Hadrons in Medium -- Theory confronts experiment
Untitled
Hadrons in Medium – Theory confronts experiment
Fabian Eichstaedt, Stefan Leupold, Ulrich Mosel∗) and Pascal Muehlich
Institut fuer Theoretische Physik, Universitaet Giessen, Giessen, Germany
In this talk we briefly summarize our theoretical understanding of in-medium selfenergies
of hadrons. With the special case of the ω meson we demonstrate that earlier calculations
that predicted a significant lowering of the mass in medium are based on an incorrect treat-
ment of the model Lagrangian; more consistent calculations lead to a significant broadening,
but hardly any mass shift. We stress that the experimental reconstruction of hadron spectral
functions from measured decay products always requires knowledge of the decay branching
ratios which may also be strongly mass-dependent. It also requires a quantitatively reliable
treatment of final state interactions which has to be part of any reliable theory.
§1. Introduction
The study of in-medium properties of hadrons has attracted quite some inter-
est among experimentalists and theorists alike because of a possible connection with
chiral symmetry restoration in hot and/or dense matter. Experiments using ultrarel-
ativistic heavy ions reach not only very high densities, but connected with that also
very high temperatures. In their dynamical evolution they run through various –
physically quite different – states, from an initial high-nonequilibrium stage through
a very hot stage of – possibly - a new state of matter (QGP) to an equilibrated ’clas-
sical’ hadronic stage at moderate densities and temperatures. Any observed signal
necessarily represents a time-integral over all these physically quite distinct states of
matter. On the contrary, in experiments with microscopic probes on cold nuclei one
tests interactions with nuclear matter in a well-known state, close to cold equilib-
rium. Even though the density probed is always smaller than the nuclear saturation
density, the expected signals are as large as those from ultrarelativistic heavy-ion
collisions.1), 2)
In this talk we discuss as an example the theoretical situation concerning the
ω meson in medium and use it to point out various essential points both in the
theoretical framework as well as in the interpretation of data (for further refs see the
reviews in3)–5)).
§2. In-medium Properties: Theory
The interest in in-medium properties arose suddenly in the early 90’s when sev-
eral authors6), 7) predicted a close connection between in-medium masses and chiral
symmetry restoration in hot and/or dense matter. This seemed to establish a direct
link between nuclear properties on one hand and QCD symmetries on the other.
Later on it was realized that the connection between the chiral condensates of QCD
and hadronic spectral functions is not as direct as originally envisaged. The only
∗) Speaker, e-mail address: mosel@physik.uni-giessen.de
typeset using PTPTEX.cls 〈Ver.0.9〉
http://arxiv.org/abs/0704.0154v1
2 U. Mosel
strict connection is given by QCD sum rules which restrict only an integral over the
hadronic spectral function by the values of the quark and gluon condensates which
themselves are known only for the lowest twist configurations. Indeed a simple, but
more realistic analysis of QCD sum rules showed that these do not make precise
predictions for hadron masses or widths, but can only serve to constrain hadronic
spectral functions.8)–11) Thus hadronic models are needed for a more specific pre-
diction of hadronic properties in medium.
For example, in the past a lively discussion has been going on about a pos-
sible mass shift of the ω-meson in a nuclear medium. While there seems to be a
general agreement that the ω acquires a certain width of the order of 40-60 MeV
in the medium, the mass shift is not so commonly agreed on. While some groups
have predicted a dropping mass,12)–14) there have also been suggestions for a rising
mass15)–18) or even a structure with several peaks.19), 20) In this context a recent
experiment by the CBELSA/TAPS collaboration is of particular interest, since it
is the first indication of a downward shift of the mass of the ω-meson in a nuclear
medium.21) Since Klingl et al.13) were among the first to predict such a downward
shift it is worthwhile to look into their approach again.
