Electromagnetic Probes of Strongly Interacting Matter: Probes of Chiral Symmetry Restoration?

0.25 MB·

by  R. Thomas

Acta Phys.Hung.A 19/1 (2005) 000–000
HEAVY ION
PHYSICS
Electromagnetic Probes of Strongly Interacting Matter:
Probes of Chiral Symmetry Restoration?
5
0
0 R. Thomas1, K. Gallmeister2, S. Zschocke1, B. K¨ampfer1
2

n 1 Institut fu¨r Kern- und Hadronenphysik, ForschungszentrumRossendorf,
a
PF 510119,D-01314 Dresden, Germany
J
1 2 Institut fu¨r Theoretische Physik, Universit¨at Giessen, D-35392 Giessen,
2 Germany

1
v Received 1 January 2005
2
0 Abstract. The QCD sum rule approach to in-medium modifications of the ω
2
meson in nuclear matter is reviewed with emphasis of its relation to 4-quark
1
condensatesandchiralsymmetry restoration. Possibleimplicationsofthe CB-
0
5 TAPS experiment for the reaction γA → A′ω(→ π0γ) are sketched and the
0 particularly important role of di-electron probes, accessible with HADES, is
/ highlighted. A brief update of a parametrization of the previous dilepton and
h
p photon probes from CERES and WA98 of heavy-ion collisions at CERN-SPS
- energies is presented.
p
e
Keywords: In-Medium Modifications, QCD Sum Rules, Chiral Symmetry,
h
: Four-Quark Condensates, Electromagnetic Probes
v
PACS: 12.40.Yx, 12.38.Lg
i
X
r
a
1. Introduction
Thechiralcondensatehq¯qiisanorderparameterforthespontaneousbreakingofchi-
ralsymmetry inthe theoryofstronginteraction(cf. [1]for introducingthis topic).
Its role is demonstrated, e.g., by the Gell–Mann-Oaks-Rennerrelationm2 ∝−hq¯qi
π
for the pion mass1 or by Ioffe’s formula m ∝ −hq¯qi for the nucleon mass. There
N
is growing evidence that the quark-gluon condensate hq¯σ Gµνqi ∝ hq¯qi [ 3] (with
µν
G as chromodynamic field strength tensor) is another order parameter. The
µν
QCD trace anomaly related to scale invariance breaking gives rise to the gluon
condensate h(α /π)G Gµνi. There are manyother condensates characterizingthe
s µν
complicatedstructureoftheQCDvacuum. Inamedium,describedbytemperature
1Theanalysisofprecisiondatahasshownthatmorethan94%ofthepionmassstemfromthe
chiralcondensate [2].
1219-7580/04/ $ 20.00
(cid:13)c 2004 Akad´emiaiKiad´o, Budapest
2 Thomas, Gallmeister, Zschocke, K¨ampfer
T and baryon density n, these condensates change, i.e., the ground state is rear-
ranged. Since hadrons are considered as excitations above the vacuum, a vacuum
changeshouldmanifestitselfasachangeofthe hadronicexcitationspectrum. This
idea triggered widespread activities to search for such in-medium modifications of
hadrons and to relate them to corresponding order parameters.
Among the promising tools are electromagnetic probes. Due to the small cross
sections, real or virtual photons can leave nuclear systems nearly undisturbed and
carry information on their creation conditions. This is evidenced in the realm of
heavy-ion collisions by photon (γ) and dielectron (e+e−) emission rates
dR 2
ω γ = −ImΠµn (k /T) , (1)
d3k µ B 0 (2π)3
dRe+e− = −ImΠµn (Q /T) α , (2)
d4Q µ B 0 12π4M2
where Πµ is the trace of the retardedcurrent-currentcorrelatorrepresentingat the
µ
same time the photon self-energy in the medium, and n is the Bose distribution
B
function; k is the photon momentum and Q denotes the 4-vector of the e+e− pair
where the electron mass is neglected and M is the invariant e+e− mass. With
virtue of(1, 2), realandvirtualphotonsareidealprobes ofthe medium properties.
Moreover, due to vector dominance [ 4] photons couple predominantly to vector
mesons. Thus, the in-medium behavior of light vector mesons ρ, ω, ··· becomes
directly accessible.
