[PPT] - New results on strangeness production from the NA61/SHINE Maciej PowerPoint Presentation

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XIII Workshop on HIC Wrocław, Jan 6, 2018

New results on strangeness production from the NA61/SHINE Maciej Lewicki

mlewicki@ift.uni.wroc.pl

University of Wrocław

Institute of Theoretical Physics

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NA61/SHINE on the Workshop

Wojciech Brylinski: Charm physics in NA61/SHINE Dag Larsen: Upgrade of the NA61/SHINE facility beyond 2020 for an expanded physics programme

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Section 1 Strangeness in Heavy Ion Collisions

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Strangeness in HIC

Most strangeness produced in the form of: The lightest (anti-)strange mesons (M ≈ 0.5 GeV):

◮ K+ – (u¯

s)

◮ K− – (¯

us)

◮ K0 – (d¯

s)

◮

¯ K0 – (¯ ds)

The lightest (anti-)strange baryons (M ≈ 1.1 GeV):

◮ Λ – (uds) ◮ ¯

Λ – (¯ u¯ d¯ s)

Strangeness neutral mesons: (M ≈ 1.0 GeV):

◮ φ – (s¯

s)

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 1 / 26

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Main strangeness carriers

in A+A collisions at high baryon density

¯ s

strangeness conservation

=

s K+

isospin symmetry

≈

K0

≪

high baryon density

¯ Λ

≪

high baryon density

K−

isospin symmetry

≈

¯ K0

≈

Λ

– sensitive to strangeness content only – sensitive to strangeness content and baryon density

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 2 / 26

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Strange definitions

Strangeness production: Ns¯

s – number of s-¯

s pairs produced in a collision. The experimental ratio: ES = Λ + K + ¯ K π ≈ 2 · Ns¯

s

π Ns¯

s ≈ K+ + K0 ≈ 2 · K+,

π ≈ 3 2

π+ + π−
Ns¯

s

π ≈ 2 3 K+ π+ ES ≈ 4 3 K+ π+

It is convenient to study the ratio ES in this form, as the identification of charged hadrons is relatively easy.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 3 / 26

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Section 2 Strangeness in Phase Transition

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Strangeness in phase transition

confined matter quark-gluon plasma K mesons (anti-)strange quarks gK = 4 gs = 12 2M ≈ 2 · 500 MeV 2m ≈ 2 · 100 MeV

TC ≈ 150 MeV

− →

Phase transition

Lightest strangeness carriers: relatively heavy kaons (M > TC) in the confined phase, relatively light strange quarks (m TC) in QGP.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 4 / 26

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Strangeness in Statistical Model of Early Stage

n = gV (2π)3

d3p

1 eE/T ± 1 ≈ gV MT

2π

3/2 e−M/T ≈ gV 2π2

4·45T3

for heavy particles for light particles K π ∝ MT3/2 T3 · e−M/T

T <Nss>/

non-strange

<

>

s u + d + g ∝ T3 T3 = const(T)

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 5 / 26

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Strangeness in Statistical Model of Early Stage

Temperature dependence

n collision energy in SMES:

sNN [GeV]

5 10 15 20 25 100 150 200 250 300

T[MeV]

Strange/non-strange particle ratio:

sNN

QGP <Nss>/

non-strange

<

>

Crossing the phase transition leads to a decrease of the strange/non-strange particle ratio – the horn-like structure

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 6 / 26

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Strangeness in Statistical Model of Early Stage

Temperature dependence

n collision energy in SMES:

sNN [GeV]

5 10 15 20 25 100 150 200 250 300

T[MeV]

Strange/non-strange particle ratio:

sNN

QGP <Nss>/

non-strange

<

>

Crossing the phase transition leads to a decrease of the strange/non-strange particle ratio – the horn-like structure – Marek’s horn.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 6 / 26

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Rafelski-Müller Dynamic Model

strangeness production in confined matter N + N → N + Y + K π + N → K + Y π + Y → Ξ + K π + Ξ → Ω + K π + N → K + Y π + Y → Ξ + K π + Ξ → Ω + K strangeness production in QGP

q1

2

k k1

2

q

2

q

2

k k1 q1

Rafelski, Müller, Phys. Rev. Lett. 48 (1982) 1066

100 fm/c 1 fm/c

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 7 / 26

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Rafelski-Müller Dynamic Model QGP <Nss>/

non-strange

<

>

sNN Equilibrium value reached in QGP ← fast strangeness production. No enhancement in the transition region ← slow strangeness production in whole hadronic region.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 8 / 26

