Resonance Production in Heavy Ion Collisions: Suppression of $Λ(1520)$ and Enhancement of $Σ(1385)$

0.27 MB·English

by  Inga Kuznetsova

Resonance Production in Heavy Ion Collisions:
Suppression of Λ(1520) and Enhancement of Σ(1385)
Inga Kuznetsova1, and Johann Rafelski1,2
1Department of Physics, University of Arizona, Tucson, Arizona, 85721, USA and
2Department fu¨r Physik der Ludwig-Maximilians-Universit¨at Mu¨nchen and
Maier-Leibnitz-Laboratorium, Am Coulombwall 1, 85748 Garching, Germany
(Dated: January 15, 2009)
We investigate the yield of Λ(1520) resonance in heavy ion collisions within the framework of a
9 kinetic master equation without the assumption of chemical equilibrium. We show that reactions
0 suchasΛ(1520)+π ↔Σ∗canfavorΣ∗production,therebydecreasingtheΛ(1520)yield. Withinthe
0 same approach we thusfind a yield enhancement for Σ(1385) and a yield suppression for Λ(1520).
2
PACSnumbers: 24.10.Pa,25.75.-q,25.75.Nq,12.38.Mh
n
a
J
I. INTRODUCTION balance, e.g. particles yields do not change in this pe-
5
riod of temporal evolution of the system. This does not
1
Hadronresonancesareobservedinasurprisinglylarge necessarily mean that we have a chemical equilibrium.
] yield when a quark-gluon plasma (QGP) fireball breaks Instead it may be a transient condition for which none
h upintohadrons[1,2,3,4,5,6]. Thisisunexpected,since of the three particles is equilibrated chemically - we will
t
- theinvariantmasssignatureformedfromdecayproducts show when this can happen.
l
c could be erased by rescattering of the strongly interact- In the study of resonance decay and regeneration we
u ing decay products [7]. Thus a direct detection of res- are using the momentum integrated population master
n onances implies an exceedingly short period of hadron equations. We assume a fireball expansion model gov-
[ scattering, and/or a final state repopulated by hadronic erned by hydrodynamic inspired flow with conserveden-
3 interactions [8]. As a result, the final resonance yield tropy content. In our considerations we presume that
v can be considerably different from statistical hadron gas the yield of pions π is so large that we can assume it
9 (SHG) benchmark expectation. It has already been re- not to be materially affected by any of the reactions we
0 ported that the short lived (compared to characteristic consider. Thus we fix pion yield in terms of an ambi-
4
hadron phase evolution times) resonances are in general ent fugacity and temperature value, and in essence the
1
. enhanced [9] compared to SHG benchmark yield. total (per unit rapidity at RHIC) yield is fixed since we
1 The new result, we obtain here, is that the long lived conserve entropy.
1
resonances, such as Λ(1520), can be considerably sup- An important assumption implied below is that the
8
0 pressed in their yield. This effect is amplified for the rapidly expanding hadron system maintains for the rel-
: casewhenthe initialhadronfugacities,andthus particle evant particles a fully thermal (Boltzmann) momentum
v yields, are above chemical equilibrium. This situation is distribution. To describe the evolution of hadron abun-
i
X expectedforahadronizingQGPphase. ThelowΛ(1520) dances in the kinetic phase we track in time the yields
r yield has been reported both in RHIC and SPS experi- of single strange hadrons after their initial formation.
a ments [1, 2]. This is implemented in terms of time dependence of the
InaglobalQGPbreakup(hadronchemicalfreeze-out) chemicalfugacitiesΥ(t), andthe time dependence ofthe
particles and resonances are formed. Many resonances hadronization temperature T(t).
have a relatively large decay rates. This implies a large We look in detail at three potential evolution scenar-
scattering formation rate. Thus resonances are excep- ios:
tionally strongly interacting particles and continue to a) a high temperature breakup at T0 180 MeV where
≃
evolve in what we call kinetic phase, even after all other the entropy content of the equilibrated QGP and HG-
particlesfreeze-out. Thiscontinuedreactionphaseisspe- phase are similar;
cific to the resonances and can last considerably beyond b) the T0 160 MeV case where chemical non-
≃
the last non-resonant (elastic) scattering. equilibriumamongproducedhadronsisalreadyrequired;
The resonance suppression, or enhancement, mecha- and
nism worksas follows. In thermal hadronicgas the reac- c) at T0 140 MeV which is favored by descriptions
≃
tion, of stable hadron production, and in which case a strong
chemical non-equilibrium situation arises.
1+2 3, (1) For the late stage of the expansion, at relatively low
↔
density the assumption of thermal momentum distribu-
can occur in both directions: the resonance decay 3 tion may not be anymore fully satisfied. In particular
→
1+2,andtheback-reaction(regeneration)resonancefor- pionsofhighmomentumcouldbeescapingfromthefire-
mation 1 + 2 3. When the reaction goes with the ball. For this reason we will consider here a second sce-
→
same rate in both directions, we have chemical detailed nario, which we call “dead channel”. In this scenario we
2
2100
N Σ Λ
(2030, 7/2, 180)
2000
(1940, 3/2, 220)
1900
14
19
1800
(1775, 5/2, 120)
(1750,1/2, 90) 34 35?
