Dilepton production from hot hadronic matter in nonequilibrium

0.62 MB·English

by  B. Schenke

Dilepton production from hot hadronic matter in nonequilibrium
B. Schenke and C. Greiner
∗ †
Institut fu¨r Theoretische Physik, Johann Wolfgang Goethe – Universit¨at Frankfurt,
Max–von–Laue–Straße 1, D–60438 Frankfurt am Main, Germany
The influence of time dependent medium modifications of low mass vector mesons on dilepton
yields is investigated within a nonequilibrium quantum field theoretical description on the basis of
the Kadanoff-Baym equations. Time scales for the adaption of the spectral properties to changing
self energies are given and, underuse of a model for the fireball evolution, nonequilibrium dilepton
yields from the decay of ρ- and ω-mesons are calculated. In a comparison of these yields with
those from calculations that assume instantaneous (Markovian) adaption to the changing medium,
quantummechanical memory effects turn out tobe important.
PACSnumbers: 11.10.Wx;05.70.Ln;25.75.-q
6
Keywords: nonequilibriumquantum fieldtheory;relativisticheavy-ioncollisions;dileptonproduction
0
0
2

n
a I. INTRODUCTION AND MOTIVATION
J

1 Relativistic heavy ion reactions, as performed at the SIS at GSI, Darmstadt, the AGS at BNL, the SpS at CERN
1
andthe RHIC at BNL,allow for studying stronglyinteracting matter under extreme conditions athigh densities and

temperatures. One of the main objectives is the creation and identification of new states of matter, most notably
2
v the quark-gluonplasma (QGP). Photons and dileptons do not undergo strong final state interactions and thus carry
6 undistorted information especially on the early hot and dense phases of the fireball. Photon spectra are a suitable
2 observable for the temperature of such a system whereas dileptons have encoded additional dynamic information
0 via their invariant mass. Particularly in the low mass region, dileptons couple directly to the light vector mesons
9
and reflect their mass distribution. They are thus considered the prime observable in studying mass (de-)generation
0
related to restoration of the spontaneously broken chiral symmetry. Additionally, vector mesons are also affected by
5
many-body effects due to coupling to baryonic resonances. For an overview see [1]. Indeed the CERES experiment
0
/ at the SPS at CERN [2, 3] has found a significant enhancement of lepton pairs for invariant masses below the pole
h
mass of the ρ-meson, giving evidence for such modifications.
p
In order to be able to extract precise information from the data, it is essential to find a thorough mathematical
-
p description for the dilepton production of an evolving fireball of strongly interacting matter. In particular it is
e necessary to consider the fact that especially during the early stages of a heavy ion reaction the system is out of
h
equilibrium - the medium and hence the properties of the regarded mesons undergo substantial changes over time.
:
v ThesescenarioshavebeendescribedwithinBoltzmann-typetransportcalculationsusingsomequantummechanically
i inspired off-shell propagation in [4, 5, 6, 7, 8, 9, 10, 11]. In principle, a consistent formulation beyond the standard
X
quasi-particleapproximationis needed- fully abinitio investigationsofoff-shellmesonicdilepton productionwithout
r
a any further approximation does not exist so far.
Moregenerallynonequilibriumquantumfieldtheoryhasbecomeamajortopicofresearchfordescribingmicroscopic
transport processes in various areas of physics (see e.g. [12] and references therein). One major question deals with
how quantum systems eventually thermalize. In [12] the quantum time evolution of φ4-theory for homogeneous
systems in 2+1 space-time dimensions for far from equilibrium initial conditions has been investigated, while earlier
works (e.g. [13]) studied the 1+1 dimensional case. It was shown in [12] that the asymptotic state in the far future
correspondstotheexactoff-shellthermalstateofthesystemobeyingtheequilibriumKubo-Martin-Schwinger(KMS)
relationsamongthevarioustwo-pointfunctions. Foracoupledfermion-bosonYukawa-typesystemin3+1dimensions,
eventual equilibrationand thermalizationin momentum occupation was shown in [14]. In addition, the full quantum
dynamics of the spectral information was analyzed in [12]. This issue was also addressed in some detail in [15]. In
a subsequent work [16] the exact solutions were examined in comparison to approximated, instantaneous off-shell
transport equations, obtained by a first order gradient expansion. In that particular case it turned out that indeed
these approximated equations are a very good substitute for the full dynamics.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
When dealing with vector mesons the important question emerges whether such a quasi instantaneous adaption of
thedynamicandspectralinformationtothechangingmedium, asalsoassumedinmoreschematicmodelcalculations
[1, 17] and Monte-Carlo kinetic transport simulations [18], is always a suitable assumption. In a ultrarelativistic
heavy ion collision the typical lifetime of the diluting hadronic phase is only 5 8 fm/c [1]. On the other hand the
−
spectralinformationshouldalwaysreactontemporalchangeswithacertain”quantummechanical”retardation. Ifthe
timescaleofthesechangesofthesystembecomescomparabletotheretardationtime,aninstantaneousapproximation
becomesinvalidandmemoryeffectsforthespectralpropertiesofthe excitationsarepresent. Suchpossible(quantum
mechanical) memory effects, i.e., potential non-Markovian dynamics, have often appeared in descriptions of the
microscopic evolution of complex quantum systems. Also a nonequilibrium treatment of photon production from a
hotQGPwasgivenin[19,20,21,22]andtheimportanceofmemoryeffectsandshortcomingsoftheS-matrixapproach
were pointed out for that case. The question of whether memory effects are important for dilepton production from
hot hadronic matter, i.e., whether such effects have influence on the measured dilepton yields, constitutes the major
motivation for the present study.
