Hadronic spectral functions in nuclear matter

Abstract

We study the in-medium properties of mesons (π,η,ρ) and baryon resonances in cold nuclear matter within a coupled-channel analysis. The meson self energies are generated by particle–hole excitations. Thus multi-peak spectra are obtained for the mesonic spectral functions. In turn this leads to medium-modifications of the baryon resonances. Special care is taken to respect the analyticity of the spectral functions and to take into account effects from short-range correlations both for positive and negative parity states. Our model produces sensible results for pion and Δ dynamics in nuclear matter. We find a strong interplay of the ρ meson and the D13(1520), which moves spectral strength of the ρ spectrum to smaller invariant masses and leads to a broadening of the baryon resonance. The optical potential for the η meson resulting from our model is rather attractive whereas the in-medium properties modifications of the S11(1535) are found to be quite small.

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arXiv:nucl-th/0309085v2 27 May 2004
Hadronic Spectral Functions in Nuclear Matter1
M. Post, S. Leupold and U. Mosel
Institut f¨ ur Theoretische Physik, Universit¨ at Giessen,
D-35392 Giessen, Germany
Abstract
We study the in-medium properties of mesons (π,η,ρ) and baryon resonances in
cold nuclear matter within a coupled-channel analysis. The meson self energies are
generated by particle-hole excitations. Thus multi-peak spectra are obtained for the
mesonic spectral functions. In turn this leads to medium-modifications of the baryon
resonances. Special care is taken to respect the analyticity of the spectral functions
and to take into account effects from short-range correlations both for positive and
negative parity states. Our model produces sensible results for pion and ∆ dynamics
in nuclear matter. We find a strong interplay of the ρ meson and the D13(1520),
which moves spectral strength of the ρ spectrum to smaller invariant masses and
leads to a broadening of the baryon resonance. The optical potential for the η me-
son resulting from our model is rather attractive whereas the in-medium properties
modifications of the S11(1535) are found to be quite small.
PACS: 21.65.+f, 24.10.Cn, 14.40.Ag, 14.40.Cs, 14.20.Gk
Keywords: Nuclear Matter, Meson Spectral Function, Baryon Resonance Spectral
Function
1Work supported by DFG and GSI Darmstadt.
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1Introduction
A wealth of experimental indications for in-medium modifications of hadrons has been
accumulated over the last years.Besides serving as a suitable testing ground for the
understanding of hadronic interactions in the nuclear medium, the search for medium
modifications has been stimulated by the works of [1, 2], which – based on very general
arguments concerning the restoration of chiral symmetry – predict dropping hadron masses
at finite density. Dilepton spectra measured in heavy-ion collisions by the NA45 [3, 4, 5, 6]
and the HELIOS [7] collaboration indicate that the spectral function of the ρ meson may
undergo a significant reshaping in a hot and dense environment with spectral strength
moving down to smaller invariant masses. In nuclear reactions medium modifications of
the P33(1232), the D13(1520) and the S11(1535) have been studied. An analysis of pion-
and photo-induced reactions has established the need to introduce a spreading potential
for the P33(1232), yielding a moderate broadening of about 80 MeV for this state [8]. The
disappearance of the second resonance region in photoabsorption reactions on the nucleus
as observed in [9, 10, 11] has been interpreted in terms of a broadening of the D13(1520)
in nuclear matter [12]. Finally, data of η photoproduction on nuclei [13, 14] have opened
up the possibility to study the in-medium properties of the S11(1535). A prime source of
information of pions and η mesons in nuclear matter is the study of pionic [15] and η-mesic
atoms [16].
At the same time numerous theoretical models have been developed in order to arrive
at an understanding of the observed phenomena. Concerning the in-medium properties of
the ρ meson, the dilepton spectra reported in [3, 4, 5, 6, 7] have triggered the development
of a variety of hadronic models. For a comprehensive review of these models see [17].
Although these works differ quite substantially in details, as a general picture a shift of
spectral strength down to smaller invariant masses is found in most of them. In one type
of models [18, 19, 20, 21, 22] this shift is generated by the excitation of resonance-hole
pairs in the nuclear medium. The formation of these states leads to additional branches of
the spectral function. Coupling the ρ to the D13(1520)N−1state moves a lot of spectral
strength down to small invariant masses [19, 20]. Another class of models [23, 24, 25, 26, 27]
takes into account the effects of the renormalization of the pion cloud generated by the
strong interaction of pions and nucleons and finds a broadening of the ρ peak. Besides
offering an appealing interpretation of the dilepton spectra, a shift of spectral strength as
offered by most hadronic models is also required by QCD sum rules [2, 26, 28].