29
1700 29 1190+η 27
MeV] 71? (1670, 3/2, 60) 27? 4?? 22?? (1520)+π }
E [ 1600 39
(938)+K+π 39 13 7?? 35? 21
48 7 (1520, 3/2, 15.6)
44 34 }
1500 (1385)+π
35
7 35?
1400 (938)+K 7
(1385, 3/2,37)
6.5
4
1300 (1190)+π
33
(1115) +π
1200
FIG. 1: (color on line) Reactions scheme for Λ(1520) and Σ(1385) population evolutions.
assume that the reaction (1) goes mainly in the direc- particle Σ(1385), Λ(1520) multiplicities during kinetic
tionofresonance3 decayandthe resonanceformationis phase. In chapter IIIC we obtain the observable ‘ob’
switched off for ratios Λ(1520) /Λ and Σ(1385) /Λ . We discuss
ob tot ob tot
our results in section IV
m (m +m )>300MeV. (2)
3 1 2
−
Without a complete kinetic modelincluding equilibra-
II. KINETIC EQUATIONS
tion and particle emission we do not know the exact en-
ergy in condition (2) and timescale (during expansion)
for which Boltzmann distribution is violated and dead A. Reactions scheme for Λ(1520) and Σ(1385)
channels appear. It is possible that reality lies between
the two cases (kinetic Boltzmann distribution and dead- In figure 1 we show the scheme of reactions which all
channels) considered here which, in our opinion, are the have a noticeable effect onΛ(1520)yield after the chem-
two most extreme limits. ical freeze-out kinetic phase. The format of this presen-
For Λ(1520)/Λ0 ratio calculations we employ and de- tation is inspired by nuclear reactions schemes. On the
velop further the approach used for Σ(1385)/Λ0 in [9]. vertical axis the energy scale is shown in MeV. There
However, in chapter IIA we investigate many further arethree classesofparticle states,whichwe denote from
reactions in which resonance Λ(1520)participates. Thus left to right as ”N” (S=0 baryon), ”Σ” (S = 1,I = 1
−
we are obliged to develop a completely numerical evolu- hyperon) and ”Λ” (S = 1,I = 0 hyperon). Near each
tion, for which the analytical study of Σ(1385)/Λ0 pro- particlebarwestate(on-l−ineinblue)itsmass,and/oran-
vides a benchmark check of our approach. In addition gularmomentumand/ortotalwidth in MeV.The states
to new and numerous reaction channels we also intro- Λ(1520) and Σ(1385) are shown along with the location
duce deformation of the reaction rates due to stimu- in energy of Λ(1520)+π and Σ(1385)+π respectively,
lated Bose enhancement of the reactions. This formal- both entries are connected by the curly bracket, and are
ism is presented in chapter IIB. We discuss the tem- highlighted (on-line in red). The inclusion of the π-mass
poral evolution of HG particle fugacities Υ(t) in IIIA ishelpingtoseethekineticthresholdenergyofareaction.
In chapter IIIB we present results for the evolution of The lines connecting the N,Σ,Λ columns are indicating
3
thereactionsweconsiderinthenumericalcomputations. Thus we find that the influence of these reactions on
All reactionsshowninfigure 1 cangoinboth directions, Σ(1385) yield is small. Another reason for a reduced ef-
asshownbythedoublearrowsplacednexttothenumer- fective depletion rate of Σ(1385) is that a lesser fraction
∗
ical value of the partial decay width Γ in MeV. of this resonance is needed to excite Σ . Thus in such
i
Λ(1520)decayswith a totaldecaywidth ofabout15.6 a reactionthe depopulation effect decreases because of a
∗
MeV, with two main channels: larger mass difference between Σ(1385) and Σ in com-
∗
parison with Λ(1520) and Σ .
Σ+ π Λ(1520), Γ 6.5MeV; (3) The reactions scheme for Λ(1520) reactions with dead
↔ ≈
N +K Λ(1520), Γ 7 MeV. channels is shown in figure 2. The difference between
↔ ≈
figure1andfigure2isthatsomeofthereactionlineshave
∗
However, Λ(1520) reacts with several heavier Σ - single-directionalarrows,asisstipulatedbythecondition
∗
resonances, (Σ Σ(1670), Σ(1750), Σ(1775), Σ(1940), Eq.(2).
≡
Σ(2030)):
∗
Λ(1520)+π Σ , (4) B. Resonances densities, time evolution equations
↔
and these reactions have a larger reaction strength
The evolution in time of the resonance yield is de-
shown in figure 1. Λ(1520) nearly behaves like a ‘stable’
scribed by a master equation, where the process of reso-
hadronic particle since:
nance formation in scattering is balanced by the natural
a) it is dominantly coupled to heavier resonances;
resonance decay:
b) its natural lifespan is larger than the hadronic
reaction rate. 1 dN3 = dW1i+2→3 dW3j→1+2, (15)
V dt dVdt − dVdt
Hereto we note that (several) Σ∗ involved in Eq. (4) Xi Xj
participate in further reactions: where subscripts i, j denote different reactions channels
when available. We further allow different subscripts
∗
Λ(1115)+π Σ ; (5)
i, j for the case where there are dead channels. Thus
↔
Σ(1190)+π ↔Σ∗; (6) dW1i+2→3/dVdt and dW3j→1+2/dVdt are invariant rates
N +K Σ∗; (7) (perunitvolumeandtime)forparticle3productionand
↔ ∗ decayrespectively. Incaseallreactionsoccurinbothdi-
Σ(1385)+π Σ ; (8)
↔ rections the total number of fusion channels is the same
∆+K Σ(1940,2030); (9)
as the total number of decay channels.
