New developments in EPOS

(1)

New developments in EPOS

Klaus Werner

with B. Guiot, M. Jafarpour, J. Jahan, I. Karpenko, T. Pierog, G. Sophys, M. Stefaniak

(2)

Current activities:

Consistent implementation of HF

Saturation and factorization

Statistical hadronization, results pp, PbPb

Low energies (BES)

EPOS+PHSD

(3)

ALICE

Nature physics 2017 Strangeness

enhancement

“rediscovered”

(4)

—————————————————————

1 EPOS Introduction

—————————————————————

(5)

Conceptually very different compared to other models (Pythia, Herwig...)

Heavily based on S-matrix theory

(S related to T via S_{f i}

=

δ_{f i}

+

i

(

2π

)

⁴δ

(

p_f

−

^pi

)

T_{f i})

Lorentz invariance

Unitarity

Analyticity (=>crossing symmetry)

(6)

Asymptotic form of T

T ( s, t ) = β ( t ) s

^α⁽^t⁾

≈ β ( t ) s

^α⁽⁰⁾⁺^α^′⁽⁰⁾^t

This form is extremely important!

After Fourier transform:

soft contribution: T

_soft

= β ( b ) ( x

⁺

x

⁻

s

_pp

)

^α⁽^b⁾

pQCD contribution: T

_ladder

= ^∑ β ( b ) ( x

⁺

x

⁻

s

_pp

)

^α⁽^b⁾

(fit of numerical calculation of pQCD calculations)

(7)

Starting point: Unitarity

2s σ

_tot

= ¹

i disc T

_el

Basic assumption :

Multiple “Pomerons”

iT

_el

= ∑

k

1 k! { iT

_Pom

× ... × iT

_Pom

}

(8)

Evaluate

1 i disc { iT

_Pom

× ... × iT

_Pom

} using “cutting rules” :

A “cut” multi-Pomeron dia- gram amounts to the sum of all possible cuts

+

+ +

Summing uncut contributions

=> Multiple scattering weights (

n-fold integrals, n up to 10⁷

)

=> Sampled via Markov chains

(9)

High multiplicity pp or AA: Many cut Pomerons

=> Many kinky strings

=> core + corona

core => hydro evolution => statistical decay

(10)

—————————————————————

2 News

—————————————————————

(11)

2.1

Microcanonical hadronization of plasma droplets

No need to match dynamical part of hydro evolution

(sudden statistical decay)

Energy and flavor conservation

(important for small systems)

Extremely fast

(12)

Grand canonical decay, T = 130 MeV V=50 fm

³

; V=1000 fm

³

0 2.5 5 7.5

0.5 1 1.5

X = E / < E >

dN/dX

(13)

Microcanonic decay

of given volume in its CMS into n hadrons

dP = C

_vol

C

_deg

C

_ident

× δ ( E − ^Σ E

_i

) δ ( ^Σ ~ ^p

i

) ∏

A

δ

_Q_A_,Σq_{A i}

n

∏

i=1

d

³

p

_i

C

_vol

= ^V

n

( 2π¯h )

³ⁿ

^, ^C

^deg

=

n

∏

i=1

g

_i

, C

_ident

= ∏

α∈S

1 n

_α

!

(nαis the number of particles of species α,S is the set of particle species)

Different from decay rate of a massive particle (using LIPS), where asymptotic states are defined over an infinitely large volume

(see Becattini et al, EPJC35:243-258,2004).

ButEi

=

^qp²_i

+

m²_i

(14)

Microcanonical decay

dP ∝ dΦ

_NRPS

= δ ( M − ^ΣE

ⁱ

) δ ( ^Σ ~ ^p

i

)

n

∏

i=1

d

³

p

i

Hagedorn 1958 methods to compute Φ

NRPS

Lorentz invariant phase space (LIPS) (James 1968)

Hagedorn methods used for decaying QGP droplets (Werner, Aichelin, 1994, Becattini 2003)

2012 (Bignamini,Becattini,Piccinini) compute Φ

_NRPS

via

the Lorentz invariant phase space (LIPS)

(15)

Hagedorn integral method can be made very efficient at large n (new), but it is VERY time consuming at small n