We givefor the firsttime explicitcalculationsfor dilepton productionfromfirstprinciple nonequilibriumtransport
equations. We approach the problem using a nonequilibrium quantum field theoretical description based on the
formalism established by Schwinger and Keldysh [23, 24, 25, 26]. By this we set up a framework incorporating
the full quantum dynamics, which is necessary for the description of the transport of true off-shell excitations. We
present a new derivation of the formula for the dynamic dilepton production rate starting from the Kadanoff-Baym
equations [27], which are nonlocal in time and hence account for the finite memory of the system. The resulting
formula for the rate involves a (half) Fourier transform over past times of the two-time Green function of the virtual
photon. The issue within this representation is that the rate for an invariant mass in the range of interest is hidden
as a tiny component in the two-time function, which contains the rate for all invariant masses. With this work we
meetthe challengeofevaluatingthis expressionforthenonequilibriumdileptonproductionrate,whichisparticularly
importantsinceitistheonlycausalapproachthatretainsallmemoryeffects. Anytreatmentsofar,involvinggradient
expansions of the Kadanoff-Baymequations, where future contributions to the Green function are treated equally to
those from the past, can not precisely describe a system that is quickly evolving with respect to the timescales on
that the regarded quantities adjust to system changes. The reader, who is already familiar with the formalism of
nonequilibrium quantum field theory may skip most of the first partof Section II and continue reading with Eq.(19).
The paper is organizedas follows. We startwith a briefintroductionof the used formalismandthe presentationof
ournew derivationofthe dileptonproductionratefor nonequilibriumsystemsinSectionsIIAandIIB. The involved
two-time Green function of the virtual photon is further discussed in Section IIC. It is then shown how the medium
modifications of the light vector mesons enter the dilepton rate via the principle of vector meson dominance. The
simulation of medium modifications of these vector mesons by introduction of a certain time dependence of their self
energies is introduced in Section IIIA. We analyze the contributions to the rate in time representation and discuss
the precision of the numerics in Section IIIB. A quantitative description of the retardation is given in Section IIIC
by introduction of time scales, which characterize the memory of the spectral function. Comparing them to typical
time scales in heavy ion reactions reveals that changes of the spectral function can not generally be assumed to
be adiabatic and that memory effects can become important. We discuss quantum mechanical interference effects
occurring within this full quantum field theoretical description in Section IIID. Dilepton yields are calculated first
for constanttemperature andvolume in SectionIIIE. Finally, convolutionof the dynamic rateswith a fireballmodel
employing a Bjorken like expansion leads to our most important result, presented in Section IIIF: Comparison to
the quantities computed in the static limit, where all meson properties adjust to the medium instantaneously, the so
called Markov limit, reveals the significance of memory effects and the consideration of the full dynamics for certain
cases such as the celebrated and continuously discussed Brown-Rho scaling [28].
II. THE NONEQUILIBRIUM PRODUCTION RATE
A. Lepton number transport equation
We utilize the Schwinger-Keldysh realtime formalism and the emerging Kadanoff-Baym equations [27] in order to
derive the dynamic nonequilibrium rate of produced electron-positron pairs, coming from the decay of light vector
mesons via virtual photons in a spatially homogeneous, yet time dependent system. A different derivation for the
dilepton rate was performed in [29] for dileptons from a pion plasma as well as in [30], starting from the dilepton
correlator. The resulting formulas will provide a powerful tool to compute the dynamic behavior of the dilepton
production rate, influenced by a changing surrounding medium.