↔
N +K(892) Σ(1940); (10) Allowing for Fermi-blocking and Bose enhancement in
↔
Σ+η Σ(1750). (11) thefinalstate,wherebydesignationparticles1and3are
↔ fermions(heavybaryons)andparticle2isaboson(often
All reactions shown above can excite Σ∗ resonances. light pion) we have for the two rates:
∗
Since the mass of Λ(1520) is near to the Σ mass, the
yield of Λ(1520) is effectively depleted by the reaction dW3j→1+2 = g3d3p3 f d3p1 (1 f )
chain dVdt Z 2E3(2π)3 3Z 2E1(2π)3 − 1
Λ(1520)+π Σ∗ N +K, etc. (12) d3p
→ → ×Z 2E (22π)3 (1+f2)(2π)4δ4(p1+p2−p3)
The balancing two step back-reactioncan also occur, es- 2
pecially once Λ(1520)has been depopulated. Thus a dy-
namicalreduceddetailedbalance yield ofΛ(1520)would 1 p3 Mj p1p2 2, (16)
×g h i
result if the system were at fixed volume rather than ex- 3 Xspin(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
panding.
The multiplicity of Σ(1385) is mostly determined by dW1i+2→3 = g1d3p1 f g2d3p2 f
its dominant decay and production in the reaction dVdt Z 2E (2π)3 1Z 2E (2π)3 2
1 2
d3p
Λ(1115)+π Σ(1385), (13) 3 (1 f )(2π)4δ4(p +p p )
↔ ×Z 2E (2π)3 − 3 1 2− 3
3
and to a lesser extent by the reaction 1 p p Mi p 2. (17)
1 2 3
×g g h i
Σ(1190)+π Σ(1385). (14) 1 2 Xspin(cid:12) (cid:12) (cid:12) (cid:12)
↔ (cid:12) (cid:12) (cid:12) (cid:12)
where g ,i = 1,2,3 is particles degeneracy. The Bose
The resonance Σ(1385) participates further in reactions i
∗ distribution function for particle 2 is
with heavier Σ ; see reaction (8), but strength of these
interactions is smaller than for similar reactions with 1
f = , (18)
Λ(1520) and smaller than the decay width of Σ(1385). 2 Υ−1eu·p2/T 1
2 −
4
2100
N Σ Λ
(2030, 7/2, 180)
2000
(1940, 3/2, 220)
1900
1800
(1775, 5/2, 120)
(1750,1/2, 90) 34 35?
1190+η 27
1700
V] (1670, 3/2, 60) 22?? 29
Me 4?? (1520)+π }
E [ 1600
29 71?
13
(938)+K+π 7??
7 27? 19 (1520, 3/2, 15.6)
}
1500 (1385)+π
39 44 48 34
7
1400 (938)+K
(1385, 3/2,37)
6.5
14 35 35? 4
1300 (1190)+π
33
7 21 35? 39
(1115) +π
1200
FIG. 2: (color on line) Reactions scheme for Λ(1520) and Σ(1385) interactions in the“dead channel” model.
and Fermi for particles 1,3 are: and,similarly,channellifespanτi, omitting the sum .
3 i
Here the rate R123 is: P
1
fj = Υ−j1eu·pj/T +1, j =1,3. (19) R1i23 =ZZZ 8dE3pE1d3Ep2(d23πp)35f1Υ−11f2Υ−21f3Υ−31
1 2 3
Huµere=Υ(1i,~0is) pinartthicelersesftugfraacmitye, oafntdheuh·eapti b=athEwi,heforer ×δ4(p1+p2−p3)eu·p3/T Xspin(cid:12)hp1p2(cid:12)Mi(cid:12)p3i(cid:12)2. (22)
d4pδ (p2 m2) d3p /E for each particle. Hence, (cid:12) (cid:12) (cid:12) (cid:12)
0 i − i → i i
Eq.(16) and Eq.(17) are Lorentz invariant, and thus as R is independent of the fugacity in the Boltzmann-limit.
presentedthese ratescanbe evaluatedinany convenient In next section IIC we will see in what way the in-
frameofreference. Normally,thisistheframeco-moving mediumdecayratevariesfromthefree spacedecayrate.
with the thermal volume element. This is due to the effect of quantum enhancement, and
For the heavy baryon (resonances), particles 2,3, we thefactthataparticleemergedinathermalbathwitha
canwork using the expansionof the relativistic distribu- finite temperature. A particle is not decaying in its rest
tion, the first term is the Boltzmann limit: frame.