The in-medium properties of the P33(1232) resonance have been studied extensively
in the literature [29, 30, 31, 32, 33, 34, 35]. While operating on different levels of so-
phistication, in most of these models the in-medium self energy is due to a change of the
dispersion relation of pions in a nuclear environment. For a quantitative description of the
resonance properties, a consistent inclusion of short-range correlations (SRC) is necessary
[29, 30, 31]. For the D13(1520) and the S11(1535) much less work has been done. In an
attempt to explain nuclear photoabsorption data reported in [9, 10, 11], a large broadening
of about 300 MeV for the D13(1520) has been obtained in a resonance fit in [12]. The
later works of [36, 37, 38] have given further support to the conjecture that an in-medium
broadening of this state yields a possible explanation of the photoabsorption data. As
alternative mechanisms effects from Fermi motion and a change of interference patterns
in the nuclear medium have been pointed out in the analysis of [39], thus questioning the
direct connection between the data and a broadening of the D13(1520). As a result in
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that analysis a smaller broadening of about 100 MeV for this state is found. In the mi-
croscopic models of [19, 35] a substantial broadening of the D13(1520) has been generated
dynamically, based either on the coupling to the Nρ channel [19] or on the coupling to
the Nπ channel [35]. Concerning the properties of the S11(1535) in nuclear matter, the
existing models [40, 41, 42] suggest rather moderate in-medium effects. This finding is well
supported from data on photoproduction of η off nuclei [43].
From a theoretical point of view it would be desirable to describe as many in-medium
effects as possible within one model in order to arrive at a combined understanding of these
phenomena. For example, reshuffling the spectral strength of the ρ meson (as suggested
from dilepton spectra) might have an immediate impact on the width of the D13(1520)
[19] and can help to explain the nuclear photoabsorption data [36, 37, 38]. Similarly, a
quantitative analysis of the optical potential of the η meson is constrained from the fact
that recent data on η photoproduction [13, 14] suggest that the in-medium modifications
experienced by the S11(1535) are relatively small. To this end we have set up a model which
generates the in-medium modifications of mesons and baryon resonances within a self-
consistent coupled channel analysis. The mesons are dressed by the excitation of resonance-
hole loops and a remarkably complicated spectrum with various peak structures is found
for the mesonic spectral functions. In a second step the in-medium self energy of the baryon
resonances arising from the dressing of the mesons is calculated. The corresponding set
of coupled-channel equations is then solved iteratively. In the course of the iterations one
leaves the regime of the low-density theorem [44], which relates the in-medium self energy
to vacuum scattering amplitudes. It is least reliable for systems close to threshold, where
already small changes of the available phase space can lead to large modifications of the
resonance and therefore the meson as well. A well-known example is the Λ(1405) coupling
to the¯K N channel, see for example [45, 46]. Another case is the ρND13(1520) system: in
a previous publication [19], a first step in this direction was done and strong effects from
the interplay of ρ and D13(1520) were reported, modifying both the ρ spectral function
and that of the baryon resonance. We have extended the model presented in [19] in several
ways: in order to guarantee the normalization of the vacuum and the in-medium spectral
functions, we employ dispersion relations to generate the real part of the self energies.
Since most baryon resonances couple strongly to the pion, a complete analysis of their in-
medium properties requires also a dressing of the pion. In order to obtain reliable estimates
for the S11(1535), which couples dominantly to the ηN channel, the η meson is included as
well. Finally, stimulated by the fact that the in-medium width of the P33(1232) needs to be
protected by repulsive short-range terms, we have developed a framework that allows for
the incorporation of such effects for resonances with negative parity, such as the D13(1520)
and the S11(1535).