We extract the number of produced electrons with momentum p at time τ from the Wigner transform of the
electron propagatorG<(1,2)=i Ψ¯(2)Ψ(1) at equal times t =t =τ, which for a generalsystem is given by (using
1 2
h i
3
notation as in [31])
d3q m q q
G<(X,p,τ)=iZ (2π)3 Xrs E−E+ nDb†p−q2,rbp+q2,sEu(p+ 2,s)u¯(p− 2,r)eiq·Xeiτ(E−−E+)
p q q
+Db†p−q2,rd−p−q2,sEv(−p− 2,s)u¯(p− 2,r)eiq·Xeiτ(E++E−)
q q
+Dd−p+q2,rd†−p−q2,sEv(−p− 2,s)v¯(−p+ 2,r)eiq·Xeiτ(E+−E−)
q q
+Dd−p+q2,rbp+q2,sEu(p+ 2,s)v¯(−p+ 2,r)eiq·Xeiτ(−E+−E−)o, (1)
by projecting on the quantity b b δ [32]:
†p,r p,s rs
(cid:10) (cid:11)
N(p,τ)= i d3XTr G<(X,p,τ) .
− Z P
(cid:8) (cid:9)
It can be easily verified that this is achieved by use of the projector
m 1
=γ0 u(p,s¯)u†(p,s¯)=γ0 (/p+m)γ0, (2)
P E 2E
p Xs¯ p
wmhoemreenwtuemussetdatePss¯u((pp+,s¯)qu¯)(pan,s¯d) =(p21m(q/p).+Fmor).fuErt+her=dEetpa+ilsq2oanntdheEs−pi=n dEecpo−mq2paorseititohneoefntehregiWesicgonrerresfupnocntdioinngsteoe
± 2 ± − 2
[33, 34, 35]. The equations of motion for G<(1,2) and G>(1,2) = i Ψ(1)Ψ¯(2) are the Kadanoff-Baym equations,
− h i
generalized to the relativistic Dirac structure [32, 33, 34, 35]:
t1
(iγ ∂µ m Σ (1))G≷(1,1)= d2(Σ>(1,2) Σ<(1,2))G≷(2,1)
µ 1 − − HF ′ Z − ′
t0
t1′
d2Σ≷(1,2)(G>(2,1) G<(2,1)), (3)
′ ′
−Z −
t0
t1
G≷(1,1′)(cid:16)−iγµ←−∂µ1′ −m−ΣHF(1′)(cid:17)=Zt0 d2(G>(1,2)−G<(1,2))Σ≷(2,1′)
t1′
d2G≷(1,2)(Σ>(2,1) Σ<(2,1)), (4)
′ ′
−Z −
t0
with the self energy Σ and its local, Hartree-like term Σ . (1,2) is the short term notation for the coordinates
HF
(t ,x ,t ,x ). Using retarded and advanced Green functions
1 1 2 2
Gret/adv(1,2)= θ( (t t ))(G>(1,2) G<(1,2)), (5)
1 2
± ± − −
an important relation can be obtained directly from the Kadanoff-Baymequations:
G≷(1,1)= ∞d2 ∞d3Gret(1,2)Σ≷(2,3)Gadv(3,1)
′ ′
Z Z
t0 t0
+ dx dx Gret(1,x ,t )G≷(x ,t ,x ,t )Gadv(x ,t ,1). (6)
2 3 2 0 2 0 3 0 3 0 ′
Z Z
It can be regarded as a generalized fluctuation dissipation relation [33, 36, 37]. The second term accounts for the
initial conditions at time t only. It can be neglected if one lets the system evolve into a specified initial state for a
0
sufficiently long time. This is done by keeping the self energy insertions time independent prior to the onset of the
dynamics at the initial time t . Fouriertransformationof (3) and(4) in the spatialcoordinates(x x ) and taking
0 1 1′
−
γ (3) (4)γ at t =t =τ leads to
0 0 1 1′
−
i∂ G<(p,τ) p (γ γG<(p,τ) G<(p,τ)γγ )
τ 0 0
− · −
m(γ G<(p,τ) G<(p,τ)γ )=γ −→C(p,τ) ←C−(p,τ)γ , (7)
0 0 0 0
− − −
4
with the collision-terms
τ
−→C(p,τ)= dt¯ Σ>(p,τ,t¯)G<(p,t¯,τ) Σ<(p,τ,t¯)G>(p,t¯,τ)
Z −
t0 (cid:0) (cid:1)
τ
←C−(p,τ)= dt¯ G>(p,τ,t¯)Σ<(p,t¯,τ) G<(p,τ,t¯)Σ>(p,t¯,τ) ,
Z −
t0 (cid:0) (cid:1)
where Σ has been effectively absorbed into the mass m. Application of the projector (2) to Eq. (7) then yields
HF
the electron production rate at time τ:
∂ N(p,τ)= Tr (γ −→C(p,τ) ←C−(p,τ)γ )
τ 0 0
− nP − o
=( 2)Re Tr (γ −→C(p,τ)) , (8)
0
− h nP oi
Due tohavingaverylongmeanfreepath, theelectronsarenotexpectedtointeractwiththemedium aftertheyhave
been produced. This is why they can be described using the free propagators
1
G<0(p,t¯,τ)=i2E (γ0Ep+γ·p−m)eiEp(t¯−τ) (9)
p
1
G>0(p,t¯,τ)=−i2E (γ0Ep−γ·p+m)e−iEp(t¯−τ). (10)
p
With that Eq. (8) becomes
τ
∂ N(p,τ)=2Re Tr γ dt¯ Σ<(p,τ,t¯)G>(p,t¯,τ) Σ>(p,τ,t¯)G<(p,t¯,τ)
τ (cid:20) (cid:26)P 0(cid:18)Z 0 − 0 (cid:19)(cid:27)(cid:21)
t0 (cid:0) (cid:1)
τ
=2Re Tr dt¯ Σ<(p,τ,t¯)G>(p,t¯,τ) ,
(cid:20) (cid:26)Z 0 (cid:27)(cid:21)
t0 (cid:0) (cid:1)
where in the second step we used that G>(p,t¯,τ) γ =G>(p,t¯,τ) and G<(p,t¯,τ) γ = 0 together with the cyclic
0 P 0 0 0 P 0
invariance of the trace. Inserting (10) finally leads to
/p+m τ
∂τN(p,τ)=2 Im Tr dt¯ Σ<(p,τ,t¯) eiEp(τ−t¯) , (11)
(cid:20) (cid:26) 2Ep Zt0 (cid:0) (cid:1) (cid:27)(cid:21)
with p = E . The full dynamic information for the production of an electron at a given time τ is incorporated in
0 p
the memory integral from the initial time t until the present τ on the right hand side of Eq. (11).