The production anddecay ratesare connected to each
N T3 other by the detailed balance relation [11, 12]:
i =Υ g x2K (x ), (20)
V i2π2 i i 2 i
Υ−1Υ−1dW1+2→3 =Υ−1dW3→1+2 =R . (23)
wherex =m /T,K (x)istheBesselfunction(nottobe 1 2 dVdt 3 dVdt 123
i i 2
mixedupwithparticle2). However,weusethecomplete
UsingdetailedbalanceEq.(23)weobtainforfugacityΥ
3
Bose distribution to describe pions.
the evolution equation [11, 12]:
We introduce in medium lifespan of particle 3:
dΥ 1 1 1 1
τ13 ≡ V−P1diNR31i/2d3Υ3, (21) dτ3 =Xi Υi1Υi2τ3i +Υ3τT + τS −Xj τ3j, (24)
5
where we have also introduced characteristic time con- of equations for baryonscloses with the equationfor Υ1′
stants of temperature T and entropy S evolution
1 = dln(x32K2(x3))T˙, (25) ddΥτ1′ =Υ1Xk τ11k ddNN11k′//ddΥΥ1k1′ −Xn Υ1′Υn2′τ11nddNN11n′//ddΥΥ1n1′
τ − dT
T 1 1
1 = dln(VT3)T˙. (26) + Υ1′(cid:18)τT + τS(cid:19). (32)
τ − dT
S
InthepresentsettingΥ =Const. byvirtueofentropy
2=π
The entropy term is negligible, τS τ3,τT since we im- conservation(seediscussionbelow)andthe sameapplies
≫
plementnearconservationofentropy. Weimplementthis to the case 2′ =π. However,if either particle 2 or2′ is a
in the way which would be exact for massless particles kaon, we need to follow the equation for Υ2,2′=K which
takingVT3 =Const.. Thusthereissomeentropygrowth is analogous to equation for particle 1 or 1′.
in HG evolution to consider, but it is not significant. In Theevolutionequationscanbe integratedoncewede-
order to evaluate the magnitude of τT we use the re- terminetheinitialvaluesofparticledensities(fugacities)
lation between Bessel functions of order 1 and 2 (not established at hadronization/chemical freeze-out. We
to be mixed up with particles 1,2) d z2K2(z) /dz = determine these for RHIC head-on Au–Au collisions at
−z2K1(z). We obtain (cid:0) (cid:1) √sNN =200GeV.Weintroducetheinitialhadronyields
inspired by a picture of a rapid hadronization of QGP
1 K1(x3) T˙ in which quarks combine into final state hadrons. For
= x , (27)
τ −K (x ) 3T simplicity we assume here that the net baryon yield at
T 2 3
central rapidity is negligible. Thus the baryon-chemical
τT > 0. We invoke a model of matter expansion of the and strangeness potentials vanish. The initial yields
type used e.g. in [15], where the longitudinal and trans- of mesons (qq¯,sq¯) and baryons (qqq,qqs) are controlled
verseexpansionisconsideredtobe(nearly)independent. aside of the ambient temperature T by the constituent
In this model we have: lightquarkfugacityγ andthestrangequarkfugacityγ .
q s
The strangeness pair-yield in QGP is maintained in
T˙ 1 2(vτ/R⊥)+1 transition to HG. This fixes the initial value of γ . In
= , (28) s
T −3(cid:18) τ (cid:19) fact, since we investigate here relative chemical equilib-
rium reactions our results do not depend significantly
whereR⊥ isthetransverseradius,v isthevelocityofex- on the exact initial value γ and/or strangeness content.
s
pansioninthetransversedimension. Allflowparameters The entropy conservation at hadronization fixes γ . For
q
(ortemperaturedependence onτ)arethe sameasin[9]. hadronization temperature T(t = 0) T = 180 MeV,
0
For a static systemwith τT →0 we see that Eq.(24) has γq = 1. However, when T0 < 180 M≡eV, γq > 1 in or-
transient stable population points whenever derto haveentropyconservedatchemicalfreeze-out. At
T = 140 MeV γ = 1.6 that is close to maximum pos-
1 1 0 q
ΥiΥi Υ =0. (29) sible value of γ , defined by Bose-Einstein condensation
Xi 1 2τ3i − 3Xj τ3j condition [10]. q
Forreactions,suchasshowninEq. (1),wehave(lower
Next we address the functional dependence on time index defines particle considered, where Y Σ,Λ is a
≡
of Υ1,Υ2. In the equation for Υ1 we have terms which hyperon)
compensate what is lost/gained in Υ see Eq.(24). Fur-
3
ther we have to allow that particle ‘1’ itself plays the Υ0(1=Y) =γq2γs, Υ0(2=π) =γq2; (33)
role of particle 3 (for example this is clearly the case for
or
Λ(1520)). That allows a chain of populations relations
as follows: Υ0 =γ3, Υ0 =γ γ ; (34)
(1=N) q (2=K) q s
′ ′
(1 +2 1)+2 3, (30)
where the particle 1 in reaction (1) is a baryon and par-
↔ ↔
ticle 2 is a meson. The particle 3 is always a strange
Then we obtain:
baryon:
ddΥτ1 =Υ3 τ1k ddNN3k//ddΥΥk3 − Υ1Υn2τ1nddNN3n//ddΥΥn3 Υ0(3=Y) =γq2γs, (35)
Xk 3 1 1 Xn 3 1 1
Note that for γ >1 we have always initially
q
1 1 1 1
+ Υ1τT + τS −Xj τ1j+Xi Υi1′Υi2′τ1i (31) ΥΥ1Υ2(cid:12) =γq2 ≥1. (36)
  3 (cid:12)t=0
(cid:12)
TheratiosofderivativeofN seeninthefirstlinearedue As a consequence initiall(cid:12)y the pair of particles 1,2 reacts
i
to the definition of relaxationtime Eq.(21). The system into 3.