The paper is organized as follows: in Section 2 we discuss the vacuum self energies of
the ρ meson and the included baryon resonances. Special emphasis is put on the effect
of dispersion relations on the baryonic spectral functions. Section 3 discusses the current
theoretical and experimental status concerning the coupling of the ρ meson to baryon
resonances, in particular the D13(1520). In Section 4 we set up the general framework
for the discussion of the in-medium self energies of mesons and baryons. The theoretical
concepts for the inclusion of short-range correlations (SRC) are given in Section 5, with
details presented in Appendix D. The results obtained for the mesons π, η and ρ, as
well as for the resonances P33(1232), D13(1520) and S11(1535) are discussed in Section 6.
In Section 7 we summarize our findings. In four Appendices we discuss some necessary
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technical issues.
2Meson and Baryon Resonances in Vacuum
In this Section we discuss the vacuum spectral functions of the ρ meson and baryon reso-
nances, which are denoted by A(q) and ρ(k), respectively. The spectral function is defined
as the imaginary part of the retarded propagator, see Appendix A and [47]. In terms of
the self energies Π+
vac(q) and Σ+
vac(k) they are given by:
A(q) = −1
π
ImΠ+
vac(q)
vac(q))2+ ImΠ+2
ImΣ+
R− ReΣ+
(q2− m2
M− ReΠ+
vac(q)
(1)
ρ(k) = −1
π
vac(k)
vac(k))2+ ImΣ+2
(k2− m2
vac(k)
.
Throughout this paper, we will denote the four-momentum of meson M by q = (q0,q)
and that of resonance R by k = (k0,k). Note that our ansatz for ρ(k) does not take into
account the full Dirac structure of the self energy. A detailed discussion of this topic can
be found in [20].
Both A(q) and ρ(k) are normalized quantities. In order to guarantee this within our
model, we calculate the retarded self energies Π+
vacand Σ+
vacin the following way:
• calculate the imaginary part of the self energy ImΠ+
means of Cutkosky’s cutting rules
vacand ImΣvac for q0> 0 by
• for mesons use the antisymmetry ImΠ+
and apply a dispersion relation to obtain the real part of the self energy .
vac(−q0) = −ImΠ+
vac(q0) (see Appendix A.1)
• as outlined in Appendix A, self energy and propagator of baryons are symmetric in
the vacuum, but not in the nuclear medium. Therefore we neglect the contribution
from negative energies to the dispersion integral already in the vacuum.
A further discussion of this topic can be found in [48]. The issue of how to obtain normalized
spectral functions is of relevance to us since within our coupled channel analysis the spectral
function of any state is allowed to influence the spectral function of any other state. This
implies that even rather small violations of the normalization can lead to uncontrollable
errors in the calculation.
Let us make a purely technical note: throughout this work we will encounter various
traces, arising from the spin summation at the meson-nucleon-resonance vertices. From
these traces we only keep the leading non-relativistic contribution. In [20] it was shown
that a non-relativistic approach leads to a very good approximation of the fully relativis-
tic results, as long as the kinematical quantities are evaluated in the rest frame of the
resonance. A non-relativistic reduction simplifies the expressions for the in-medium self
energies, see Chapter 4. In particular, a consistent relativistic description of the short-
range interactions is a formidable task [49, 50] and - albeit in principle desirable - beyond
the scope of this work.
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?
?
?
?
?
?
Figure 1: Self energy Πvacresulting from the coupling of the ρ to pions. The diagram on
the left corresponds to the usual decay of a ρ to two pions. The diagram on the right gives
rise to an energy independent real mass shift.
2.1
ρ Meson
The propagator of the ρ meson in vacuum has both a four-transverse and a four-longitudinal
part, of which in the presence of current conserving couplings only the former gets dressed
by a self energy Π+
vac[17, 23, 51]:
Dµν,+
ρ
(q) =
1
q2− (m0
ρ)2− Π+
vac(q)Pµν
T+qµqν
q2
1
(m0
ρ)2
. (2)
Here Pµν
T
is the four-transverse projector:
PT
µν(q) = gµν−qµqν
q2
. (3)
The self energy Π+
the following Lagrangian [23]:
vac(q) arises from the coupling of the ρ to pions, which is described by
Lρπ
ρµν
= (Dµπ)⋆(Dµπ) − m2
= ∂µρν− ∂νρµ
ππ⋆π −1
Dµ = ∂µ+ igρρµ
4ρµνρµν+1
2(m0
.