The central quantity that contains all the information about the properties of
an ω meson in medium is the spectral function
Amed(q) = −
q2 − (m0ω)
2 −Πvac(q)−Πmed(q)
, (2.1)
with the bare mass m0ω of the ω. The vacuum part of the ω selfenergy Πvac is
dominated by the decay ω → π+π0π−.22) For the calculation of the in-medium part
one can employ the low-density-theorem13), 19), 20) which states that at sufficiently
small density of the nuclear medium one can expand the selfenergy in orders of the
density ρ
Πmed(ν, ~q = 0; ρ) = −ρT (ν) , (2.2)
where T (ν) is the ω-nucleon forward-scattering amplitude. We note that a priori it
is not clear up to which densities this low-density-theorem is reliable.23)
To obtain the imaginary part of the forward scattering amplitude via Cutkosky’s
Cutting Rules Klingl et. al12), 13) used an effective Lagrangian that combined chiral
SU(3) dynamics with VMD. The ω selfenergy was evaluated on tree-level which
needs as input the inelastic reactions ωN → πN (1π channel) and ωN → 2πN (2π
channel) to determine the effective coupling constants. The amplitude ωN → πN
is more or less fixed by the measurable and measured back reaction.25) This is in
contrast to the reaction ωN ↔ ρN which – in the calculations of ref.12), 13) – is not
constrained by any data and which dominates the 2π channel. Furthermore, Klingl
et. al12), 13) employed a heavy baryon approximation (HBA) to drop some of the
tree-level diagrams generated by their Lagrangian. All the calculations were made
for isospin-symmetric nuclear matter at temperature T = 0. The scattered ω was
taken to have ~q = 0 relative to the nuclear medium.
We have repeated these calculations without, however, invoking the HBA.∗) For
∗) For further details of the present calculations we refer to ref.26)
Hadrons in Medium 3
the 2π channel which decides about the in-medium mass shift of the ω in the calcu-
lations of ref.12), 13) we find considerable differences – up to one order of magnitude
in the imaginary part of the selfenergy – when comparing calculations using the full
model with those using the HBA.26) We thus have to conclude already at this point
that the HBA is unjustified for the processes considered here and leads to grossly
incorrect results.
We show our resulting in-medium spectral function of the ω (where HBA was
not employed) in figure 1. Note that in the medium the peak is shifted to 544 MeV
which is due to the large effects of a relativistic, full treatment of the imaginary and
real parts of the amplitudes obtained in the present model. This has to be compared
with the results obtained by Klingl et al.12) Since Klingl et al. find an in-medium
peak at about 620 MeV it is obvious that in the relativistic calculation the physical
picture changes drastically. It is also obvious that the correct treatment of the same
 0.001
 0.01
 0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1
ω [GeV]
vacuum
Fig. 1. Spectral function of the ω meson in the vacuum and at normal nuclear density.
Lagrangian as used in ref.12) on tree-level leads to an unrealistic lowering of the ω
spectral function.
It is, therefore, worthwhile to look into another method to calculate the ω self-
energy that takes experimental constraints as much as possible into account and –
in contrast to the tree-level calculations of ref.12) – respects unitarity. A first study
in this direction has been performed by Lutz et al.19) who solved the Bethe-Salpeter
equation with local interaction kernels. These authors found a rather complex spec-
tral function with a second peak at lower energies due to a coupling to nucleon
resonances with masses of about ≈ 1500 MeV. We have recently used a large-scale
K-matrix analysis of all available γN and πN data27)–29) that does respect unitarity
and thus constrains the essential 2π channel by the inelasticities in the 1π chan-
nel.20) By consistently using the low-density-approximation we have obtained the
result shown in Fig. 2. Fig. 2 clearly exhibits a broadened ω spectral function with
only a small (upwards) shift of the peak mass. In agreement with the calculations of
Lutz et al.,19) although with less strength, it also exhibits a second peak at masses
around 550 MeV that is due to a coupling to a N*(1535)-nucleon hole configuration.
Such a resonance-hole coupling is known to play also a major role in the determi-
4 U. Mosel
0.5 0.6 0.7 0.8 0.9 1.0
 [GeV]
Fig. 2. ω spectral function for an ω meson at rest, i.e. q0 =
q2 (from ref.20)). The appropriately
normalized data points correspond to the reaction e+e− → ω → 3π in vacuum. Shown are
results for densities ρ = 0, ρ = ρ0 = 0.16 fm
−3 (solid) and ρ = 2ρ0 (dashed).
nation of the ρ meson spectral function;23), 24) in the context of QCD sum rules it
has been examined in ref.17) It is obviously quite sensitive to the detailed coupling
strength of this resonance to the ωN channel which energetically opens up only at
much higher masses.