6
As already noted, we do not need to follow the evo-
lution in time for the pion yield, which is fixed by con- 2.2
servationof entropyper unit rapidity, asincorporatedin 2.1 T =140 MeV, γ =1.6
Eq.(28). Thus it is (approximately) a constant of mo- T0=160 MeV, γq max=1.52
tion. This can be seen recalling that the entropy per 2 0 q max
T =180 MeV, γ =1.45
pion is nearly 4 within the domain of temperatures con- 0 q max
1.9
sidered. Thus the conservation of entropy implies that
pion number is conserved. With VT3 Const., this fur- 1.8
≃
ther implies that during the expansion 1.7
π
Υ =γ2 =Const., +f 1.6
π q 1
1.5
which we keep at the initial value.
1.4
1.3
C. In medium lifespan calculations
1.2
In our calculations we take into account the influence 1.1
ofthemediumonresonancelifespanandtheeffectofthe
1
motion of the decaying particle with respect to the ther- 0.6 0.8 1.0 1.2 1.4 1.6
γ
mal rest frame. In [11, 12] it was noted that the decay q
rate R , Eq.(22), of particle 3 (density) in a thermally
123 FIG. 3: (color on line) The Bose enhancement factor 1+
equilibrated system can be cast into a form which in- fπ(E1∗) in Σ(1385) rest frame as a function of light quark
volves the free space decay rate: fugacity γq for the reaction Σ(1385) ↔ Λπ at T = 140 MeV
m ∞ p2dp Υ−1eE3/T (blue, solid line), at 160 MeV (green, dash-dot line) and 180
R = 3 3 3 3 Φ(p ), (37) MeV (red, dashed line). The dots show the initial value of
123 τ0 Z0 E3 Υ−31eE3/T ±1 3 fugacities for thethree possible hadronization cases.
where function Φ(p ) for reaction (13) Σ(1385) Λ+π
3
↔
is
∗
energy E =250MeV (Eq.(39)) in the Σ rest frame and
2
1 Υ eb+e−a2 ea1 Υ e−b the Bose enhancement effect is possible in the oversatu-
Λ π
Φ(p )= ln − .
3 b(eE3/T+ΥπΥΛ) (cid:0)(ΥΛe−b+e−a(cid:1)2(cid:0))(ea1 Υπeb)(cid:1) rated hadronic gas after QGP hadronization.
−
For the low temperatures considered here we can as-
sume that Σ resonances almost do not move. Thus the
∗ ∗ ∗
E E E E p p
a = 1 3, a = 2 3, b= 3. (38) enhancement effect in the thermal bath frame is close to
1 2
m T m T m T
3 3 3 the enhancement in the Σ(1385) rest frame. The decay
rate increases by Bose enhancement factor 1+f (here
Herep∗ =p1 =p2 andE1∗,2 =qp∗2+m21,2 arethemag- fπ =fπ(E2∗,T)). In figure 3 we show Bose enhanπcement
nitudeofthemomentumand,respectively,theenergy,of factor as a function of light quark fugacity γ for tem-
q
particles1and2intherestframeoftheparticle3. From perature T =140MeV (blue, solid line), T =160 MeV
0 0
energy conservation: (green, dash-dot line), T =180 MeV (red, dashed line).
0
The large dots show Bose enhancement factor for our
E∗ = m23±(m21−m22), initial γq determined from entropy conservation in fast
1,2 2m hadronization. The fugacity γ = 1.6 is close to maxi-
3 q
p∗2 = E2 m2 mum expected value at T0 = 140 MeV. The maximum
1,2− 1,2 fugacitiesforeachtemperaturecorrespondtoBose-Ein-
m2 m2+m2 (m2 m2)2
= 3 1 2 + 1− 2 . (39) stein singularity. The Bose enhancement effect is largest
4 − 2 4m23 for maximum γq and it diminishes for small γq. At fixed
entropythegreatestenhancementisforsmallestambient
Here fugacities for Λ and π correspond to those for
temperature, see the dot on solid line in figure 3.
particles 1 and 2, respectively. For the temperatures
In figure 4 we show the corresponding decrease in the
of interest (hadronization of QGP and below) m and
Λ
lifespan, the ratio τ /τ as a function of temperature T
m >>T. With sufficient accuracy we can write 3 0
Σ
inthe reactionΣ(1385) Λπ. We considertemperature
↔
1 ea1+b Υπ rangefromcorrespondinghadronizationtemperatureun-
Φ(p ) ln − . (40)
3 ≃ beE3/T (cid:0)(ea1−b Υπ(cid:1)) til T = 70 MeV. We assumed, that Υπ is a constant.
− Fugacities of heavy resonances do not influence the re-
There are no significant medium effects upon decay rate sult. The lowest τ /τ ratio is for γ = 1.6 at T = 140
3 0 q 0
of Σ(1385) and Λ resonances. However the pions have MeV when we have maximum value of γ for given tem-
q
7
solvethissystemofequationsnumerically,usingclassical
1.1 fourth order Runge-Kutta method.
Particle fugacities, except Υ , change rather rapidly.
