The $\rho$ Spectral Function in a Relativistic Resonance Model

Abstract

We calculate the spectral function $A_\rho$ of the $\rho$ meson in nuclear matter. The calculation is performed in the {\it low density} approximation, where the in-medium self energy $\Sigma_{med}$ is completely determined by the vacuum $\rho N$ forward scattering amplitude. This amplitude is derived from a relativistic resonance model. In comparison to previous non-relativistic calculations we find a much weaker momentum dependence of $\Sigma_{med}$, especially in the transverse channel. Special attention is paid to uncertainties in the model. Thus, we compare the impact of different coupling schemes for the $RN\rho$ interaction on the results and discuss various resonance parameter sets. Comment: 47 pages, 19 figures, some discussion and formulae added, minor typos removed, accepted for publication in Nuclear Physics A

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arXiv:nucl-th/0008027v2 9 Jan 2001
UGI-00-14
The ρ Spectral Function in a Relativistic
Resonance Model∗
M. Post, S. Leupold and U. Mosel
Institut f¨ ur Theoretische Physik, Universit¨ at Giessen,
D-35392 Giessen, Germany
Abstract
We calculate the spectral function Aρof the ρ meson in nuclear
matter. The calculation is performed in the low density approxima-
tion, where the in-medium self energy Σmedis completely determined
by the vacuum ρN forward scattering amplitude. This amplitude is
derived from a relativistic resonance model. In comparison to previous
non-relativistic calculations we find a much weaker momentum depen-
dence of Σmed, especially in the transverse channel. Special attention
is paid to uncertainties in the model. Thus, we compare the impact
of different coupling schemes for the RNρ interaction on the results
and discuss various resonance parameter sets.
PACS: 21.65.+f, 12.38.Lg, 14.40.Cs, 13.60.Lg
Keywords: Rho Spectral Function, Nuclear Matter, Nucleon Reso-
nance, Breit-Wigner Model
∗Work supported by BMBF and DFG.
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1Introduction
The question of how vector meson properties change at finite baryon density
or temperature has triggered a lot of theoretical and experimental work in
the last decade. Of particular interest are the in-medium properties of ρ
and a1 meson, which are chiral partners. One of the most evident signa-
tures of the spontaneous breaking of chiral symmetry in vacuum is the large
mass difference between these mesons. The claim is that the in-medium
mass spectra of ρ and a1meson can be related to the partial restauration
of chiral symmetry. If a chirally symmetric phase is reached at high den-
sity/temperature, a degeneracy of their spectral functions is expected [1, 2].
Whereas only a few studies exist on the in-medium properties of the a1me-
son [3, 4], numerous theoretical works have been performed for the ρ meson
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
It has been argued by several groups that the mass of the ρ meson is
related to the expectation value of the scalar quark condensate. The latter is
supposed to decrease at finite nuclear density due to the restauration of chiral
symmetry, thereby inducing a mass shift on the ρ meson. The most famous
of these models is that of Brown and Rho [5]. It predicts a downward shift
for the ρ meson mass of about 15%−20% at normal nuclear matter density.
Similar results have been found in [6] using the QCD sum rule approach
under the assumption that the width of the ρ meson remains small in the
nuclear medium (cf. [7] for a thorough discussion of that point).
In other approaches the properties of the ρ meson in nuclear matter are
calculated by taking into account conventional nuclear many body effects. At
low nuclear densities ρN the self energy Σmedcan be calculated in the ρNT
approximation (low density approximation) and is completely determined by
the knowledge of the vacuum ρN forward scattering amplitude T. Since this
quantity is not directly accessible by experiment, one needs to construct a
model which incorporates experimental constraints as far as possible. This
was first pointed out in [8]. Information on ρN scattering comes mostly
from πN → ππN processes. One of the most successful descriptions of these
reactions has been achieved by Manley et al within an isobar model [18, 19].
A resonance model for the ρN scattering amplitude allows the incor-
poration of experimental constraints in a straightforward fashion. Such a
strategy was first suggested in [9], where only p−wave resonances were taken
into account. However, as shown in our previous paper [10], the D13(1520)
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resonance plays a dominant role in ρN scattering and can therefore not be
neglected in a complete analysis. In both models the resonance parameters
were taken from the PDG [20] and the calculation was performed within a
non-relativistic framework. An alternative approach has been chosen in [11],
where the baryonic resonances are not introduced as excited three-quark
states, but are generated dynamically in a coupled channel approach from
meson-nucleon and meson-∆ rescattering. In [12] the ρN scattering ampli-
tude is derived from an effective chiral Lagrangian theory, neglecting, how-
ever, resonant contributions.
Going beyond the low density approximation, various groups have con-
sidered the change in the 2 π decay of the ρ as induced by the modified π
spectrum in the nuclear medium, see e.g. [13, 14, 15, 16, 17]. In [16, 17]
resonance contributions have also been included.