ρ)2ρµρµ
,
From Lρπ one derives two Feynman diagrams for Π+
evaluation of these diagrams we use the results from [23], where the divergent integrals have
been treated by a Pauli-Villars regularization in order to preserve the gauge invariance of
the self energy:
vac(q), see Fig. 1. Concerning the
ReΠ+
vac(q) = −
g2
ρ
24π2q2
?
G(q,mπ) − G(q,Λ) + 4(Λ2− m2
?
π)/q2+ lnΛ
mπ
?
(4)
ImΠ+
vac(q) = −sgn(q0)
g2
48πq2
ρ
θ(q2− 4m2
π)
?
1 −4m2
?3/2?
π
q2
?3/2
−(5)
−θ(q2− 4Λ2)
?
1 −4Λ2
q2
.
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Here Λ is a regularization parameter. The function G is defined as


y =
q2− 1
G(q,m) =


4m2



y3/2arctan(1/√y) fory > 0
−1
2(−y)3/2ln
????
√−y + 1
√−y − 1
????
fory < 0
(6)
.
Note that the real part of the self energy is related to ImΠ+
relation:
vacby a subtracted dispersion
ReΠ+
vac(q) = q2P
?∞
4m2
π
dq′2
π
ImΠ+
q′2(q2− q′2)
vac(q′)
.(7)
The subtraction is made at the point q = 0 in order to satisfy the condition ReΠ+
0 as required from gauge invariance [23, 26]. For the imaginary part the Pauli-Villars pre-
scription acts like a form factor which improves the convergence of the dispersion integral.
There are three free parameters in this expression, m0
by fitting the phase shift of ππ scattering in the vector-isovector channel and the pion
electromagnetic form factor, with the result:
vac(q = 0)=
ρ, gρand Λ, which are determined
m0
ρ= 0.875GeV , gρ= 6.05 , Λ = 1GeV.(8)
As shown in [23], with these parameters a good fit of both observables is obtained. Sum-
marizing, this model for the ρ meson in vacuum is a good starting point for an in-medium
calculation since it provides a spectral function, which is both normalized and in good
agreement with observables.
2.2 Baryon Resonances
The self energy of a baryon resonance arises from the coupling to meson-nucleon channels,
see Fig. 2. After calculating the corresponding decay width, we obtain ReΣ+
dispersion analysis.
Following standard Feynman rules, the decay width of a nucleon resonance with in-
variant mass
frame given by:
vacby a
√k2into a pseudoscalar meson ϕ = π, η of mass mϕis in the resonance rest
ΓNϕ(k) =
1
2j + 1IΣ
?f
mϕ
?2
F2(k,mϕ)qcm
8πk2Ωϕ
.
The coupling constants f are obtained by fits to the corresponding hadronic partial decay
widths and j denotes the spin of the decaying resonance. For a complete list of the included
resonances see Table 1. By qcmwe denote the momentum of the decay products in the
rest frame of the resonance. The isospin factor IΣis derived form the isospin part of the
Lagrangian which is given in Eq. 82 in Appendix C. One finds that IΣ= 1 for ∆ resonances
with isospin
π meson is considered. For the decay into the isoscalar η this factor is 1 and there is no
coupling to ∆ resonances. The quantity Ωϕis the non-relativistic trace arising from spin
3
2and IΣ= 3 for N∗resonances with isospin1
2if the decay into an isovector
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?????
R
N
R
Figure 2: Self energy of a baryon resonance from the decay into a π, η or ρ meson.
summation at the meson-nucleon-resonance vertices. Explicit expressions for Ωϕ, listed
according to the quantum numbers of the resonances considered in this work, can be found
in Table 3, Appendix C.2.
When considering the decay into one stable and one unstable particle, an integration
over the spectral function of the unstable particle is necessary. For a resonance with fixed
mass
√k2and spin j decaying into the Nρ channel, for example, one finds:
ΓNρ(k) =
(
√k2−mN)
?