π
Figure 5 shows the computed Υ(t) as a function of tem-
perature T(t). We present here the scenario in which all
1.0 reactions evolve in both directions, for the initial condi-
tion γ = γ . The time, corresponding to the tempera-
s q
tureshownatthebottom,isshownatthetopoffigure5,
5)
8 in each frame. On the left we have hadronization at 140
30.9
vacΣ(1 MeV, in the middle at 160 and to the right at 180 MeV.
τ/85) Enoatchtimfraem. FeohraΥsΣth∗ewseamsheowscatwleosipzoesfsoibrlteeemvpoeluratitounreexuanmit-,
3
Σ(10.8 ples, for Σ1750 (dash-dotdark line) and Σ(1775)(dashed
τ line). These resonances have significant influence on the
Λ(1520)yield. ThesolidlinesareforΥ (upper,red
Λ(1520)
line) and Υ (lower, light blue line). The dash-dot
Σ(1385)
0.7
T0=140 MeV, ϒq=1.6 and dashed light lines are for ΥΣ(1190) and ΥΛ0, respec-
T=160 MeV, ϒ=1.27 tively. The upper dotted line is for Υ andlowerdotted
T00=180 MeV, ϒqq=1 line is for ΥK. N
0.6 AnimportantfeatureisthattheΥsofmassivehadron
80 100 120 140 160 180 (resonances)increaseveryfastwhenT decreases. Thisis
T [MeV] so since in absence of a rapid re-equilibration reactions,
multiplicityofgivenresonancemustbeconserved. Then,
FIG. 4: (color on line) The ratio of the in medium lifespan
according to Eq. (20) Υ 1/K (m /T), and thus for
τ3 with the vacuum lifespan τ0 as a function of temperature i ∝ 2 i
T for the reaction Σ(1385) ↔ Λπ. The dashed (red) line is large mi Υi exp(mi/T). We would expect Υi > Υj,
∝
forhadronization atT0 =180MeV,γq =1.0; thedot-dashed when mi > mj, and T decreases. This behavior is just
line (green) for hadronization at 160 MeV, γq = 1.27; solid like we found for the case of large charm fugacity [10].
line (blue) is for hadronization at 140 MeV and γq =1.6. However, because of the decay and regeneration reac-
tions there are some deviations from this expectation in
figure 5.
perature. If we compare this value of τ /τ = 0.65 with For T = 180 MeV in most cases Υ > Υ Υ (t > 0).
3 0 0 3 1 2
inverseBoseenhancementfactor1/(1+f (E∗,T))=0.54 Massive resonances decay to lower mass particles. The
π 2
for this T and γ (see figure 3) we see that these values result is defined by resonance mass, its decay width
q
are near to each other (difference is about 20% ) as ex- and decay products. For example Υ is smaller
Σ(1775)
pected for m >>T. For smaller T, γ decay time goes than Υ and Υ , because of its large decay
Σ q Σ(1750) Λ(1520)
to its vacuum value. width. Therefore excitation of Σ(1775) by Λ slightly
The same calculations are applicable for heavier Σ∗. dominates over Σ(1775) decay to Λ(1520) even in this
When the difference of mass of the initial and final state case, when for most resonances the decay is dominant.
resonance decreases, the Bose enhancement effect in- For smaller initial hadronization temperatures Υ
Λ(1520)
creases, since it involves small momenta. The largest becomes smaller than Υ , and even smaller than
Σ(1775)
effect is for reaction Σ(1670) Λ(1520)+π. On the Υ in some range of temperatures. This suppres-
Σ(1385)
↔ ∗
other hand, for the reactions which satisfy condition (2) sion occurs because of Σ(1775), and others Σ regener-
the enhancement effect becomes so small that we do not ation. Because of large Υ , Υ < Υ Υ , the
π Σ(1775) Λ(1520) π
need to include it in our calculations. Σ(1775) production by Λ(1520) is dominant in the full
range of T considered here.
III. NUMERICAL RESULTS
B. Final Λ(1520) and Σ(1385) multiplicities
A. Evolution of fugacities
In this section we consider the evolution of the multi-
plicity of resonances Λ(1520), Σ(1385), Σ(1775) during
In order to evaluate the Λ(1520) and Σ(1385) mul-
the kinetic phase. We use the Boltzmann yield limit,
tiplicities we must integrate Eq.(24), or Eq.(31), or
Eq.(20). By the symbol X(T) we refer to a particular
Eq.(32)foreachparticleinvolvedinfigure1,andperform
resonance,and X is the initial multiplicity for that res-
similaroperationsforreactionswithdeadchannelsinfig- 0
onance. The dynamic yield of this resonance may be
ure 2. This system of equations includes equations for
∗ expressed as
Λ(1520), Σ(1385), five equations for Σ s, equations for
K(892) and ∆ and equations for ground states Λ(1115), X(T) ΥX(t)T(t)3K2(mX/T(t))
= (41)
Σ(1190),N,K.Allreactionsinfigures1areincluded. We X Υ T3K (m /T )
0 X0 0 2 X 0
8
t [fm] t[fm] t [fm]
2.96 1.25 0 3.64 2.15 0.94 0 3.7 2.33 1.29 0.59 0
3
10
ϒ
Σ(1775)
ϒ
Σ(1750)
ϒ
Λ(1520)
ϒ
Σ(1385)
ϒ
102 Σ(1190)
ϒ
Λ0
ϒ
N
ϒ ϒ
K
1
10
ϒ = γ2 = 1.613
ϒπ = γq2 = 2.56 π q ϒ=γ2 = 1
π q
0
10
100 120 140100 120 140 160100 120 140 160 180
T [MeV] T [MeV] T [MeV]
FIG. 5: (color on line) The fugacities Υ for selected particles are shown as a function of temperatureT(t),for T0 =140 MeV
on theleft, for T0 =160 MeV in themiddle and for T0 =180 MeV, on the right. See text for further details.
