QGP lifetime in the 1d Bjorken model with a transition from an ideal QGP to a pion gas

Klaus Reygers, reygers@physi.uni-heidelberg,de, 4 June 2015

Clear["Global`*"]

Input : energy density ϵ0 at the time τ0 = 1 fm/c

Define QGP degrees of freedom for 2 und 3 quark flavors

gQGP2fl = 37;

gQGP3fl = 47.5;

Define degrees of freedom of the pion gas:

gHGEff =12;

gpi = 3;

Energy densities to be considered (in GeV/(fm^3) at tau = 1 fm/c)

General dependence between energy density and temperature for an ideal QGP:

ϵ=gQGP ((π^2)/30)(T^4);

The other way round: temperature as a function of energy density:

T0 = T/.Solve[ϵ0==ϵ ,T][[4]]

Example: initial temperature for ϵ0 = 5 GeV/(fm^3), hbar c = 0.197 GeV fm, 1/(fm^3) = (0.197 GeV)3

f=(0.197^3);

N[T0/.{ϵ0->5 f , gQGP->gQGP2fl}] (* in fm/c, resulting temperature in GeV *)

Time when mixed phase starts:

τc= τ0((((T0/Tc)))^3);

QGP lifetime:

τQGP=τc - τ0;

Example : QGP lifetime for Tc=160 MeV and ϵ0 = 5 GeV/(fm^3) at τ0 = 1 fm/c

τQGP/.{τ0->1, ϵ0->5 f , Tc-> 0.16, gQGP->gQGP2fl}

Time when mixed phase ends

τh=τc (gQGP/gHG);

Table  with energy densities at 1 fm/c, QGP lifetime, lifetime of the mixed phase

ed={3, 5, 14} (* GeV/fm^3 *)

tab1=Table[{ed[[i]], τc - τ0, τh-τc} /.{τ0->1, ϵ0->ed[[i]] f , Tc-> 0.155, gQGP->gQGP2fl,gHG->gHGEff},{i,1,Length[ed]}]

TableForm[tab1,TableHeadings-> {None,{"energy density (GeV/(fm^3))","lifetime QGP (fm/c)","lifetime mixed phase (fm/c)"}},TableAlignments->"."]