LIPS method very fast for small n, gets time consuming at large n

around n ≈ 30 − 40 both methods work

(=> checks)

(16)

Hagedorn integral method The phase-space integral:

φ

_NRPS

( M, m

1

, . . . , m

n

)

= ( 4π )

ⁿ

Z

n

∏

i=1

p

²_i

δ ( E −

n

∑

i=1

E

i

) W ( p

1

, . . . , p

n

)

n

∏

i=1

dp

i

, with the “random walk function” W (angular integral)

W ( p

1

, . . . , p

n

) : = ¹ ( 4π )

ⁿ

Z δ

n

∑

i=1

p

i

× ~ ^u

ⁱ

n

∏

i=1

dΩ

i

New: Very efficient procedures to compute W for large n

with high precision

(17)

We obtain (Werner, Aichelin 94) φ ( M, m

₁

, . . . , m

_n

) =

Z

1 0

dr

₁

. . . Z

1

0

dr

_n₋₁

ψ ( r

₁

, ..., r

_n₋₁

)

ψ = ( 4π )

ⁿ

T

ⁿ⁻¹

( n − 1 ) !

n

∏

i=1

p

_i

E

_i

W ( p

₁

, . . . , p

_n

) ,

with z

i

= r

^1/i_i

, x

i

= z

i

x

i+1

, s

i

= x

i

T, t

i

= s

i

− ^s

ⁱ−¹

, E

i

= t

i

+ m

i

, T = M − ^∑

ⁿi=1

m

i

Suited for MC via Markov chains if W is known

(18)

2.2

Grand canonical limit

For very large M we should recover the “grand canonical limit” for single particle spectra:

f

_k

= ^g

^k

^V

( 2π¯h )

³

^exp

− ^E

^k

T

,

The average energy is E ¯ = ∑

k

g

_k

V ( 2π¯h )

³

Z

^∞

0

E

_k

exp

− ^E

^k

T

4π p

²

dp

Changing variables via E_kdE_k = pdp, and using K₁(z) = zR∞

1 exp(−zx)√

x²−1dx, and 3 K₂(z) =z²R∞

1 exp(−zx)√

x²−1³dx

(19)

=>

E ¯ = ∑

k

4πg

k

V ( 2π¯h )

³

^m

2k

T

3TK

₂

( ^m

^k

T ) + m

_k

K

₁

( ^m

^k

T ) . The microcanonical decay of an object of mass M and volume V should converge (for M → ^∞ ) to the GC single particle spectra

with T obtained from M = E. ^¯

(20)

2.3

Comparing GC et MiC decay We consider a complete set of hadrons ( ≈ 400, PDG list)

We check the effect of

energy conservation

flavor conservation

(21)

GC decay, E/V= 0.57 GeV/fm

³

T=167 MeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

GrCan

V=87.5 fm

³

(22)

GC+MiC decay, E/V= 0.57 GeV/fm

³

M=200 GeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

M = 200 GeV

GrCan + MiCan

V=350 fm

³

× ¹ ₄

good test for Metropolis proposal

(23)

GC+MiC decay, E/V= 0.57 GeV/fm

³

M=100 GeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

M = 100 GeV

GrCan + MiCan

V=350/2 fm

³

× ¹ ₂

(24)

GC+MiC decay, E/V= 0.57 GeV/fm

³

M=50 GeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

M = 50 GeV

GrCan + MiCan

V=350/4 fm

³

× ¹

(25)

GC+MiC decay, E/V= 0.57 GeV/fm

³

M=25 GeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

M = 25 GeV

GrCan + MiCan

V=350/8 fm

³

× ²

(26)

GC+MiC decay, E/V= 0.57 GeV/fm

³

M=12.5 GeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

M = 12.5 GeV

GrCan + MiCan

V=350/16 fm

³

× ⁴

(27)

GC+MiC decay, E/V= 0.57 GeV/fm

³

M=6.25 GeV

10

^-4

10

^-3

10

^-2

10

^-1

1 10

0 0.5 1 1.5 2

p (GeV/c)

dn/dp (GeV/c)