1.2 1.1
1.1
1
0
0 )
) 0
5 2
38 1 15
1 ( 0.9
( Λ
Σ
Σ (1385)/0.9 T0111, 446M000e,,,V 111 ...γ662q,7 d.ch. Λ (1520)/ 0.8 T0, M14e0V, 1γ.q6
0.8 160, 1.27, d.ch. 140, 1.6, d.ch.
180, 1 160, 1.27
180, 1, d.ch. 0.7 160, 1.27, d.ch.
180, 1
0.7 180, 1, d.ch.
0.6
0.6
100 120 140 160 180 100 120 140 160 180
T [MeV] T [MeV]
FIG. 6: The ratio Σ(1385)/Σ(1385)0 on left and Λ(1520)/Λ(1520)0 on right as a functions of temperature T(t) for different
initial hadronization temperatures T0 = 140, 160 and 180 MeV (blue/bottom, black/middle and red/top lines, respectively).
Solid lines are for calculations with dead channels, dashed lines are for calculations without dead channels.
9
2 2
10 10
Σ* → N+K Σ* → N+K
N+K → Σ* N+K → Σ*
Σ* → Λ(1520) +π dt Σ* → Λ(1520) +π
dt Λ(1520) +π → Σ* dV Λ(1520) +π → Σ*
dV Λ0+π→ Σ* W/ Σ+η → Σ*
W/ Σ* → Λ0+π d Σ* → Σ+η
d
1
10
1
10
Σ(1750)
Σ(1775)
0
10
0
10
100 120 140 100 120 140
T [MeV] T [MeV]
FIG. 7: The rates for main channels of Σ(1775) (on the left) and Σ(1750) (on the right) decay and production as a functions
of temperature T in the case when all reactions go in both directions and T0 = 140 MeV. Solid lines are for reaction Σ∗ ↔
Λ(1520)+π; dash-dot lines are for reaction Σ∗ ↔ N +K; dashed lines are for reaction Σ(1775) ↔ Λ0+π on the left and
Σ(1750)↔Σ+η on theright; blue and red lines are for decay and backward fusion reaction, respectively.
Figure6showsthis yieldasafunctionofT(t)forX = (on-lineblue)andbackwardfusionreaction(on-linered),
Σ(1385) (left) and X = Λ(1520) (right). We consider respectively.
three initial conditions, temperature T0 = 140,160,180 As temperature decreases, all rates dW/dtdV are in-
MeV,withcorrespondingγq =1.6,1.27,1.0,respectively. creasing rapidly. This is mainly because fugacities Υ
Thesolidlinescorrespondforthemodelwithdeadchan- increase nearly exponentially when number of particles
nels and dashed one are for case when all reactions are is conserved, see figure 5. We see that at the begin-
symmetric in both directions. The thin dotted vertical ning of the kinetic phase allreactions go in the direction
lineatT =120MeVmarksthekineticfreeze-outtemper- of Σ(1775) production, since Σ(1775) production rate is
ature, assumed before in [9]. The main resultis that the larger than its decay rate for all channels. Then at first
resulting relative yields for Λ(1520) and Σ(1385) behave Σ(1775) Λ0 + π decay rate becomes dominant over
↔
qualitatively different from each other. In particular, as Σ(1775)production rate in this channel, followed by the
the temperature decreases, for the case T0 = 140 MeV same for Σ(1775) N +K channel.
↔
we observea strong yield suppressionfor Λ(1520),and a For the reaction Σ(1775) Λ(1520)+π backwardre-
strong enhancement for Σ(1385) (as compared to initial action is always dominant. ↔As result, during the kinetic
SHM yields). phase always more Λ(1520) resonances are excited into
To better understand the mechanism of Λ(1520) sup- Σ(1775) than they are produced by Σ(1775) decay. The
pression, we analyze in some detail the case of Σ(1775) reasonforthis isthe decayofΣ(1775)tothe otherchan-
and Σ(1750) decay and production rates dW/dVdt. We nels, as long as ΥΣ(1775) < ΥΛ(1520)Υπ. The lighter is
assume here that these reactions can go in both direc- the total mass of decay products, the earlier the decay
tions. In figure 7 we show the reactions rates for the reaction becomes dominant. This is due to the fact that
principalchannelsofdecayandproductionasafunctions the fugacity of Υ for heavier particles increases faster
of temperature T for Σ(1775)(left) and Σ(1750) (right), withexpansion. Therefore,thedecayratebecomesdom-
for the case of initial temperature T = 140 MeV which inantearlier,whenthedifferencebetweeninitialandfinal
0
provides the largest Λ(1520) suppression. Solid lines are mass is larger. The net result is Λ(1520) suppression by
for the reaction Σ Λ(1520)+ π, dash-dot lines are Σ(1775) excitation.