0
B. The electron self energy Σ
The medium as the source for the production of dileptons enters via the dressing of the virtual photon propagator
in the electron self energy (see Fig. 1). This dressing will finally be given by vector mesons.
Π
1 2
0
G
FIG. 1: Feynman graph for the electron self energy Σ(1,2)
5
Π is the self energy of the virtual photon and we have
d3k
iΣ<(p,t ,t )= e2γ D<µν(k,t ,t )G<(p k,t ,t ) γ , (12)
1 2 − µ(cid:18)Z (2π)3 γ 1 2 0 − 1 2 (cid:19) ν
with D<µν(k,t ,t ) being the propagator of the dressed virtual photon with momentum k. Inserting this with the
γ 1 2
explicit form of the free electron propagator (9) into Eq. (11) yields
τ d3k
∂τN(p,τ)=2 Re(cid:20)e2Z dt¯Z (2π)3(i)Dγ<µν(k,τ,t¯)eiEp(τ−t¯)eEp−k(τ−t¯)
t0
1 1
Tr (/p+m)γµ(γ0Ep k+γ (p k) m)γν , (13)
×2Ep2Ep−k (cid:8) − · − − (cid:9)(cid:21)
where evaluation of the trace leads to
d3k 1 1
∂ N(p,τ)=2 e2 p (k p) +p (k p) g (p (k p)µ+m2)
τ Z (2π)3EpEp−k (cid:2) µ − ν ν − µ− µν µ − (cid:3)
τ
×Re(cid:20)Z dt¯iDγ<µν(k,τ,t¯)ei(Ep+Ek−p)(τ−t¯)(cid:21). (14)
t0
Defining p =k p and p+ =p as the four-momenta of the outgoing electron and positron, we rewrite Eq. (14) to
−
−
dR 2e2
E E (τ)= p+p +p+p g (p+p +m2)
+ −d3p+d3p− (2π)6 (cid:2) µ −ν ν −µ − µν − (cid:3)
τ
×Re(cid:20)Z dt¯iDγ<µν(k,τ,t¯)ei(E++E−)(τ−t¯)(cid:21), (15)
t0
with E = E and E = E . R denotes the number of lepton pairs per unit four-volume, produced with the
k p + p
specifie−d mome−ntum configuration. We now show that Eq. (15) is indeed the generalization of the well known
thermalproductionrateforleptonpairs[38,39,40]. UsingtheFouriertransforminrelativetime coordinates,defined
by
dω
D<µν(k,τ,t¯)= D<µν(k,τ,ω)e iω(τ t¯), (16)
γ Z 2π γ − −
and taking t , we have for the stationary case:
0
→−∞
dR 2e2
E+E−d3p+d3p− =(2π)6 (cid:2)p+µp−ν +p+νp−µ −gµν(p+p−+m2)(cid:3)
dω τ
×Re(cid:20)iZ 2πDγ<µν(k,ω)Z dt¯ei(E++E−−ω)(τ−t¯)(cid:21) (17)
−∞
iD<µν(k,ω) is realand time independent in the stationary case. The realpartof the last integralis simply πδ(E +
γ +
E ω). With that and the virtual photon momentum kµ =(E,k), E =E +E , the rate becomes
+
−− −
dR e2
E E = p+p +p+p g (p+p +m2) D<µν(k)
+ −d3p+d3p− −(2π)6 (cid:2) µ −ν ν −µ − µν − (cid:3) γ
e2 1
= p+p +p+p g (p+p +m2) Π<µν(k)
−(2π)6 µ −ν ν −µ − µν − M4 γ
(cid:2) (cid:3)
2e2 1 1
=−(2π)6 p+µp−ν +p+νp−µ −gµν(p+p−+m2) M4eβE 1ImΠrγetµν(k), (18)
(cid:2) (cid:3) −
where we used Eq. (6) for D<µν(k) inits stationarylimit in the firststep as wellas Π< =2in ImΠret, whichfollows
γ B
from the Kubo-Martin-Schwinger (KMS) relation [37, 41, 42], in the second step. Eq. (18) is the well known rate of
dilepton production derived in e.g. [40]1.
