The impact of the sigma meson on the freeze-out,pion condensation and proton puzzle at the LHC

Viktor Begun

Jan Kochanowski University, Kielce, Poland

in collaboration with Wojciech Broniowski, Francesco Giacosa

arXiv:1506.01260

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 1 / 13

The phase diagram of strongly interacting matter

http : //www0.bnl.gov/newsroom/news.php?a = 24281

0 0.2 0.4 0.6 0.8 1µ

B (GeV)

0

0.05

0.1

0.15

0.2

Tch

(G

eV)

s/T3=7

percolation

E/N = 1.08 GeV

ALICE

RHIC

SPS

GSI

AGS

Cleymans, EPJ Web Conf. (2015)

Thermal model gives the freeze-out curve

The fit of the LHC data gives the parameters that fall out to the "wrong" side

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 2 / 13

Problems of thermal models with the mean multiplicities at the LHC

ALICE Collaboration, Phys. Rev. Lett. (2012)

m(D

ata-

Fit)/

-3

-2

-1

0

1

2

3 =2.76 TeVNNsPb-Pb 3= 0 MeV, V=5380 fm

bµFit: T=156 MeV,

+/ -/ +K -K s0K 0K* q p p R -U

+U

-1

+1 d HR3 HR3

Stachel, Andronic, Braun-Munzinger, Redlich,

J. Phys. Conf. Ser. (2014)

The prediction of thermal models gave too high ratios to pions, especially proton topion ratio

The best fit of the LHC data gives three standard deviations for protons

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 3 / 13

Problems of hydrodynamic models with the pion spectra at the LHCIP - Glasma + MUSIC:

10-1

100

101

102

103

104

105

0 0.5 1 1.5 2 2.5

(1/2/)

dN

/dy

p T d

p T

pT [GeV]

ALICE 0-5%

IP-Glasma+MUSIC /+ + /-

K+ + K- p + —p

Gale, Jeon, Schenke, Tribedy, Venugopalan,

Phys. Rev. Lett. (2013)

pion enhancement

AdS + hydro + cascade:

0 1 2 3 4p

T[GeV/c]

0.001

0.01

0.1

1

10

100

1000

dN

/dY

/dp

T

AdS+hydro+cascade

ZF+hydro+cascade

FS+hydro+cascade

Light Particle Spectra

Pb+Pb @ √s = 2.76 TeV

π+, π

-

K+, K

- x 0.1

Schee, Romatschke, Pratt, Phys. Rev. Lett. (2013)

pions well described, protons?

Hydro with dynamical freeze-out:

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

0 0.5 1 1.5 2 2.5

dN

/dyd

pT2 [

1/G

eV

2]

pT [GeV]

0-5 %

10-20 %

20-30 %

30-40 %

40-50 %

π+

Tdec = 100 MeV

LHC 2760 AGeV

idealLH-LQLH-HQHH-LQHH-HQALICE

Molnar, Holopainen, Huovinen, Niemi, Phys. Rev. C (2014)

pion enhancement

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 4 / 13

Cracow single freeze-out thermal model

The phase-space distribution of the primordial particles has the form:

fi = gi

∫d3p

(2π)3

1

γi−1 exp(

√m2 + p2/T ) ± 1

, where γi = γN i

q+N iq̄

q γN i

s+N is̄

s exp(µBBi + µSSi

T

),

and N iq , N i

s are the numbers of light (u,d) and strange (s) quarks in the ith hadron. Itincludes all well established resonances from PDG. Resonance decay according to theirbranching ratios.

Single-freeze out model (Broniowski, Florkowski, Phys. Rev. Lett. (2001))Monte-Carlo implementations, THERMINATOR 1 & 2 (Kisiel, Taluc, Broniowski, Florkowski, Comput. Phys.

Commun. (2006); Chojnacki, Kisiel, Florkowski, Broniowski, Comput. Phys. Commun. (2012))

The spectra are calculated from the Cooper-Frye formula at the freeze-out hyper surface

dNdyd2pT

=

∫dΣµpµf (p · u), t2 = τ2

f + x2 + y2 + z2, x2 + y2≤ r2

max,

assuming the Hubble-like flow: uµ = xµ/τf .

There is only one additional parameter in the model, because the product πτf r2max is

equal to the volume (per unit rapidity), while the ratio rmax/τf determines the slope of thespectra.