Concentratingon the iso-scalarpartthe causalcurrent-currentcorrelator,here
specified for the ω meson,
i
Πω(q,n)= d4xeiqxhΩ|Tjω(x)jωµ(0)|Ωi (3)
3Z µ
with the current jω = u¯γ u+∗d¯γ d /2,2 an operator product expansion and a
µ µ µ
Borel transformation of(cid:0)the twice-subt(cid:1)racted dispersion relation results in
1 ∞dsImΠω(s,n)e−s/M2 =−c M2− ∞ cj +Πω(0,n), (4)
π Z s 0 (j−1)!M2(j−1)
0 Xj=1
where Πω(0,n) = 9n/(4m )3 is a subtraction constant with the meaning of Lan-
N
dau damping or ω forward scattering amplitude, and the coefficients c contain
j
condensates and Wilson coefficients; M is the Borel mass.
Through the chain of equations (1 - 4) electromagnetic signals off hadron sys-
tems and QCD condensates are interrelated. Chiral symmetry restoration, in this
context, means a dropping chiral condensate which modifies the electromagnetic
emission pattern [ 5]. As mentioned above there exist many condensates which are
2Hereandafterwardsanasteriskindicatesasignchangefortheρmeson.
3ndenotes thebaryondensityandmN isthenucleonmass.
Electromagnetic Probes 3
modified in the nuclear medium. Therefore, the resulting picture becomes more
complex. As we are going to point out, in truncated sum rules different hadrons
are sensitive to different condensates.
Our paper is organized as follows. In section 2 we discuss in some detail the
QCD sum rule for the ω meson and try to draw first consequences from the recent
CB-TAPS experiment. We contrast in section 3 the crucial dependence of the ω
properties on 4-quark condensates with the QCD sum rules for the nucleon which
is sensitive to the chiral condensate. In section 4 we present a brief update of
the description of data on electromagnetic probes at CERN-SPS energies. The
summary can be found in section 5.
2. QCD Sum Rule for ω Meson
The truncated QCD sum rule (4) for the ω meson can be arranged as [ 6]
s c c c
m2(n,M2,s )= c0M2h1−(cid:16)1+ Mω2(cid:17)e−sω/M2i− M22 − M34 − 2M46 ,
ω ω c c c c Πω(0,n)
c0 1−e−sω/M2 + M12 + M24 + 2M36 + 6M48 − M2
(cid:0) (cid:1) (5)
where the quantity m2(n,M2,s ) is the normalizedmoment ofthe imaginarypart
ω ω
of the current-currentoperator
sωdsImΠω(s,n)e−s/M2
m2(n,M2,s )= 0 . (6)
ω ω sRωdsImΠω(s,n)s−1e−s/M2
0
R
Sumrules,evenwithresonance+continuumansatzasexploitedin(5,6),constrain
some integral strength but not individual parameters without further specification
of ImΠ.
The first coefficients c in (4) refer to the decomposition Πω(Q2 = −q qµ) =
j µ
Πω +Πω +Πω +Πω +... in the 2-flavor sector with
scalar d=4,τ=2 d=6,τ=2 d=6,τ=4
1 α Q2 3
Πω = − 1+ s Q2ln − (m2 +m2) (7)
scalar 8π2 π µ2 8π2 u d
(cid:16) (cid:17)
1 α 1 1 1 α
+ 1+ s m hu¯ui+m hd¯di + h sG2i
2 3π Q2 u d 24Q2 π
(cid:16) (cid:17)
(cid:0) (cid:1)
1 1
− πα hu¯γ γ λauu¯γµγ λaui+hd¯γ γ λadd¯γµγ λadi