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Section 3 Strangeness at NA61/SHINE

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NA61/SHINE — facility

T

F-L

T

F-R

PSD T

F-F

MTPC-R MTPC-L VTPC-2 VTPC-1 Vertex magnets T arget GAP TPC Beam S4 S5

S2 S1 BPD-1 BPD-2 BPD-3 V1 V1 V0 THC CEDAR

z x y

p

FTPC-1 VD

FTPC-2/3

Beam detectors: position charge mass time TPCs: electric charge momentum dE/dx ToF: tof PSD: EF – energy of projectile spectators reaction plane

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 9 / 26

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Particle identification — tof-dE/dx

20 40 60 80 100 120

dE/dx [a.u.]

0.8 1 1.2 1.4 1.6 1.8

]

2

)

2

[(GeV/c

2

m

0.4
0.2

0.2 0.4 0.6 0.8 1 1.2 1.4

+

π

+

p

+

e

Be+Be @40A GeV/c

K

20 40 60 80 100 120

dE/dx [a.u.]

0.8 1 1.2 1.4 1.6 1.8

]

2

)

2

[(GeV/c

2

m

0.4
0.2

0.2 0.4 0.6 0.8 1 1.2 1.4

π

p e

Be+Be @40A GeV/c

K-

Very good separation. Very efficient PID in mid-rapidity region.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 10 / 26

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Particle identification — dE/dx

Probability PID. Applicable in forward-rapidity region.

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

500 1000

pions protons kaons deuterons electrons sum

[12.59; 15.85) ∈ p [0.20; 0.30) ∈

T

p charge = 1

dE/dx [a. b.] 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

%

5 − 5

σ / ∆

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

100 200 300 400

pions protons kaons deuterons electrons sum

[12.59; 15.85) ∈ p [0.20; 0.30) ∈

T

p charge = -1

dE/dx [a. b.] 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

%

5 − 5

σ / ∆

Ar+Sc @30A GeV/c Ar+Sc @30A GeV/c

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 11 / 26

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Event selection

The PSD is located most downstream on the beam line and measures the projectile spectator energy EF of the non-interacting nucleons of the beam nucleus. The energy measured by the PSD is used to select events classes corresponding to the collision "violence" (≈ centrality).

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 12 / 26

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Section 4 Results on Strangeness

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Results on strangeness production

Results from NA61/SHINE on identified hadrons produced in strong and electromagnetic processes in primary interactions: Ar+Sc

[CPOD 2017, arXiv:1712.02417]

Be+Be

[Nucl. Phys. A 967, 35 (2017)]

p+p

[Eur. Phys. J. C74 (2014) 2794, Eur. Phys. J. C77 (2017) 671]

World data on Pb+Pb, Au+Au, C+C, Si+Si and p+p: NA49

[Phys.Rev. C77, 024903 (2008)], [Phys.Rev. C66 (2002) 054902], [Phys.Rev. C86 (2012) 054903] [Eur. Phys. J. C68 (2010) 1], [Eur. Phys. J. C45 (2006) 343]

ALICE

[Phys. Lett. B736 (2014) 196], [Eur. Phys. J. C71 (2011) 1655], [Phys. Rev. Lett. (2012) 109]

STAR [Phys. Rev. C79 (2009) 034909], [Phys. Rev. C96 (2017) 044904] BRAHMS [Phys. Rev. C72 (2005) 014908] p+p world data [Z. Phys. C65 (1995) 215], [Phys. Rev. C69 (2004) 044903]

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 13 / 26

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Results stand for primary particles produced in strong and electromagnetic processes. Corrections: Results are corrected for:

◮ biases in event selection ◮ reconstruction inefficiency ◮ weak decays ◮ secondary interactions ◮ detector geometrical acceptance.

MC used for corrections: EPOS 1.99 model and GEANT3.2+NA61/SHINE detector simulation. Uncertainties: There are two sources of statistical uncertainties in results:

◮ data uncertainties ◮ MC corrections uncertainties (insignificant).

The systematic uncertainties comes from:

◮ limited precision of simulation and detector description.