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 5 / 13

Spectra of pions in Cracow model at the LHC. Linear scale

The fits to the ratios of hadronabundances (Petran, Letessier, Petracek,

Rafelski, Phys. Rev. C (2013)) yield γq whichis very close to the critical pionchemical potential

µπ = 2T lnγq ' 134 MeV

' mπ0 ' 134.98 MeV

It may suggest that a substantial partof π0 mesons form the condensate.

The calculations of the pion spectrasupport the formation of thecondensate at the LHCV.B., Florkowski, Phys. Rev. C (2015)

0

500

1000

1500

2000

2500

3000

3500

0.1 0.2 0.3 0.4

0

500

1000

1500

2000

2500

3000

3500

12πpT

d2N

dpTdy

[ (GeV

/c)−

2]

pT (GeV/c)

chemical equilibrium

primordial + secondary pions

secondary pions only

12πpT

d2N

dpTdy

[ (GeV

/c)−

2]

chemical non-equilibrium

Pb+Pb√sNN = 2.76 TeV

c = 0%÷ 5%

V.B., Florkowski, Rybczynski, Phys. Rev. C (2014)

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 6 / 13

Can the LHC data be explained by the updated sigma?

The recent PDG reviews report much lower mass and width of the f0(500) or thesigma meson

The lower mass of the σ would result in it’s higher multiplicity. It decays into pions,therefore it could add some of the missing pions

106 R. Kamiński

Figure 2 presents differences in values estimated for the position of theσ pole (real and imaginary parts) before 2012 and after. As is seen, theRoy’s and GKPY equations led to dramatic changes in these estimations.Following these new results the S ππ amplitudes can be parameterized withmuch higher precision.

Fig. 2. Present and previous ranges of the real and imaginary parts of the σ poleestimated by the Particle Data Group (PDG2012 and PDG2010 respectively). Inthe middle of the circle, there are positions of this pole calculated by the Madrid–Kraków group [2] using the Roy’s (left cross) and GKPY equations (the right one).

To check how the GKPY equations can modify amplitudes determinedpreviously without crossing symmetry condition, these dispersion relationshave been applied to the S- and P -wave amplitudes from analysis presentedin [11]. The mass and the width of the σ meson in this analysis had values ofabout several hundred MeV what significantly differed from those obtainedby the Bern and Madrid–Kraków groups. After application of the GKPYequations the σ pole shifted and placed in vicinity of the position foundby both these groups. Results of this practical application of the GKPYequations were presented in [12].

Question of uniqueness of results obtained by the Bern and Madrid–Kraków groups and confirmed later in [12] can be easily proved using simpletrigonometric arguments. As presented in [12], due to some trigonometricrelations fulfilled by the ππ phase shifts in the S-wave, the crossing symmetrycan be fulfilled only when the real and imaginary parts of the σ pole are invicinity of those indicated by both groups. This proof can definitely finishdiscussions and eliminate doubts about the uniqueness of the results.

Kaminski, Acta Phys. Polon. Supp. (2015); Garcia-Martin, Kaminski, Pelaez, Ruiz de Elvira, Phys. Rev. Lett. (2011)

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 7 / 13

Non-interacting hadron-resonance gas

In thermal models the calculations are performed using the sum of contributions of all(stable and resonance) hadrons to the partition function

ln Z =∑

k

ln Zstablek +

∑k

ln Z resk

In practice, one uses the list of existing particles from the PDG.In the limit where the decay widths of resonances and chemical potentials areneglected, one has

ln Zstable,resk = gk V

∫d3p

(2π)3ln

[1 ± e−Ep/T

]±1

where gk is the spin-isospin degeneracy, V – volume, ~p – momentum, Mk – the mass of the

resonance, Ep =√~p2 + M2

k – the energy, and the ± corresponds to fermions or bosons.

As a better approximation for the partition function, one can take into account the finitewidths of resonances:

ln Z resk = gk V

∫∞

0dk (M) dM

∫d3p

(2π)3ln

[1 − e−Ep/T

]−1

For narrow resonances one can approximate dk (M) with a (non-relativistic or relativistic)normalized Breit-Wigner function peaked at Mk .