-1

π K p Λ Ξ Ω

M = 6.25 GeV

GrCan + MiCan

V=350/32 fm

³

× ⁸

(28)

2.4

Hadronization on hyper-surface

Hypersurface element:

dΣ

_µ

= ε

µνκλ

∂x

^ν

∂τ

∂x

^κ

∂ϕ

∂x

^λ

∂η dτ dϕ dη

x

⁰

= τ cosh η, x

¹

= r cos ϕ, x

²

= r sin ϕ, x

³

= τ sinh η with r = r ( τ, ϕ, η ) , repre- senting the FO condition.

d Σ _µ

(29)

GC particle production via Cooper-Frye

E dn d

³

p =

Z

dΣ

_µ

p

^µ

f ( up ) , assuming that “matter” is a thermalized resonance gas

(adding δ f for viscous hydro, close to equilibrium)

(30)

Our approach:

Flow of momentum vector dP

^µ

and conserved charges dQ

A

through the surface element:

dP

^µ

= T

^µν

dΣ

_ν

, dQ

_A

= J

_A^ν

dΣ

_ν

.

(with A ∈ {^{C, B, S}}^, corresponding electric charge, baryon number and strangeness)

dP ^µ

(31)

Momentum and charges are conserved :

Z

ΣFO

dP

^µ

= P

_ini^µ

, Z

ΣFO

dQ

_A

= Q

_{A ini}

r

τ

(32)

Construct an effective mass by summing surface elements:

M =

Z

surface area

dM , with

dM = ^q dP

^µ

dP

_µ

,

knowing for each element four-velocity

U

^µ

= dP

^µ

/dM, The four-velocity U

^µ

is NOT

equal to the fluid velocity u

^µ

!

(33)

The effective mass decays microcanonically

Particles are distributed on the hyper-surface

x

^µ

( τ, ϕ, η )

according to the distribution dM ( τ, ϕ, η )

and they are boosted according to the four-velocity

U

^µ

( τ, ϕ, η )

(34)

2.5

Hadronization in pp

What are the effective masses produced in pp?

Where are they produced in space and time?

(35)

Effective mass vs η

Event with 6 Pomerons

(~ 3

×

minimum bias )

E

core

≈ ^2300GeV

(36)

Effective mass vs η

Event with 6 Pomerons

(~ 3

×

minimum bias )

E

core

≈ ^2300GeV

(37)

Effective mass vs η

Event with 12 Pomerons

(~ 6

×

minimum bias )

E

core

≈ ^3700GeV

(38)

Effective mass vs η

Event with 12 Pomerons

(~ 6

×

minimum bias )

E

core

≈ ^3700GeV

(39)

Effective mass vs η

Event with 2 Pomerons

(~ minimum bias )

E

core

≈ ^430GeV

(40)

Effective mass vs η

Event with 2 Pomerons

(~ minimum bias )

E

core

≈ ^430GeV

(41)

Decaying object extended in space-time

η

τ Does is decay as single

effective mass M?

(42)

Decaying object extended in space-time

η

τ ∆η _... or as several in-

dependent objects of width ∆η

We will try several

choices of ∆η

(43)

Xi to pion ratio

10

^-3

10

^-2

1 10 10

²

10

³

<dn

_ch

/dη(0)>

ratio to π ALICE (black)

EPOS 3.264 Ξ

inf 4.5 1.8

different choices of ∆η

∆ η = ^∞ : almost flat

∆ η = 1.8 : drops

quickly around

dn/dη = 10

(44)

Omega to pion ratio

10

^-4

10

^-3

1 10 10

²

10

³

<dn

_ch

/dη(0)>

ratio to π ALICE (black)

EPOS 3.264 Ω

inf 4.5 1.8

different choices of ∆η

∆ η = ^∞ : drops slightly

∆ η = 1.8 : drops

quickly around

dn/dη = 10

(45)

3 Summary

New microcanonical hadronization procedure

(big and small systems)

–

Very efficient, possible for “big” systems

–

Coincides with GC for M>10-50GeV

–

Results for yields vs multipicity depend on corre- lation width ∆η

–

New developments in EPOS