Qualitatively, all these models predict a strong broadening of the ρ me-
son. The mass, however, remains nearly unchanged. This finding is in good
agreement with QCD sum rules, see e.g. [7, 12].
So far experimental signals indicating a change of the ρ spectrum in the
nuclear medium, have been obtained by the CERES [21, 22, 23] and the
HELIOS collaboration [24]. In these experiments the dilepton spectra display
a downward-shift of strength in the region of the ρ meson as compared to
theoretical predictions. Agreement between theory and experiment can be
reached by introducing medium modifications for the ρ meson [25]. On the
other hand, in [26] the claim is made, that the CERES data do not necessarily
hint to a medium modification of the ρ meson, but can also be explained with
its vacuum properties.
Certainly, in order to gain a more quantitative understanding of the ρ
meson properties in nuclear matter, further theoretical studies, especially on
the momentum dependence of the spectral function, and improved experi-
mental statistics, are mandatory. The sensitivity on details of the spectral
function is even enhanced in photo-nuclear reactions, as was demonstrated
in [27].
Up to now all existing calculations [9, 10, 16, 17] have treated the res-
onance contribution in a non-relativistic manner. As we will show, this in-
troduces substantial uncertainties concerning the momentum dependence of
the ρ spectral function. We therefore do not only lift the non-relativistic
approximation, but also study the dependence of the results on various cou-
pling schemes. Furthermore, we analyse the influence of different resonance
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parameter sets on the results. We thus examine up to which extent safe con-
clusions about the propagation of ρ mesons in nuclear matter can be drawn
from a resonance model.
The paper is organized as follows. In Sect. 2 we will present an overview
of the formalism employed to describe the spectral function. Furthermore,
an arbitrariness in the non-relativistic approach is exposed and thus the need
for a relativistic model is motivated. The basic ingredients of the relativistic
calculation will be introduced in Sect. 3. A problem related with the de-
scription of the resonance propagator and its solution is presented in Sect.
3.1. In Sect. 4 we analyze the quality and properties of different resonance
parameter sets and their influence on the results. In Sect. 5 the results
of the relativistic calculation are presented and then compared to the non-
relativistic calculation. The impact of various coupling schemes for the vertex
RNρ is studied in Sect. 6. We summarize our results in Sect. 7.
2 Overview of the Model
In this section we explain the formalism on which our calculation of the in-
medium spectral function of the ρ meson is based and introduce some basic
notation. We then review the non-relativistic scheme which has been em-
ployed in all works up to now [9, 10, 16, 17] and use some simple kinematical
considerations to point out why a fully relativistic framework is needed.
2.1Definition of the Basic Quantities
The spectral function is defined as the imaginary part of the propagator. Due
to Lorentz invariance the vacuum spectral function of the ρ meson depends
only on one kinematic variable – its invariant mass – and both transverse
and longitudinal modes have the same spectral distribution. The presence
of nuclear matter breaks Lorentz invariance. Consequently transverse and
longitudinal propagation modes of the ρ meson receive different in-medium
modification. Also, the corresponding spectral functions depend both on the
energy ω of the ρ meson and its three-momentum q:
AT/L
ρ
(ω,q) =1
π
Im ΣT/L(ω,q)
ρ+ Re ΣT/L(ω,q))2+ Im ΣT/L(ω,q)2
(ω2− q2− m2
.
(1)
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mρis the pole mass of the ρ meson. ΣT/Lis the transverse or longitudinal
part of the self energy of the ρ meson and can be decomposed into a vacuum
and an in-medium part:
ΣT/L(ω,q) = Σvac(q2) + ΣT/L
med(ω,q). (2)
The imaginary part of the former is to lowest order in the coupling con-
stant given by the 2-pion decay width of the ρ meson [12, 14]:
Im Σvac(q2) =√q2Γππ(q2),(3)
and
Γππ(q2) =m2
ρ
q2Γ0
?
p
pmρ
?3
.(4)
Here q2is the squared invariant mass of the ρ meson, Γ0its decay width on
the pole mass and pmρ(p) denotes the 3-momentum of the pions measured
in the rest frame of a decaying ρ meson with mass mρ(√q2). For invariant
masses of the ρ meson up to around 1 GeV - which we consider in this paper
- the real part of Σvac(q2) shows only small variations and can be neglected
[12].
At finite nucleon density ρN the in-medium self energy ΣT/L
the following expression:
medis given by
ΣT/L
med(ω,q) =
?
Ω
d3pN
(2π)32EN
TT/L
tot (q,pN) (5)
In our notation q = (ω,q) and pN= (EN,pN) are the 4-vectors of ρ meson
and nucleon, respectively. The integration is restricted to the volume of
the Fermi volume, denoted here by Ω. By performing the integration in
the rest frame of nuclear matter Ω is simply the Fermi sphere, and the pN
integration reaches from 0 to the Fermi momentum pF. TT/L
forward scattering amplitude, as discussed below. Note that Σmed(ω,q) is a
Lorentz invariant quantity, since both the forward scattering amplitude TT/L
and the integrations measure
At sufficiently low nuclear densities it is advantageous to expand the self
energy in terms of the Fermi momentum. This is most easily done in the rest
tot
is the ρN
tot
d3pN
(2π)32ENare Lorentz scalars.
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