1 ThedifferenceintheoverallsignisduetotheoppositesigninthedefinitionoftheGreenfunctionsin[40].
6
We return to the nonequilibrium formula (15) and project on the virtual photon momentum using
dN dR
= δ4(p++p− k)d3p+d3p−.
d4xd4k Z d3p+d3p −
−
This leads to
dN 2e2 d3p+ d3p
d4xd4k(τ,k,E)=(2π)6 Z E+ Z E−−δ4(p++p−−k)(cid:2)p+µp−ν +p+νp−µ −gµν(p+p−+m2)(cid:3)
τ
Re dt¯iD<µν(k,τ,t¯)eiE(τ t¯) (19)
× (cid:20)Z γ − (cid:21)
t0
for the production rate of dilepton pairs of momentum k = (E = E +E ,k). In the following numerical study we
+
will consider the mode k = 0 exclusively, i.e., the virtual photon resting w−ith respect to the medium. For this case,
and taking the electron mass m to zero, Eq. (19) can be simplified to
dN 2e2 2 τ
(τ,k=0,E)= π(k k k2g )Re dt¯iD<µν(k=0,τ,¯t)eiE(τ t¯) . (20)
d4xd4k (2π)63 µ ν − µν (cid:20)Z γ − (cid:21)
t0
This expression is easily understood. The dynamic information is inherent in the memory integral on the right hand
side that runs over all virtual photon occupation numbers, Fourier transformed at energy E from the initial time to
the present. Hence this memory integral determines the full nonequilibrium dilepton production rate at time τ.
C. The in-medium virtual photon self energy Π
The dilepton production rate (20) involves the virtual photon occupation number, expressed by the propagator
D<. We introduce the dynamic medium dependence by dressing this virtual photon propagator with the medium
γ
dependent ρ- or ω-meson. This dressing enters via the photon self energy Π< through the fluctuation dissipation
relation (cf. (6)) for D<µν:
γ
D<µν =Dretµα Π< Dadvβν, (21)
γ γ ⊙ αβ ⊙ γ
where implies the integration over intermediate space-time coordinates. In the medium the vector mesons and
⊙
virtualphotonshavetwopossiblepolarizationsrelativetotheirmomentuminthemedium. Thisleadstotwodifferent
self energies Π (transverse) and Π (longitudinal). Introduction of the projectors P and P allows us to split the
T L L T
propagatorsand the self energy into a 3-longitudinal and a 3-transversalpart, relatively to the particle’s momentum
[39]:
Pµα Pµα Pβν Pβν
Dretµα = T L ; Dadvβν = T L (22)
γ −k2 Πret − k2 Πret γ −k2 Πret − k2 Πret
− T − L − T ∗ − L ∗
and
Π< = P Π< P Π<. (23)
αβ − Tαβ T − Lαβ L
The projectors fulfill the usual projector properties P2 = P and P P = P P = 0. With that, Eq. (21)
(T/L) (T/L) T L L T
becomes
D<µν =Dretµα ( P Π< P Π<) Dadvβν
γ γ ⊙ − Tαβ T − Lαβ L ⊙ γ
= D< Pµν D< Pµν, (24)
− γ,T T − γ,L L
with
1 1
D< = Π< . (25)
γ,T/L k2 Πret ⊙ T/L⊙ k2 Πret
− T/L − T/L∗
In the case k=0, considered later, the longitudinal and transverse parts become identical and it follows
kµkν
D<µν = D< Pµν D< Pµν = D< (Pµν +Pµν)= D< gµν .