4m2
π
2
dq2ΓNρ(k,q)Aρ(q)(9)
ΓNρ(k,q) =
1
2j + 1IΣ
?f
mρ
?2
F2(k,q)qcm
8πk2(2ΩT+ ΩL).
where ΓNρ(k,q) stands for the width of a resonance for decay into a ρ meson with invariant
mass
?q2. The isospin factor IΣis the same as for the pions. Explicit expressions for the
In Table 1 we give a list of all resonances and their decay channels. For most states
the sum Γπ N+Γη N+ΓρNdoes not exhaust the total width. As an approximation we put
the remaining width into the ∆π channel and take the energy dependence to be s-wave for
negative parity states and p-wave for positive parity states. In contrast to the other decay
channels we do not modify Γ∆πwhen going to the nuclear medium. The corresponding
Lagrangians are given in Appendix C.2 and lead to traces Ω∆, which we do not explicitly
denote here. In analogy to the N ρ width, we find for Γ∆π:
spin-traces ΩT/Lare found in Table 3, Appendix C.2.
Γ∆π(k) =
(√k2−mπ)2
?
(mN+mπ)2
dm2Γ∆π(k,m)ρ∆(m)(10)
Γ∆π(k,m) =
1
2j + 1IΣ
?
f
m∆
?2
F2(k,mπ)qcm
8πk2Ω∆
.
The isospin factor IΣis 1 both for the decay of isospin-3
stands for the decay into a pion and a ∆ with invariant mass m. Since the ∆ resonance is
a broad particle, we need to integrate over its spectral function ρ∆(m).
We write the form factor F(k,q) at the resonance-meson-nucleon vertex in the following
form:
2and isospin-1
2states and Γ∆π(k,m)
F(k,q) ≡ Fs(k)Ft(q). (11)
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mΓtot
ΓNπ
ΓNρ
Γ∆π
ΓNη
ΓNω
JIlϕ lV
Λs
ρ
P11(1440)1.462 0.3910.27000.088001/2 1/2pp0.8
P33(1232)1.2320.120.1200003/2 3/2pp0.8
P13(1720)1.717 0.1210.0110.11 0003/2 1/2pp1.0
P13(1879) 1.879 0.498 0.130.217000.151 3/2 1/2pp1.1
F15(1680)1.684 0.1390.096 0.011 0.014005/2 1/2fp0.9
F35(1905)1.881 0.329 0.041 0.282 0.006005/2 3/2fp1.4
F15(2000)1.903 0.4940.039 0.369 0.086 005/2 1/2fp1.4
S11(1535)1.534 0.1510.077 0.00500.06601/2 1/2ss0.8
S31(1620)1.672 0.1540.014 0.044 0.095001/2 3/2ss0.9
S11(1650)1.659 0.173 0.154 0.005 0.008 0.00601/2 1/2ss0.9
S11(2090)1.928 0.415 0.043 0.203 0.167 0.00201/2 1/2ss1.5
D13(1520)1.524 0.1240.073 0.026 0.025003/2 1/2ds0.9
D33(1700)1.762 0.5980.081 0.046 0.471 003/2 3/2ds1.3
D33(1940) 2.057 0.460 0.081 0.162 0.217 003/2 3/2ds1.8
D13(2080)1.804 0.4470.104 0.114 0.229 003/2 1/2ds1.6
Table 1: List of all resonances which are taken into account in our calculation. Apart from
mass and width into the individual decay channels, we also give spin and isospin as well
as the lowest orbital angular momentum needed in pseudoscalar (ϕ) or vector (V ) meson
scattering on a nucleon to form the resonance. All quantities are given in GeV. In those
cases where the given partial widths do not add up to the full decay width, the remaining
width is assigned to the ∆π channel. In the last row we denote the cutoff of the form factor
Fs(k2) of Eq. 13 at the ρN R vertex.
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?
?
?
?
?
?
?
?
???
Figure 3: Form factor FF1 of Eq. 13 (solid line) and form factor FF2 of Eq. 14 (dashed
line) for the D13(1520) resonance. The cutoff parameter in FF2 is taken to be Λ = 1 GeV,
the parameters s0and Λ of form factor FF1 are listed in Table 1.