For nucleus-nucleus collisions, the event classes are defined by forward energy measured by PSD. Results for p+p collisions refer to all inelastic interactions.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 14 / 26

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mT spectra and inverse slope parameter

[GeV]

+

K

m

T

m 0.0 0.2 0.4 0.6 0.8 1.0 ]

1

)

2

[(GeV/c

T

dydm n

2

d

T

m 1

3

10

2

10

1

10 1 10

2

10

3

10

N A 6 1 / S H I N E p r e l i m i n a r y 75A GeV/c Pb+Pb Be+Be p+p 0) ≈ y (

+

K

[GeV]

+

K

m

T

m 0.0 0.2 0.4 0.6 0.8 1.0 ]

1

)

2

[(GeV/c

T

dydm n

2

d

T

m 1

3

10

2

10

1

10 1 10

2

10

3

10

N A 6 1 / S H I N E p r e l i m i n a r y 75A GeV/c Pb+Pb Be+Be p+p 0) ≈ y (

K

mT spectra at mid-rapidity were fitted with an exponential function 1 mT d2n dmTdy = A exp

−mT

T

which well describes K spectra for all beam momenta and all reactions

The energy dependence of the inverse slope parameter was predicted to be sensitive to the phase transition between confined matter and QGP.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 15 / 26

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Inverse slope parameter

"step" plot

[GeV]

NN

s 1

2

10

4

10 T [MeV] 200 400

+

K

≈ y

[GeV]

NN

s 1

2

10

4

10 T [MeV] 200 400

K

≈ y

Inverse slope parameter T (vague temperature analogy) – a parameter in fit to transverse mass spectra:

dN mTdmT ∼

= C exp

− mT

T

Especially convenient to study in case of kaons – insignificant effect of collective flow.

A plateau in the phase transition region (

SNN ≈ 10 GeV) ← predicted by SMES.

Be+Be points slightly above p+p and both significantly lower than heavy ions.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 16 / 26

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Inverse slope parameter

Extrapolation of Ar+Sc points to T(y ≈ 0) falls close to Pb+Pb, while smaller systems show significantly smaller values.

y 0.5 1 1.5 2

T [MeV]

50 100 150 200 250

GeV/c A 30

y 0.5 1 1.5 2

T [MeV]

50 100 150 200 250

GeV/c A 40

y 0.5 1 1.5 2

T [MeV]

50 100 150 200 250

GeV/c A 75

Ar+Sc

+

K

_

K Pb+Pb

+

K

_

K p+p

+

K

_

K Be+Be

+

K

_

K C+C

+

K

_

K Si+Si

+

K

_

K

NA61/SHINE NA49 Preliminary

Ar+Sc Ar+Sc Ar+Sc

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 17 / 26

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Energy dependence

"horn" plot [GeV]

NN

s 1

2

10

4

10 0) ≈ (y

+

π /

+

K 0.1 0.2

SPS NA61/SHINE AGS SPS NA49 RHIC

Pb+Pb Au+Au p+p

LHC

Rapid change in strangeness production observed in Pb+Pb – the horn. Plateau-like energy dependence in p+p data.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 18 / 26

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Collision energy dependence of strangeness production

Rafelski-Müller

QGP <Nss>/

non-strange

<

>

sNN

SMES

sNN

QGP <Nss>/

non-strange

<

>

Qualitatively, data follows dependence predicted by SMES. The dependence predicted by the Rafelski-Müller model is in contradiction with heavy-ion data.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 19 / 26

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PHSD model with & without Chiral Symmetry Restoration

Without CSR – prediction of PHSD qualitatively resembles predictions of the Rafelski-Müller model. With CSR – enhanced strangeness production in the confined phase. The strange quark mass used in the string decay Schwinger-formula in assumed to decrease with energy density, while still in the confined phase. Palmese et al. , PRC94 (2016) 044912

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 20 / 26

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Summary – Model predictions with transition to QGP

QGP

equilibrium

SMES PHSD RM

<Nss>/

non-strange

<

>

sNN

High energies – all three models with phase transition predict the strange/non-strange particle ratio close to the one for the equilibrium QGP. At low collision energies (in the confined matter):

◮ RM model predicts the slow and suppressed strangeness production. ◮ PHSD cures this problem reducing the strange quark mass in string decays. ◮ SMES overcomes it by postulating the statistical particle production at early

stage of collisions.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 21 / 26

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System size dependence of strangeness production

[GeV]

NN

s 1 10

2

10 0.2

〉

+

π 〈 〉

+

K 〈

SPS NA61/SHINE SPS NA61/SHINE WORLD (p+p) AGS SPS NA49 RHIC

Pb+Pb Au+Au p+p Ar+Sc

/

[GeV]

NN

s 1

2

10

4

10 0) ≈ (y

+

π /

+

K 0.1 0.2

SPS NA61/SHINE AGS SPS NA49 RHIC

Pb+Pb Au+Au

LHC

p+p Be+Be

SPS NA61/SHINE

Ar+Sc placed in between light and heavy systems. Be+Be almost overlaps with p+p.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 22 / 26