γ − γ,T T − γ,T L − γ,T T L − γ,T(cid:18) − k2 (cid:19)
7
Inthis casethe productionratedepends onlyonthe transversepartofthevirtualphotonpropagator. Theappearing
factor of 3 accounts for the two transverse and one longitudinal directions:
dN 2 e2 τ
d4xd4k(τ,k=0,E)=3(2π)5(3E2)Re(cid:20)Z dt¯iDγ<,T(k=0,τ,¯t)eiE(τ−t¯)(cid:21) (26)
t0
For dilepton production Πret e2 and E is the invariant mass of the virtual photon. For the cases,we are interested
in, it holds Πret E and we∝can approximate
| |≪
D< =Dret Π< Dadv. (27)
γ,T γ,0⊙ T ⊙ γ,0
For k 0
→
Dret(k 0,t)= θ(t)t=Dadv(k 0, t) (28)
γ,0 → − γ,0 → −
and we may calculate the virtual photon propagator using the transport equation
τ t¯
iD< (k=0,τ,¯t)= dt dt (τ t ) iΠ<(k=0,t ,t ) (t¯ t ). (29)
γ,T Z 1Z 2 − 1 T 1 2 − 2
t0 t0 (cid:0) (cid:1)
Theappearingundampedphotonpropagatorsleadtodivergingcontributionsfromearlytimes,i.e.,forlowfrequencies.
In the later numerical calculation these contributions turn out to be at least of the order 105 larger than the actual
(higher frequency) structure. Due to the naturally limited numerical accuracy, the higher frequency structure would
get lost among these early time contributions. In order to cure this numerical problem, we introduce an additional
cutoff Λ for the free photon propagators,i.e., we perform the replacement:
Dγre,t0(τ −t1)=(τ −t1)→(τ −t1)e−Λ(τ−t1) (30)
andanalogouslyforDadv(t t¯). IntheperformedcalculationsweemployΛ 0.3GeV.Theexponentialfactorslead
γ,0 2− ≈
to a reduction of the rate, which we will overcome by renormalizing the final result by multiplication with (ω2+Λ2)2,
ω4
a factor we get from Fourier transforming the convolution (29), assuming equilibrium. This will not affect the time
scales we are interested in, and comparison of the dynamically computed rate for the stationary case (constant self
energy) with the analytic thermal rate shows perfect agreement.
Vector meson dominance (VMD) [43, 44] allows for the calculation of Π<, using the identity between the electro-
T
magnetic current and the canonical interpolating fields of the vector mesons [45]:
e
J = m2ρ ..., (31)
µ −g ρ µ−
ρ
which leads to
e2
Π< = m4D< (32)
αβ g2 ρ ραβ
ρ
fortheselfenergy. Whentreatingtheω-meson,weusethecorrespondingselfenergyandpropagator. Weagainapply
the generalized fluctuation dissipation relation (6) to calculate
D< =Dret Σ< Dadv, (33)
ρ,T ρ,T⊙ ρ,T⊙ ρ,T
with the ρ-meson self energy Σ< . The transverse parts of the retarded and advanced propagators Dret(k,t ,t ) =
ρ,T ρ,T 1 2
Dadv(k,t ,t ) of the vector meson in a spatially homogeneous and isotropic medium follow the equation of motion
ρ,T 2 1
t1
∂2 m2 k2 Dret(k,t ,t ) dt¯Σret(k,t ,t¯)Dret(k,t¯,t )=δ(t t ). (34)
− t1 − ρ− ρ,T 1 2 −Z ρ,T 1 ρ,T 2 1− 2
(cid:0) (cid:1) t2
In the following we will omit the index T for convenience.
The dynamic medium evolution is now introduced by hand via a specified time dependent retarded meson self
energy Σret(τ,ω) with system time τ (see Section IIIA). From that the self energy Σ<, needed for solving Eq. (33),
follows by introductionof anassumedbackgroundtemperature ofthe fireball. The fireball, constituting the medium,
8
generatesthe time dependentselfenergyΣret and,assumingaquasithermalizedsystem,the ρ-mesoncurrent-current
correlator Σ< is given via
Σ<(τ,ω,k)=2in (T(τ))ImΣret(τ,ω,k), (35)
B
which follows from the KMS relation
Σ<(ω,k)= e βωΣ>(ω,k), (36)
−
∓
being valid for thermal systems [37, 41, 42]. n is the Bose-distribution. The assumption of a quasi thermalized
B
backgroundmediumisofcourseratherstrong,butnecessaryinordertoproceed: Inprinciple,forafullnonequilibrium
situation the self energies Σ< and Σ> for the ρ-meson have to be obtained self-consistently via e.g. coupling to
resonance-holepairs[46],beingoutofequilibriumthemselves. (Forarealizationoftruenonequilibriumdynamicsofa
homogeneoussystemwithin aΦ4-theorysee [12]andwithin a coupledfermion-mesonsystem [14].) For anexpanding
and inhomogeneous reaction geometry this is still not possible today. In any case, explicit calculation of Σ< in the
two-timerepresentationwillcauseevenstrongermemoryeffects. Additionally,inordertosimulatearealisticsituation
by application of a fireball model, a temperature needs to be defined and hence local equilibrium has to be assumed.