2 sQ4 µ 5 5 µ 5 5
(cid:0) (cid:1)
1 2 1
−∗πα hu¯γ γ λaud¯γµγ λadi− πα hu¯γ λaud¯γµλadi
sQ4 µ 5 5 9 sQ4 µ
1 1
− πα hu¯γ λauu¯γµλaui+hd¯γ λadd¯γµλadi
9 sQ4 µ µ
(cid:0) (cid:1)
1 1
+g m2hm u¯σ Gµνui+m2hm d¯σ Gµνdi ,
s12Q6 u u µν d d µν
(cid:0) (cid:1)
4 Thomas, Gallmeister, Zschocke, K¨ampfer
α 1
Πω = s qµqνhSˆTˆ G αG i (8)
d=4,τ=2 π Q4 µ αν
(cid:0) (cid:1)
2 10α 1
− − s i qµqνhSˆTˆ u¯γ D u+d¯γ D d i,
(cid:18)3 27π (cid:19) Q4 µ ν µ ν
(cid:0) (cid:1)
82α 1
Πω = − s qµqνqλqσhSˆTˆ G ρD D G i (9)
d=6,τ=2 27π Q8 µ ν λ ρσ
(cid:0) (cid:1)
8 134α 1
+ + s i qµqνqλqσhSˆTˆ u¯γ D D D u+d¯γ D D D d i,
(cid:18)3 45π (cid:19) Q8 µ ν λ σ µ ν λ σ
(cid:0) (cid:1)
1 1
Πω = −∗ qµqνhg2SˆTˆ u¯γ γ λaud¯γ γ λad i (10)
d=6,τ=4 3Q6 s µ 5 ν 5
(cid:0) (cid:1)
1 1
− qµqνhg2SˆTˆ u¯γ γ λauu¯γ γ λau+d¯γ γ λadd¯γ γ λad i
6Q6 s µ 5 ν 5 µ 5 ν 5
(cid:0) (cid:1)
1 1
− qµqνhg2SˆTˆ u¯γ λau u¯γ λau+d¯γ λad i
24Q6 s µ ν ν
(cid:0) (cid:0) (cid:1)(cid:1)
1 1
− qµqνhg2SˆTˆ d¯γ λad u¯γ λau+d¯γ λad i
24Q6 s µ ν ν
(cid:0) (cid:0) (cid:1)(cid:1)
5 1
− qµqνhig SˆTˆ u¯ D ,G˜ γαγ u+d¯ D ,G˜ γαγ d i
12Q6 s (cid:18) h µ ναi+ 5 h µ ναi+ 5 (cid:19)
7 1
− qµqνhg2SˆTˆ m u¯D D u+m d¯D D d i
3Q6 s u µ ν d µ ν
(cid:0) (cid:1)
(h···i is a short hand notation for the Gibbs average over the nuclear matter
states hΩ|···|Ωi, SˆTˆ denote operators causing symmetric and traceless structures,
α =g2/4π, andD denotes the covariantderivative;for further details cf. [ 6, 7]).
s s µ
We havespelledoutthesefirsttermsto evidence thateveninlowordermuchmore
(and partially poorly known) condensates determine the sum rule. The genuine
chiralcondensates hu¯ui and hd¯di enter c with the small factors m and are irrel-
2 u,d
evant,bothinvacuumandinnuclearmatter. ThisisexhibitedinFig.1,wherethe
influence of various terms is discussed. In most previous approaches the 4-quark
condensateshavebeenfactorized,i.e.,expressedbysquareofthechiralcondensate.
Within such an approximation, statements that the in-medium change of the chi-
ral condensate drives in-medium modifications of light vector mesons are correct.
However, strictly speaking it is the change of 4-quark condensates which drive the
change of m , together with other subleading terms such as the gluon condensate.
ω
It should be emphasized that most sum rule approaches calculate the param-
eter m2 via (5) by requiring independence of the Borel mass M by choosing
ω
an appropriate continuum threshold s and identify m with the pole mass of
ω ω
the physical particle. This is justified in zero-width approximation. In general,
however, one needs a model or data for deducing conclusions on ImΠ. To illus-
trate this feature we choose as model of ImΠω a Breit-Wigner distribution, i.e.,
ImΠ=s1/2Γ(s)[(s−m2)2+sΓ(s)2]−1withΓ(s)=Γ 1−(s /s)2/ 1−(s /m2)2.