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System size dependence

<W> 1 10

2

10

3

10 0) ≈ (y

+

π /

+

K 0.1 0.15 0.2 0.25

p+p Be+Be Pb+Pb NA61/SHINE preliminary NA49

c GeV/ A 30

WNM

〉 W 〈

1 10

2

10

〉

+

π 〈 / 〉

+

K 〈

0.00 0.05 0.10 0.15 0.20

Ar+Sc Pb+Pb p+p

GeV/c A 75

Be+Be resembles closely the p+p data. Ar+Sc much closer to Pb+Pb

− → Qualitative difference between

7Be+9Be and 40Ar+45Sc.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 23 / 26

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System size dependence of strangeness production - SMES

<W> 1 10

2

10

3

10 0) ≈ (y

+

π /

+

K 0.1 0.15 0.2 0.25

p+p Be+Be Pb+Pb NA61/SHINE preliminary NA49 c GeV/ A 30 SMES WNM

SMES predicts very different system size dependence of K+/π+ ratio than the one measured by the NA61/SHINE experiment. System size dependence predicted by SMES is due to diminishing effect of the canonical strangeness suppression with increasing volume within statistical models.

Poberezhnyuk, Gaździcki, Gorenstein, Acta Phys.Polon. B46 (2015) 10

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 24 / 26

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System size dependence of strangeness production - PHSD

> 1 10

2

10

3

10 0) ≈ (y

+

π /

+

K 0.1 0.15 0.2 0.25

p+p Be+Be Pb+Pb NA61/SHINE preliminary NA49 c GeV/ A 30

<W> 1 10

2

10

3

10 0) ≈ (y

+

π /

+

K 0.1 0.15 0.2 0.25

p+p Be+Be Pb+Pb NA61/SHINE preliminary NA49 c GeV/ A

158

A 150

<W> 1 10

2

10

3

10 0) ≈ (y

+

π /

+

K 0.1 0.15 0.2 0.25

] ] ] ] p+p Be+Be C+C Si+Si Pb+Pb NA61/SHINE preliminary NA49

c GeV/ A

158

A 150 WNM pHSD

PHSD predicts increase of strangeness production with system size at low (<10 GeV) collision energies and decrease at high (>10 GeV) collision energies. PHSD predictions in disagreement with data at high energies.

Palmese et al., PRC94 (2016) 044912

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 25 / 26

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Summary

1 Models of strangeness production in phase transition were reviewed.

Dynamical and statistical approaches were compared. The latter being favored by the data.

2 NA61/SHINE’s results on p+p, Be+Be (preliminary) and Ar+Sc (preliminary) were

presented:

◮ transverse mass mT spectra of kaons and inverse slope parameter T, ◮ mean multiplicities of kaons.

3 Energy dependence of strangeness production in light and heavy systems

was reviewed.

4 System size dependence was studied in the light of new results on

intermediate size nuclei: Be+Be and Ar+Sc.

5 Clear qualitative difference between Be+Be and Ar+Sc measurements

bserved.

6 It was found that none of existing models can reproduce measured system

size dependence.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 26 / 26

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Thank you for your attention!

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BACKUP SLIDES

SLIDE 36

Predictions of SMES

step kink horn

Plateu in "temperature" dependence on collision energy. Enhancement of entropy production in QGP phase (per participating nucleon). Suppresion of strangeness production in QGP phase.

Predictions of the Statistical Model of the Early Stage. "Step" and "horn" discussed in presented analysis.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 1 / 8

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Predictions of SMES

step kink horn

[GeV]

NN

s 1

2

10

4

10 T [MeV] 200 400

+

K

≈ y [GeV]

NN

s 1 10

2

10 0.2

〉

+

π 〈 〉

+

K 〈

SPS NA61/SHINE SPS NA61/SHINE WORLD (p+p) AGS SPS NA49 RHIC

Pb+Pb Au+Au p+p Ar+Sc

Experimental results – confirming SMES predictions. Signatures of PT happen all at the same √sNN.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 1 / 8

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Predictions of Hadron Resonance Gas model

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 2 / 8

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Particle yield extrapolation to full pT acceptance

Examples for Be+Be

[GeV/c]

T

p 0.2 0.4 0.6 0.8 1 ]

1

dy [(GeV/c)