The framework, we have established, allows us to calculate the dynamic evolution of vector mesons’ spectral
properties, occupation, and the nonequilibrium dilepton production rate, given an evolving medium with assumed
timedependentquasi-thermalproperties. Atthispointitisalsoworthwhiletopointoutthatworkinginthetwo-time
representation has great advantages over the mixed representation, in which all quantities are expressed by their
Wigner transforms. The problem in this case is that the Wigner transforms of the many appearing convolutions are
nontrivial. They can be expressed by a gradient expansion to infinite order [47]. Explicit calculations are usually
carriedout within a first order gradientapproximation,which is not applicable in our case because the system is not
evolving slowly with respect to the relevant time scales (see Sections IIIC and IIIF for details on the time scales
involved). Allmemoryislostby applicationofthis approximation,becausewhenusingthe mixedrepresentation,full
Fouriertransformationsfromrelativetimetoitsconjugatefrequencyareinvolved,whichimplytreatingcontributions
from the future and from the past on equal grounds. This violates causality in a quickly evolving system. Hence,
calculations within the two-time representation allow for the most exact investigation of the evolving system under
consideration.
III. NONEQUILIBRIUM DILEPTON PRODUCTION FROM AN EVOLVING MEDIUM
A. Vector meson self energies
We are now able to calculate the evolution of the spectral properties of vector mesons in a changing medium as
well as the corresponding dilepton production rates. We will treat mass shifts described by Brown-Rho scaling [28],
broadening, as caused by pion scattering, and scattering of the mesons with nucleons, leading to further broadening
andexcitationofresonances[46,48,49,50,51,52,53]. Themediumeffectsareintroducedviaaspecifictimeevolving
self energyfor the vectormesons. The mainpurpose ofthis work is to investigatemedium modifications dynamically
for the first time, and compare the results to those obtained from instantaneous, Markoviancalculations.
In the following we first employ simplified self energies for possible broadening such as
ImΣret(τ,ω)= ωΓ(τ), (37)
−
with a k- and ω-independent width Γ, which, in the static case, leads to a Breit-Wigner distribution of the spectral
function
1 1 ωΓ
A(ω,k)= ImDret(ω,k)= . (38)
−π ρ π(ω2 k2 m2)2+(ωΓ)2
− −
The self energy (37) is given in a mixed time-frequency representation. The time dependence is being accounted for
by introduction of the system time τ, for which we will discuss two possible choices below. ω stems from the Fourier
transformation in relative time (t t ).
1 2
−
Modificationsofthe massareintroduceddirectlyasviaalocaltermintheselfenergy. Fromthissortofselfenergy
several numerical issues emerge. In Eq. (34) the self energy enters in time representation
Σret(t ,t )=(Σ>(t ,t ) Σ<(t ,t ))θ(t t ), (39)
1 2 1 2 1 2 1 2
− −
9
ItsFouriertransforminrelativetime(t t ),Σret(τ,ω),isgivenbyaconvolutionoftheθ-function’sFouriertransform
1 2
and (Σ> Σ<)(τ,ω): −
−
dω¯ 1
Σret(τ,ω)=i (Σ> Σ<)(τ,ω¯)
Z 2πω ω¯ +iǫ −
−
dω¯ 1 1
=i (Σ> Σ<)(τ,ω¯)+ (Σ> Σ<)(τ,ω), (40)
PZ 2π ω ω¯ − 2 −
−
where we used the standard relation
1 1
= iπδ(ω ω¯). (41)
ω ω¯ +iǫ Pω ω¯ − −
− −
With (Σ> Σ<)(τ,ω)= 2iωΓ(τ), one has
− −
+c dω¯ ω¯
Σret(τ,ω)=2 Γ(τ) iωΓ(τ),
PZ 2π ω ω¯ −
−c −
whichhastheimaginarypart(37). Inadditionwealsogetan”unwanted”dispersiverealpartthatcausesan(infinite)
mass shift. The introduced cutoff c cures that infinity and we can renormalize the mass by the replacement
m m2 ReΣret(ω,τ),
→ −
p
with
+c dω¯ ω¯
ReΣret(τ,ω)=Re 2Γ(τ)
(cid:20) PZ 2π ω ω¯(cid:21)
−c −
2 1 1 ω2
= Γ(τ)c F ,1, ,
−π (cid:18)2 1(cid:18)−2 2 c2(cid:19)(cid:19)
2 ω2 ω4
Γ(τ)c 1 , (42)
≈−π (cid:18) − c2 − 3c4(cid:19)
with the hypergeometricfunction F , Taylorexpandedto secondorderinthe lastline. To maketherenormalization
2 1
ω-independent, we replace ω by the physical pole mass m, causing only minimal inaccuracies far from the peak
position.