↔
for reaction Σ N +K, dashed lines are for reaction In figure 8 we show the yield of Σ(1775) normalized
Σ Λ0+π. Tw↔osetoflines arepresentedforthe decay by its initial yield at hadronization: Σ(1775)/Σ(1775)
0
↔
10
as a function of T(t). Like in the other figures above,
solid lines are for the dead channels and dashed lines
are for case when reactions go in both directions, solid
(blue) lines are for T = 140 MeV, solid (black) lines 1.2
0
for T =160 MeV, and solid (red) lines are for T =180
0 0
MeV.EachofthelinescanbeidentifiedbytheirinitialT-
value. Weseethatwhenallreactionsgoinbothdirection
1
the ratio Σ(1775)/Σ(1775) increases at first similar to
0
Σ(1385)/Σ(1385) and ∆(1230)/∆(1230) ratios [9]. 0
0 0 )
5
Compared to these ratios, Σ(1775)/Σ(1775)0 ratio 7
7 0.8
reaches its maximum value earlier, and after the max- 1
(
imum, the yield of Σ(1775) decreases faster. The reason Σ
for this behavior is that the mass of Σ(1775) is larger. 5)/
7
The phase space occupancy ΥΣ(1775), and therefore its 7 0.6
1
dcaeycayratreastefso,rinΣc(r1e3a8s5e)faasntder∆t(h1a2n30t)h.e Tfuhgearceiftoyreanddecadyes- Σ ( T0, MeV γq
140, 1.6
Σ(1775) to some channels and its total decay rate be-
0.4 140, 1.6, d.ch.
come dominant earlier (see figure 7). Although the total 160, 1.27
decay width ofΣ(1775)is approximatelythe same as for 160, 1.27, d.ch.
∆(1230), the maximum value of this ratio is smaller. 180, 1
Said differently, the maximum yield of Σ(1775) does 0.2 180, 1, d.ch.
not have time to reach the value as high as that for
∆(1230). We thus learn that the time evolution of the 100 120 140 160 180
yield of resonances with large decay width depends not T [MeV]
only on their decay width, but also on mass difference
between initial and final states. Similar time evolution
∗ FIG. 8: The ratio Σ(1775)/Σ(1775)0 as a functions of tem-
occursforthe otherΣ ,whichquantitativelydepends on
peratureT(t)fordifferentinitialhadronization temperatures
their mass, decay products masses and decay width.
T0 =140,160 and180 MeV(blue/bottom,black/middleand
∗
For most Σ s, the decay products in the channel red/top lines), respectively. Solid lines are for calculations
Λ(1520)+π are heavier than the decay products in oth- with dead channels, dashed lines are for calculations without
ers channels, which are thus favored by phase space. dead channels.
For most resonances in our range of temperature, the
decay into Λ(1520)+ π remains weak. The exception
is Σ(1750) which decays also to Σ + η, see figure 7.
(m + m > m + m ). Σ(1750) begins to de- we took into account the Bose enhancement of interac-
Σ η Λ(1520) π ∗
caydominantlytoΛ0(1520)atrelativelylowtemperature tion rates, reaction (14), and Σ production. Σ(1385)
∗
contribution to Σ production is small, compared to the
T = 116 MeV, and continues to be produced by Σ+η
influence of the first two effects. The time (i.e. tem-
fusion.
perature) evolution of Σ(1385) practically does not de-
As a result, allowing all reactions to go in both direc-
pend on the presence of dead channels, and the maxi-
tions, the ratio Λ(1520)/Λ(1520) has a minimum. This
0 mumenhancementofΣ(1385)isevenlesssensitive. This
is specifically due to Σ(1750)) decay back to Λ(1520) at
∗
in fact indirectly confirms that Σ has a small influence
small temperatures as described above. However, when
on Σ(1385) multiplicity. Thus we confirm that:
wesatisfyEq.(2)fordeadchannelstheonlydecayoccurs
a) for T = 180 MeV Σ(1385) evolves with the system
inthebeginningofkineticthedead-channelmodelphase. 0
following the ambient temperature;
ΛIn(1th52a0t)ca+seπtheΥΣΣ∗∗sexacreeesdmsatlhleer,raantedftohrebraatcekwofarredacretiaocn- b) for T0 = 160 MeV Σ(1385) shows some increase in
→ yield;
tionbylargeramount,comparedtothe scenariowithout
c) for T = 140 MeV there is a strong yield increase of
dead channels. This amplifies the effect of Λ(1520) sup- 0
∗ Σ(1385).
pression. In this case, Σ decay to lighter hadrons right
after they are produced by Λ(1520). We can see that for While there is little sensitivity in the yield of Σ(1385)
T0 =140MeVandT0 =160MeVΛ(1520)yieldisalways to issue of particle momentum distribution (little dif-
decreasing in the here considered temperature range. ference between the two models considered, dashed and
For Σ(1385) multiplicity we find a result quite differ- solid lines), the Σ(1385) yield is highly sensitive to ini-
ent from Λ(1520) behavior discussed here, but similar tial hadronization condition. While for Σ(1385) the
to what we obtained in [9] by a very different method yield increases with decreased hadronization tempera-
in a smaller basis set of states. In particular, the ture, for Λ(1520) the opposite is true, and in particu-
Σ(1385) yield is enhanced, but the maximum value of lar the smallest final Λ(1520) yield corresponds to the
Σ(1385)/Σ(1385) we find is a few percent higher, since smallest hadronization temperature for both models.
0