0 0 0 0 0
We assumethatthe sumrule(5)gavem =0.77GepV. Thenthe exppression(6)de-
ω
termines the correlationΓ (m ) as exhibited in Fig. 2. Such a correlationhas been
0 0
Electromagnetic Probes 5
1.0 1.0
0.9 in vacuum, default: 0 = 3 in medium, default: N = 3
0.8 0.8
(0) [GeV]m0000....4567 default =20 =10 <qq>=0 2 0=>G< Q=00 0=c1 3- 01+=c4 -3 c=-104 m(n=n) [GeV]000..46 default =4N =2N ... K=0... den.dep. <qq> 2 >den.dep. <G q,g A2 Q0 q,g A4 flavor.mix 4-q (1) c = +0.54 (1) c = -0.54 Landau/3
0.3
0.2 0.2
0.1
0.0 0.0

Fig. 1. Importance of various quantities for the parameter m in the QCD sum
ω
rule(5). Left(right)panel: forvacuum(nuclearmatter). Defaultvaluesarehq¯qi =
0
(−0.245 GeV)3 and h(α /π)G2i = (0.33 GeV)4. The 4-quark condensates enter
s 0
c in the combination −112πα κ hq¯qi2[1+ κN σN n] with σ = 45 MeV and
3 81 s 0 0 κ0 mqhq¯qi0 N
m = 5.5 MeV. Furthermore, c contains a term ∝ Q2 responsible for the ρ−ω
q 3 0
mass splitting in vacuum. Aq,g are contributions to c which can be traced back to
2,4 3
moments of the parton distribution in the nucleon. ”flavor.mix” and K··· indicate
···
poorly known terms in c stemming from flavor mixing condensates. The term c
3 1
is determined by m2 . The complete coefficient c is presently unknown. In the
u,d 4
default calculation this term is discarded. To estimate its influence we assume ad
hoc c (n)=±10−3GeV8(1+c(1)n) with c(1) =±0.5/n .
4 4 4 0
notedforthefirsttimein[8]. Thisexampleshowsthatthesumruleapproachneeds
further ingredients to pin down interesting parametersof ImΠ. Another possibility
to employ the sum rule is to determine m from data via (6) with experimental
ω
input ImΠω. In this respect the electromagnetic probes are superior, as ImΠ is
directly determined by rates, see (1, 2).

0.10
0.08
V]0.06
Ge
[00.04
0.02
0.00
0.78 0.79 0.80
m0 [GeV]
Fig. 2. Emerging correlation Γ (m ) from (6) when assuming m = 0.77 GeV.
0 0 ω
The plot is for M=1 GeV and s =0.135 GeV.
0
6 Thomas, Gallmeister, Zschocke, K¨ampfer
The CB-TAPS collaboration [ 9] has studied the reaction γA → A′ω(→ π0γ).
Count rates as a function of the invariant mass m have been measured for LH
π0γ 2
and Nb targets. It is remarkable that for the Nb target the strength is noticeably
shifted to smaller values of m for small values of the ω momentum, while for
π0γ
larger ω momenta the difference for both targets diminishes. Even without knowl-
edge of acceptance corrections, phase space factor etc. one may use the count rate
as estimator of ImΠω4 and deduce a small down shift of m from (6). Supposing
ω
(i)thevalidityofthesumrule(5)asitstands,and(ii)assumingthatthemeasured
decays ω → π0γ happen to a large extent in nuclear matter, and (iii) neglecting
poorly known higher terms, the unavoidable conclusion is that in-medium 4-quark
condensates arereduced atleastby 30%. (The actualnumber depends onthe used
value of m and details of the chosen Borel window. We here assumed that the
ω
in-medium ω mass is 0.77 GeV, as in vacuum.) Such a strong density dependence
of 4-quark condensates is needed to counterbalance the strong Landau damping [
10] which tends to up-shift the ω mass parameter in nuclear matter. One severe
problemis the finalstate interactionof the π0 which may cause a diminished value
of m even when the parent ω meson has its vacuum mass [ 11]. Nevertheless,
π0γ
thefactthatthein-mediumω massparameterisnotlargerthaninvacuumhasthe
mentioned consequence for the 4-quark condensate. More quantitative results are
possible for the reaction πA → A′ω(→ e+e−) as envisaged in the HADES physics
program [ 12]: Due to a similar relation as (2), ImΠ is directly accessible without
facing the problem of final state interaction.