The form factor Ft(q) is a usual monopole form factor:
Ft(q) =
Λ2
Λ2
M− m2
M− q2
M
.(12)
The values taken for ΛMare listed in Table 2 in Appendix B. We multiply the resonance-
nucleon-meson vertex with a monopole type form factor since this vertex is also used in
baryon-baryon interactions, where the large space-like 4-momenta acquired by the exchange
particle need to be cut off. For the decay of a resonance into a stable final state we have
q2= m2
to evaluate Ftalso for time like 4-momenta q2≈ Λ2
put the form factor equal to unity for q2≥ m2
the space-like region. For Fs(k) we take different parameterizations at the RNρ and the
RNϕ vertices. When considering an RNρ vertex we choose [52]
Mand therefore Ft(q) = 1. When going to the nuclear medium, we will be forced
M. In order to avoid poles of Ft(q), we
M. This does not affect the action of Ftin
Fs(k) =
Λ4+1
4(s0− m2
2(s0+ m2
R)2
Λ4+ (k2−1
R))2
,(13)
while at the RNϕ vertex we take [52, 53]:
Fs(k) =
Λ4
Λ4+ (k2− m2
R)2
.(14)
In the following we will refer to the form factor of Eq. 13 as FF1 and to the form factor
of Eq. 14 as FF2.
We have plotted both FF1 (solid line) and FF2 (dashed line) in Fig. 3. As a function
of k2the form factor of Eq. 13 is asymmetric with respect to the resonance mass mR.
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It is equal to unity both at k2= m2
than unity within the interval {s0,m2
the cutoff Λ and the threshold parameter s0. We give values for the cutoff Λ in Table 1
and take s0= (mN+ 2mπ)2. In the following Section we will give arguments in support
of the somewhat unconventional form factor Eq. 13. Asymmetric form factors like Blatt-
Weisskopf type form factors are quite commonly used in the literature, e. g. in [54].
For positive energies, the imaginary part of the self energy ImΣ+
the decay width via
Rand at the decay threshold k2= s0< m2
R} and smaller outside. The exact shape depends on
R, larger
vacis obtained from
ImΣ+
vac(k2) = −
√
k2Γ(k2).(15)
The real part of the self energy is calculated using a dispersion integral:
ReΣ+
vac(k) = P
∞
?
ωmin
dω
π
ImΣ+(ω,k)
ω − k0
− cvac(k)(16)
with
cvac(k) = P
∞
?
ωmin
dω
π
ImΣ+(ω,k)
ω −?m2
R+ k2
.
Here P denotes the principal value. The energy ωmin=
the threshold for the decay into Nπ. By mRthe mass of the resonance is denoted. As
can be inferred from Fig. 4, the suppression from the form factor F(k,q) is sufficient to
produce a decreasing width, such that the dispersion integral converges. The subtraction is
convenient to ensure that the physical mass of the resonance is recovered. In principle the
dispersion integral extends over negative energies as well. We omit this contribution since
in the nuclear medium no symmetry exists which relates ImΣ+(k0) to ImΣ+(−k0). This
issue is addressed in Appendix A. We have checked that in the vacuum the contributions
from negative energies to the dispersion integral can safely be neglected. Also, in cold
nuclear matter we do not expect that antibaryons are important.
?(mN+ mπ)2+ k2follows from
2.3Results for ReΣ and ImΣ
In this Section we present results for Σvacand compare two resonances – the P33(1232) and
the D13(1520) resonance, both of which have according to the PDG [55] an on-shell width
of about 120 MeV.
In Fig. 4 we show the decay width of both resonances as a function of their invariant
mass. Note the different scales on the y axes. The Nρ width (solid line) of the D13state
displays a strong energy dependence, which is of kinematical origin. The resonance is below
the nominal threshold for the decay at mN+ mρ= 0.938 + 0.77 ≈ 1.7 GeV. Therefore the
decay into this channel can only proceed via the low mass tail of the ρ meson, which in
turn generates a steep increase of the width as the available phase space opens up.
Let us now turn to the results for the self energy and the spectral function, which
are depicted in Figs. 5 and 6. From Fig. 5 we find that around the resonance peak
the spectral functions of the D13(1520) and the P33(1232) (solid lines) obtained by a full
calculation including the real part of the self energy do not differ much from those obtained
by neglecting ReΣ (dashed lines). In both cases we observe a slight squeezing of the
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??
?
?
??
?
?
?
?
??
?
?
?
?
Figure 4: Left: Partial decay widths of the D13(1520). The Nρ width ΓNρas following from
form factor FF1 of Eq. 13 is indicated by the solid line, the result from form factor FF2
of Eq. 14 by the dotted line. Right: Width of the P33(1232) resonance in vacuum. The ρ
component is negligible over the energy interval shown here and has not been plotted.
resonance peak in the spectral function due to the real part, which is more pronounced for
the D13(1520). Going away from the resonance peak, we observe an additional shoulder
for the D13(1520). This shoulder has most probably no direct influence on observables
since it is located far away from the resonance peak and is likely to be overshadowed by
the contribution of other resonances. Hence we are not concerned about this structure.