As mentioned earlier, there is general consensus among different theories, that
the on-shell width of the ω meson in medium reaches values of about 50 MeV at
saturation density. To illustrate this point we show in Fig. 3 the width as a function
of omega momentum relative to the nuclear matter restframe both for the transverse
and the longitudinal polarization degree of freedom. It is clearly seen that the
0.0 0.1 0.2 0.3 0.4 0.5 0.6
|q| [GeV]
Fig. 3. On-shell width of the ω in nuclear matter at nuclear matter density ρ0 (from ref.
20)). The
open (solid) points give the width for the transverse (longitudinal) degree of freedom.
transverse width increases strongly as a function of momentum. At values of about
Hadrons in Medium 5
500 MeV, i.e. the region, where CBELSA/TAPS measures, the transverse width
has already increased to about 125 MeV and even the polarization averaged width
amounts to 100 MeV.
§3. Spectral Functions and Observables
Apart from invariant mass measurements, there is another possibility to exper-
imentally constrain the in-medium broadening of the ω-meson. The total width
plotted in Fig. 3 is the sum of elastic and inelastic widths. In general, the inelastic
width alone is determined by the imaginary part of the selfenergy and the latter
determines the amount of reabsorption of ω mesons in the medium. In a Glauber
approximation the cross section for ω production on a nucleus reads
dσγ+A→ω+X
d3x ρ(~x)
dσγ+N→ω+X
ℑΠ(p, ρ(~x ′))
(3.1)
The ratio of this cross section on the nucleus to that on the nucleon then deter-
mines the nuclear transmission T which depends on the imaginary part of the omega
selfenergy ℑΠ
T (A) ≈
d3x ρ(~x) exp
ℑΠ(p, ρ(~x ′))
. (3.2)
Using in addition the low-density-approximation
ℑΠ(p, ~x) = −pρ(~x)σinelωN (3
one obtains the usual Glauber result
T (A) =
d3x ρ(~x) exp
dz′ ρ(~x ′)σinelωN
. (3.4)
We show the calculated transmission T in Fig. 4 together with the data obtained
by CBELSA/TAPS. The measured cross section dependence on massnumber A is
reproduced very well30) if the inelastic ωN cross section is increased by 25% over the
usually used parametrization. This may indicate a problem with the usually used
cross section, or - more interesting - it may indicate a breakdown of the low-density-
approximation.
It is, furthermore, important to realize that the spectral functions themselves are
not observable. What can be observed are the decay products of the meson under
study. It is thus obvious that even in vacuum the invariant mass distribution of the
decaying resonance (V → X1X2), reconstructed from the four-momenta of the decay
products (X1,X2), involves a product of spectral function and partial decay width
into the channel being studied
dRV→X1X2
∼ A(q2)×
ΓV→X1X2(q
Γtot(q2)
. (3.5)
Since in general the branching ratio also depends on the invariant mass of the decay-
ing resonance this dependence may distort the observed invariant mass distribution
6 U. Mosel
0 50 100 150 200
Fig. 4. Transparency of nuclei for ω production. Calculations and preliminary data are normalized
to 12C. Dashed lines reflect error estimates obtained from the spread of the data. Data are
from CBELSA/TAPS.31)
compared with the spectral function itself. This effect is obviously the more impor-
tant the broader the decaying resonance is and the stronger the widths depend on
While these branching ratios are usually well known in vacuum there is con-
siderable uncertainty about their value in the nuclear medium. This uncertainty is
connected with the lack of knowledge about the in-medium vertex corrections, i.e.
the change of coupling constants with density. Even if we assume that these quan-
tities stay the same, then at least the total width appearing in the denominator of
the branching ratio has to be changed, consistent with the change of the width in
the spectral function. This point has only rarely been discussed so far, but it has
far-reaching consequences.