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 8 / 13

Interacting hadron gas

The 2→ 2 reactions are incorporated according to the formalism of Dashen, Ma,Bernstein, and Rajaraman. The mass distribution is given by the physical phase shifts δ:

dk (M) =dδ(M)

πdM

One can get it for the relative radial wave function of a pair of scattered particles withangular momentum l, interacting with a central potential, which has the asymptotic

ψl (r) ∝ sin[kr − lπ/2 + δ]

where k = |~k | is the length of the three-momentum, and δ is the phase shift. If we confineour system into a sphere of radius R, the condition

kR − lπ/2 + δ = nπ with n = 0, 1, 2, . . .

must be met, since ψl (r) has to vanish at the boundary. Analogously, in a free system

kR − lπ/2 = nfreeπ

In the limit R →∞, upon subtraction,

δ = (n − nfree)π

Differentiation with respect to M yields the distribution dδ/(πdM)

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 9 / 13

The experimental ππ phase shifts and their derivatives

The isospin-spin channel (0, 0) that is responsible for the emergence of the f0(500) pole isattractive, while the channel (2, 0) is repulsive.

(a) 11 (ρ)

00 (f0)

20

0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.5

1.0

1.5

2.0

2.5

δIJ

(a) 00+20

00

20

0.3 0.4 0.5 0.6 0.7 0.8 0.9

-1

0

1

2

3

f IJ/π

dδIJ/d

M[G

eV-

1]

V.B., Broniowski, Giacosa, arXiv:1506.01260

The derivative of the phase shift for the resonance (0, 0) is almost cancelled by thederivative of the (2, 0) channel until f0(980) takes over above M ∼ 0.85 GeV.

This is achieved with the multiplication of the isotensor channel by the isospin degeneracyfactor (2I + 1) = 5, which occurs for isospin-averaged quantities.

The phase shifts are from Garcia-Martin, Kaminski, Pelaez, Ruiz de Elvira, Phys. Rev. Lett. (2011), Phys.Rev. D (2011)

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 10 / 13

The case of K ∗0(800)

The attractive πK channel with I = 1/2 and J = 0 is capable of the generation of a polecorresponding to the K ∗0(800) or the κ resonance. It is not included in the PDG, but isnaturally expected to exist.

(a)

(1/2,0)

(3/2,0)

0.7 0.8 0.9 1.0 1.1 1.2 1.3

0.0

0.5

1.0

δIJ

(b)

(1/2,0)+(3/2,0)

(1/2,0)

(3/2,0)

0.7 0.8 0.9 1.0 1.1 1.2 1.3

-2

-1

0

1

2

M [GeV]f IJ/π

dδIJ/d

M[G

eV-

1]

V.B., Broniowski, Giacosa, arXiv:1506.01260

One can see that there is a partial cancellation and one can neglect the K ∗0(800).

The phase shifts are from Estabrooks, Martin, Brandenburg, Carnegie, Cashmore, et al., Nucl.Phys. B106, 61 (1976).

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 11 / 13

The error one would make by including the sigma

The cancellation occurs at the level of the distribution functions dk (M), therefore it persistsin all isospin-averaged observables.

101 102 1030

2

4

6

+

-

Con

tribu

tion

from

[

%]

s1/2NN [GeV]

Freeze-out line

LHC

101 102 1030

1

2

3

p s

Con

tribu

tion

from

[

%]

s1/2NN [GeV]

Freeze-out line

LHC

Left: the pions coming from the σ decay. Right: the contribution to pressure,entropy, energy density and the "interaction measure" ∆ = (ε − 3p)/T 4.

The σ is implemented as a Breit-Winger pole with Mσ = 484 and Γσ/2 = 255 MeV.

The famous K/π horn is affected, as well as all ratios to pions.V.B., Broniowski, Giacosa, arXiv:1506.01260

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 12 / 13

Conclusions

The contribution of the resonance f0(500) or sigma meson to isospin-averagedobservables like thermodynamic functions, pion yields, etc. is cancelled by therepulsion from the isotensor-scalar channel

The cancellation occurs from a “conspiracy” of the isospin degeneracy factor andthe derivative of the phase shifts

There is no cancellation mechanism in correlation studies of pion pair production

We thus clearly see the potential importance of the σ in studies of pion correlations

The ratios of particle multiplicities involving pions are affected up to 6%

The cancellation enhances the proton-pion puzzle at the LHC and opens even morespace for possible novel interpretations

Viktor Begun (UJK) Cracow School of Theoretical Physics June 22, 2015 13 / 13