3. QCD Sum Rules for the Nucleon
In contrastto the QCD sum rule for the light vector mesons, the chiralcondensate
is an important quantity for the nucleon mass. The detailed reminder of the QCD
sum rules for the nucleon will be published elsewhere. We present here only a
few of the striking results to expose the interplay of chiral condensate and 4-quark
condensates. In the left panel of Fig. 3 the dependence of the nucleon mass in
vacuum on various quantities entering the sum rule is exhibited. One observes a
nearly linear dependence on the genuine chiral condensate supporting the validity
of the Ioffe formula. The impact of 4-quark condensates is sizeable (their strength
isscaledbyκvac),whilethegluoncondensateisofminorrelevance. Thein-medium
N
changesofthechiralcondensatecausesubstantialchangesofthescalarnucleonself-
energy, see right panel of Fig. 3. Further, the density dependence of the 4-quark
condensates parameterized by a strength factor κmed becomes important in the
N
nuclear medium. As noticed in [ 13], the known phenomenology of the in-medium
nucleon is obtained only by assuming a weak density dependence of the 4-quark
condensates. This looks like a contradiction to results of the previous section. It
mustbenoted,however,thatthenucleonsumruleisenteredbymanymore4-quark
4Indeed, the rate ω → π0γ is given by dRω→π0γ/d4q ∝ Φ(q)ImΠω(q), where Φ is a phase
spacefactor. Theabsolutenormalizationcancelsin(6).
Electromagnetic Probes 7
condensates. InTableIwelistforthevacuumthepossible4-quarkcondensatesand
their weightin ω and nucleon sum rules. It is conceivable that the net effect of the
4-quark condensates leads to some cancellation for the nucleon, while the ω meson
sum rule is sensitive to the few entering 4-quark condensates. For a calculation of
the 4-quark condensates for the nucleon within a chiral quark model we refer the
interested reader to [ 14].
1.4 1.4
V]
V] 1.2 Ge 1.2
[GeN 1 ∗M[N 1
M y
g
Nucleonvacuummass 0000....2468 default vac.01¯qqhivac.05¯qqhivac.15¯qqhi vacκ0=Nvacκ.05=Nvacκ.10=N αvac2G0hi→sπαvac.205Ghisπαvac.2G15ihsπ .t09=− .t11=− Nucleonscalarselfener 0000....2468 default ..densdep¯qq0hi→n ...densdep05¯qqhin ...densdep15¯qqhin medκ0=Nmedκ.05=Nmedκ.10=N α..22densdepEB0h(−)i→snπα...22EBdensdep20(−)ihsnπ α..22densdepEB0h(+)i→snπα...22densdep20EBh(+)isnπ
0 0
Fig.3. Importanceofvariouscontributionstothenucleonmass. Defaultvaluesare
as for the ω meson, and the interpolating fields have a default mixing angle of t=
−1. Thefactorizedandsummedup4-quarkcondensatesarefine-tunedwithafactor
κvac = 0.8 to the vacuum nucleon mass. Left panel: Nucleon in vacuum. Right
N
panel: Nucleon in medium at saturation density. The linear density dependence of
in-medium 4-quark condensates is regulated by the factor κmed, similar to the case
N
ofω. The variationsofgluoncondensate,whosedensity dependence is writtenhere
in terms of the color-electromagneticfields E and B, cause minor effects.
4. Electromagnetic Probes in Heavy-Ion Collisions at CERN-
SPS Energies
To lowest order in temperature, the vector and axialvector correlators become de-
generateat chiralrestorationtemperature and the vector-axialvectormixing modi-
fies the dilepton emissivity to a smooth thermal continuum above a certainduality
threshold [ 1]. Extending the duality to soft modes one arrives at the hypothe-
sis that the emissivity of strongly interacting matter can be parameterized by the
Born term qq¯annihilation rate, irrespectively whether the matter is in a confined
or deconfined state. (The reasoning is that the e+e− rate, in lowest orders in T
and n, is related to the inverse reaction e+e− → hadrons via detailed balance, i.e.
ImΠµ(s) = −sαR(s) with R(s) = σ(e+e− → hadrons)/σ(e+e− → µ+µ−). The
µ
cross features of σ(e+e− → hadrons) in turn can be reproduced by the Born term
for σ(e+e− → qq¯) above the ρ resonance region. The crucial point is therefore the
assumption of a medium-caused reshuffling of electromagnetic strength from the ρ
8 Thomas, Gallmeister, Zschocke, K¨ampfer
regiontosmallerenergies.) Thisansatzworksastonishinglywell[15]. Here,wegive
an update of our parametrizationsince new CERES data [ 16] are at our disposal.
