) 260-264.

v 1=⟨ px

pT⟩

Z,

x

By

JJFaradayFaraday

JJHallHall

c

cc

cEx

JJHallHall

px 0

px 0

xy

z

BByy JJFaradayFaraday

++

++