For example, for the ρ meson the partial decay width into the dilepton channel
goes like
Γρ→e+e− ∼
, (3.6)
where the first factor on the rhs originates in the photon propagator and the last
factor M comes from phase-space. On the other hand, the total decay width of the
ρ meson in vacuum is given by (neglecting the pion masses for simplicity)
Γtot ≈ Γρ→ππ ∼ M , (3.7)
so that the branching ratio in vacuum goes like
Γρ→e+e−
. (3.8)
This strong M -dependence distorts the spectral function, in particular, for a broad
resonance such as the ρ meson. This effect is contained and clearly seen in theoretical
simulations of the total dilepton yield from nuclear reactions (see, e. g., Figs. 8− 10
Hadrons in Medium 7
in32)); it leads to a considerable shift of strength in the dilepton spectrum towards
lower masses.
For the semileptonic decay channel π0γ that has been exploited in the CBELSA
TAPS experiment again a strong mass-dependence of the branching ratio shows up
because just at the resonance the decay channel ω → ρπ opens up.
In both of these cases the in-medium broadening changes the total widths in the
denominator of the branching ratios even if the partial decay width stays the same as
in vacuum. Such an in-medium broadening of the total width, which should be the
same as in the spectral function, will tend to weaken the M -dependence of the total
width and thus the branching ratio as a whole. In medium another complication
arises: the spectral function no longer depends on the invariant mass alone, but
– due to a breaking of Lorentz-invariance because of the presence of the nuclear
medium – in addition also on the three-momentum of the hadron being probed.
Again, this p-dependence of the vector meson selfenergy has only rarely been taken
into account (see, however, refs.20), 24), 33)). In addition, final state interactions do
affect hadronic decay channels. A quantitatively reliable treatment of these FSI thus
has to be integral part of any trustable theory that aims at describing these data.
§4. Conclusions
QCD sum rules establish a very useful link between the chiral condensates, both
in vacuum and in medium, but their connection to hadronic spectral functions is
indirect. The latter can thus only be constrained by the QCDSR, but not be fixed;
for a detailed determination hadronic models are needed. We have pointed out in
this talk that the low-density-approximation nearly always used in these studies does
not answer the question up to which densities it is applicable. First studies23) have
shown that this may be different from particle to particle.
While the in-medium properties of all vector mesons ρ, ω, and φ are the subject
of intensive experimental and theoretical research, in this talk we have concentrated
on the ω meson for which recent experiments indicate a lowering of the mass by
about 60 MeV in photon-produced experiments on nuclei. A tree-level calculation,
based on an effective Lagrangian, that predicted such a lowering, has been shown
to be incorrect because of the heavy-baryon approximation used in that calculation.
A correct tree-level calculation with the same Lagrangian gives strong contributions
from the ω → 2πN channel, which, however, is unconstrained by any data; in effect,
the spectral function is softened by an unreasonable pole mass shift. This problem
might partially be based on the fact that all the inelastic processes ωN → πN and
ωN → 2πN are only treated at tree-level. Here an improved calculation is needed,
which incorporates coupled-channels and rescattering, e.g. a Bethe-Salpeter19) or a
K-matrix approach.27), 34), 35)
We have indeed shown that a better calculation that again starts out from an
effective Lagrangian and takes unitarity, channel-coupling and rescattering into ac-
count yields a significantly different in-medium spectral function in which the pole
mass hardly changes, but a broadening of about 60 MeV at nuclear saturation density
takes place, which increases with momentum, primarily in the transverse channel.
8 U. Mosel
Finally, we have pointed out that any measurement of the spectral function
necessarily involves also a branching ratio into the channel being studied. The ex-
perimental in-medium signal thus contains changes of both the spectral function and
the branching ratio.
Acknowledgements
The authors acknowledge discussions with Norbert Kaiser and Wolfram Weise.
They have also benefitted a lot from discussions with Vitaly Shklyar. This work has
been supported by DFG through the SFB/TR16 ”Subnuclear Structure of Matter”.
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