InFig.4,boththeinvariantmassdi-electronspectraandthetransversemomentum
spectra are exhibited. In contrastto the parametrizationof previously known data
by a single space-time averaged effective radiation temperature hTi = 170 MeV,
the new data within larger phase space coverage enforce the inclusion of a second
thermal radiation component hTi = 120 MeV which may arise from the enlarged
sensitivity to the late stages prior to freeze-out.
The physics motivation behind such a parametrization is the following obser-
vation [ 15]: Integrating the radiation off an expanding fireball with volume V(t)
andtemperatureT(t)forcertainevolutionaryscenariosresultsinspectrawhichcan
be described by a temperature hTi and an effective space-time volume. Enlarging
the phase space coverage by lowering the minimum single-electron transverse mo-
mentum threshold reveals that the final spectrum appears as superposition of two
effective temperatures. The new high-statistics NA60 data will shed further light
on this issue.
10−2 10−3
p > 200 MeV p > 200 MeV
−1]10−3 pCTTo >c k1ta0i0l MeV −2V]10−4 pCTTo >c k1ta0i0l MeV
V e
e Cocktail+Therm G Cocktail+Therm
> [GNch10−4 dN> [ch10−5
d10−5 <10−6
M /< Q /T
N/d10−6 Q dT10−7
d N/ M=0.2..0.7 GeV
10−7 d10−8
0 0.5 1 1.5 0 0.5 1 1.5 2
M [GeV] Q [GeV]
T
Fig.4. ComparisonofourparametrizationwithnewCERESdata[16]. Leftpanel:
invariant mass (M) spectra, right panel: transverse momentum (Q ) spectra for
T
the massbin 0.2···0.7GeV. The minimum single-electrontransversemomenta are
200 and 100 MeV/c. Cocktails are without ρ contribution.
As consistency check one may try to describe the WA98 photon data [ 17] by
exactlythesamesourceparametrization. Fig.5showsthatthisnewparametrization
delivers indeed a better description of the data than the previous one [ 15].
One might envisage an intimate contact of photon and di-electron rates, as
suggestedby(1,2). Thethermaldi-electronratedependsonImΠµ( M2+Q~2,Q~)
µ
q
and is sensitive to the CERES data for M >150 MeV due to strong contributions
of π0 → γe+e− and η → γe+e− and detector acceptance as well. The photon
rate,incontrast,is relatedto ImΠµ( ~k2,~k), i.e. tests a differentkinematic region.
µ
p
Electromagnetic Probes 9
Therefore, one can not establish a direct link of di-electron and photon rates, in
general. In fact, our di-electron rate employs ImΠµ = −5αM2, while for the
µ 3
photon rate we use ImΠµ = −2παα T2 1log(3/4πα )+C (k /T) with C
µ s 2 s tot 0 tot
from equations 1.8 - 1.10 in [ 18]. (cid:2) (cid:3)
103
102
2] 101 hard
−V thermal
e 100 sum
G
) [ 10−1 WA98
E 10−2
p/
3d 10−3
(
N/ 10−4
d
N 10−5
1/ 10−6
10−7
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
p [GeV]
T
Fig. 5. Adopting the same source parametrization as in Fig. 4 for describing the
WA98 photon data [ 17]. The thermal component employs the complete leading
order rate with collinear enhancement factor [ 18]. For the calculation of the hard
Drell-Yan like component see [ 15].
5. Summary
In summary we argue that the CB-TAPS experiment on ω photo-production at
nucleigivefirsthintsonastrongdensitydependenceofcertain4-quarkcondensates.
Such 4-quark condensates contain chirality flipping terms and may be considered
as further order parameters of chiral symmetry restoration. In contrast to QCD
sum rules for the nucleon, the genuine chiral condensate is unimportant for light
vector mesons. Further, basing on vector-axialvector mixing near the chiral limit
we present an upgraded parametrization of interesting electromagnetic probes in
heavy-ion collisions at CERN-SPS energies by a realizationand extension of chiral
symmetry restoration as degeneracy of vector and axialvector correlators.
It should be emphasized that real and virtual photon rates are directly related
totheimaginarypartofthecurrent-currentcorrelator,thuslinkingdatatoconden-
sates via QCD sum rules. In such a way the change of the QCD vacuum structure
is accessible at finite temperature and density.
10 Thomas, Gallmeister, Zschocke, K¨ampfer
Acknowledgment(s)
The work is supported by BMBF, GSI, DAAD, and EU.
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