AnotherpotentialproblemisthebehavioroftheselfenergyΣ<(τ,ω)fornegativefrequencies. Asalreadydescribed,
forathermalizedsystem,itisgivenbyΣ<(τ,ω)=2in (T(τ),ω)ImΣret(τ,ω). Fornegativefrequenciesthepropagator
B
D<(τ,ω)contains afactorof(1+n(ω)) since there holds the symmetry relationD<(τ, ω)=D>(τ,ω)forthe scalar
−
bosoncaseandD>(τ,ω)=2i(1+n )A(τ,ω)in equilibrium. The firsttermin parenthesesleads todivergentvacuum
B
contributions, which are uninteresting for the investigation of the positive frequency behavior and cause numerical
problems. To see this, we first give the limits of the expression Σ<(τ,ω) in frequency representation:
e ω/T(τ) for ω 0
−
iΣ<(τ,ω)=∝2T(τ)Γ(τ) for ω =≫0 (43)
 2ωΓ(τ) for ω 0
− ≪

The Fourier transform thus becomes a δ -distribution for the negative frequency part. In order to cure this problem
′
we first split the self energy into positive and negative frequency parts
2ω Γ(τ)
iΣ<(τ,ω)= | | 2ωΓ(τ)θ( ω), (44)
eω/T(τ) 1− −
| | −
=:iΣ<,eff(τ,ω)
| {z }
andneglectthe negativefrequencypart, violatingthe mentioned symmetryrelationforD< andD> (alsofor Σ< and
Σ>). This violation however only means an omittance of the vacuum contributions for negative frequencies and does
not cause changes for positive frequencies. The same approachmay be taken with other types of self energies having
the same symmetry property under the exchange ω ω.
↔−
10
Coupling of resonances to the ρ N-channel has been treated by [46, 51, 53]. For the k = 0 mode, the full self
−
energy for coupling to JP = 3− -resonancesis givenby [53] (also see [54], who treatedcoupling of pions to resonance
2
hole excitations)
ρ(τ) f 2 ω3E¯Γ (τ)
ImΣ(τ,ω,k=0)= RNρ g R ωΓ(τ). (45)
− 3 (cid:18) mρ (cid:19) I(ω2−E¯2− ΓR(4τ)2)2+(ΓR(τ)ω)2 −
E¯ = m2 +k2 m is the energy necessary for the meson to scatter from a nucleon at rest, with m and m the
q R − N R N
masses of the resonanceand the nucleon respectively. Γ is the width of the resonanceand g is the isospinfactor (2
R I
for isospin 1 and 4 for isospin 3 resonances). ρ(τ) is the time dependent density of the system and appears due to
2 3 2
application of the ρ- -approximation, where stands for the forward scattering amplitude. The important part for
our purpose is the stTructure of the denominatTor, which represents a characteristic pole structure. We will replace ω2
in the numerator by a constant factor since it causes divergence of ReΣret, only cured by application of subtracted
dispersionrelationscorrespondingto counterterms. Thecutoffc alsopreventsdivergencesbut leavinginthe factorof
ω2 causes inaccurate numerical results. We show the spectral function for the case of the N(1520) resonance, where
f 7.0 [46] and the width of the resonance is Γ =120 GeV, in Fig. 2 for different densities at k=0.
RNρ R
≈
3.0
= 0
2.5
= 0
] 2.0 vacuum
2
- V
e
G
1.5
[

A
1.0
0.5
0.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
[GeV]
FIG.2: Spectralfunctionof theρ-mesoncoupledtoanN(1520)-hole pairfordifferentdensities. E¯ =582MeV,ΓR =120 MeV
We now discuss the system time τ. Defining τ = t1+t2, as first done in [27], seems to be a sensible choice, but
2
when investigating the spectral function, one is dealing with retarded quantities and we do not want them to collect
information from the future, i.e., we want to retain causality. This is why we choose τ =t instead of the symmetric
1
form τ = t1+t2, because in this case at a certain time τ information of Σret that is located in the future of τ enters
2
the spectral function. This cannot happen with τ =t as demonstrated in Appendix A.
1
For the self energy Σ<(τ,ω) one should stick to the symmetric choice in order to fulfill the symmetry relation
Σ<(t ,t )=Σ>(t ,t ). However,doingthisreducesaccuracyinthenumericalcalculation,duetoadditionalnecessary
1 2 2 1
Fourier transformations. Comparison of calculations using either Σ<(t ,ω) or Σ<(t1+t2,ω) reveals minor differences
1 2
in the rate when changing the temperature with system time t or t1+t2 respectively, but on average, and hence in
1 2
the final yield, the differences cancel, as could be expected.
B. Contributions to the rate in time representation
Before we calculate nonequilibrium rates and yields, we investigate how the present rate is created over time, i.e.,
we focus on which contributions to Eq. (26) come from which times in the past. Figs. 3 and 4 show the integrand
for fixed energy ω = 750 MeV and fixed time τ for relative times τ t. The first case shown (left of Fig. 3) is an
−
equilibrium scenario with free ρ-mesons embedded in an environmentat constant temperature. One can see that the
maincontributioncomesfromtheverynearpast,butthattherearealsocontributionsfromearlytimesaswellaslarge