T

n/dp

2

d 0.2 0.4 0.6 0.8 150 GeV/c 75 GeV/c 40 GeV/c 30 GeV/c + X K → Be+Be ≈ y

+

[GeV/c]

T

p 0.2 0.4 0.6 0.8 1 ]

1

dy [(GeV/c)

T

n/dp

2

d 0.2 0.4 0.6 0.8 150 GeV/c 75 GeV/c 40 GeV/c 30 GeV/c + X

K

→ Be+Be ≈ y

In order to obtain the dn/dy yields of K mesons, the data is extrapolated with the exponential function in mT beyond the detector acceptance. dn/(dpTdy) spectra were fitted with the corresponding function in pT. The function integral outside the acceptance region (<10%) is added to the measured yield.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 3 / 8

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Fitting rapidity distribution

Two symmetrically placed gaussians are used to construct the fitting function: ffit(y) = A σ0 √ 2π exp

−(y − y0)2

2σ2

+

A σ0 √ 2π exp

−(y + y0)2

2σ2

Shape parameters: y0 and σ are fixed to values obtained in NA49’s Pb+Pb.

The amplitude A is the only free parameter. Varying the shape parameters provides an estimate of a systematic error.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 4 / 8

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Rapidity Distribution

30A GeV/c 40A GeV/c 75A GeV/c

y

2 − 1 − 1 2

dn/dy

1 2 3 4 5 +

K

_

K

y

2 − 1 − 1 2

dn/dy

1 2 3 4 5 +

K

_

K

y

2 − 1 − 1 2

dn/dy

1 2 3 4 5 +

K

_

K

Pb+Pb spectra shape fits Ar+Sc data surprisingly well. Measurements of tof will add data in y ≈ 0 region in the near future.

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 5 / 8

SLIDE 42

dE/dx distribution

Functions are fitted to experimental data by considering the parameters depending on the absorbing material as free fit parameters:

−dE

dx

trunc

= E0 1 β2

K + ln(γ) − β2 − δ(β, XA, a)
E0 contains all the constant factors.

K adjusts for the shape of the curve around the minimum. Parameters fitted to the data: E0, K, XA, a

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 6 / 8

SLIDE 43

Truncated mean dE/dxTr distribution

The basic peak shape is assumed to be a sum of asymmetric Gaussians: dE dx

total

=

i=d,p,K,π,e

Ni 1

l nl
l

nl √ 2πσi,l exp  − 1 2

x − xi

(1 ± δ) σi,l 2  with widths σi,l (l ≡ npts ≡ # of clusters): σi,l = σ0 √ l xi x1 α (lβ arbitrary fixed at β = − 1

2)

0.00 0.02 0.04 0.06

σil

20 40 60 80 100 120 140 160

npts

0.00 0.05 0.10 0.15 0.20 0.25

σ0 = σil · √ l

Entries 10000 / ndf 2 χ 694.0652 / 126 Constant 3.1985 ± 236.7132 Mean 0.0008 ± 2.3017 Sigma 0.0006 ± 0.0629

Truncated mean <dE/dx> 2.1 2.2 2.3 2.4 2.5 2.6 entries/max 100 200 300 400 500 600 700 800 900

Entries 10000 / ndf 2 χ 694.0652 / 126 Constant 3.1985 ± 236.7132 Mean 0.0008 ± 2.3017 Sigma 0.0006 ± 0.0629

npts=10 npts=20 npts=40 npts=80 npts=160

=694.065

2

χ =0.0628715, σ =2.30172, µ 10: =368.85

2

χ =0.0457802, σ =2.29654, µ 20: =250.676

2

χ =0.0325206, σ =2.29292, µ 40: =155.22

2

χ =0.0234032, σ =2.29092, µ 80: =107.346

2

χ =0.0167061, σ =2.29035, µ 160:

Truncated Mean Distribution

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 7 / 8

SLIDE 44

Strangeness suppression in Q-state

gs

W, gs Q – numbers of internal dof of (anti)strangeness carriers in W-, Q-state.

The entropy carried by strange (and antistrange) particles: Ss = gs g S For massless particles of j-th species: Sj = 4Nj, Ns + N¯

s = S

4 gs g And the strangeness to entropy ratio: Ns + N¯

s

S = 1 4 gs g Estimate (for massless dof): Q-state: gs

Q/gQ ≈ 0.22, W-state: gs W/gW ≈ 0.5

Numerical calculations with true masses considered: energy dependent

Maciej Lewicki (UWr) Strangeness@NA61/SHINE Jan 6, 2018 8 / 8