The real part of the self energy of both states – indicated by the solid and the dashed
lines in Fig. 6 – is comparable around resonance, with a slightly larger energy variation
for the D13(1520). Away from the resonance peak we observe a strong energy variation in
the self energy of the D13(1520) which is responsible for the additional shoulder found in
the spectral function. The dashed-dotted line in Fig. 5 shows the spectral function of the
D13(1520) as resulting from the use of a smaller partial decay width ΓNρ= 12 MeV instead
of ΓNρ= 26 MeV. Using the smaller width the shoulder nearly disappears. We will come
back to the issue of the proper choice for ΓNρin the next Section.
Next we address the question as to why form factor FF2 should be discarded at the
RNρ vertex. Therefore consider the results for ImΣ, ReΣ and the spectral function
ρ, which are depicted by the dotted lines in Figs. 4, 5 and 6. All three curves display
unsatisfying features: the Nρ decay width obtained with FF2 rises very quickly to values
above 1 GeV, around the resonance peak the real part of the self energy has a strong energy
dependence ∂ReΣ/∂k2and we observe a significant squeezing of the resonance peak in the
spectral function ρ. The sum of these effects provides enough evidence to abandon form
factor FF2 and take form factor FF1 instead.
These three effects are connected to each other in the following way: the rapid increase
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?
?
?
???
??
?
????
?
?
??
?
?
??
??
??
??
?
?
??
?
?
Figure 5:
both cases the solid lines indicate the results as following from using FF1 for ΓNρ. The
dashed line shows the spectral function without ReΣ. In the left figure, the dash-dotted
curve is the spectral function of the D13(1520) if a value of 12 MeV for ΓNρis used. In the
dotted curve the form factor FF2 of Eq. 14 is taken at the RNρ vertex.
Spectral function ρ of the D13(1520) (left) and the P33(1232) resonances. In
of ΓNρtranslates into a strong energy dependence of ReΣ. If ImΣ is nearly constant
around the pole of the resonance, one expects ReΣto be small since the contributions from
below and above the pole approximately cancel. Turning this argument around implies that
a rapid variation with energy leads to a sizeable ReΣ. This leads to a squeezing of the
peak which can be understood by expanding ReΣ to first order in k2, thus producing the
quasi-particle approximation. One gets for the spectral function:
ρ(k2) ≈
−1
π
z2ImΣ
R)2+ z2ImΣ2
?−1?????
(k2− m2
1 −∂ReΣ
(17)
z =
?
∂k2
k2=m2
R
.
The factor z < 1 effectively measures the influence of ReΣ and indicates that – depending
on the energy variation of ReΣ– strength is shifted away from the resonance peak to larger
invariant masses. This explains the pronounced peak at invariant resonances masses
around mR+ 0.6 GeV. Now one can also understand why form factor FF1 can cure this
problem: its functional form limits the energy variation of the Nρ width of the D13(1520),
which we have identified as the main source of trouble.
It follows from Eq. 17 that one could recover the nominal width of the resonance
peak by increasing ImΣ. The price to pay is that then the peak height is reduced,
because in any case the area under the peak is reduced since z < 1. A priori it is not
clear how this problem should be handled, since in principle peak height and peak width
can be additionally influenced by interference with background terms. In this work we
have decided to preserve the peak height. Adjusting instead to the peak width leads to
further complications for the coupled channel problem at hand: considering for example
the D13(1520), the z factor is essentially generated by the ρN width. We have checked
√k2
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?
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??
?
?
Figure 6: Real part of the self energy of the D13(1520) resonance (solid and dotted lines)
and the P33(1232) resonance (dashed line). The solid line is obtained using form factor
FF1 of Eq. 13 and the dotted line follows from form factor FF2 of Eq. 14.
numerically that by increasing the Nρ width – leading to a smaller z factor – one cannot
restore the original peak width. Instead, one had to change the πN or the π∆ partial
width (or both of them). In a numerical simulation we have found that both had to be
multiplied by a factor of 1.7 in order to restore the original peak width. Clearly, this
would influence the results for the π self energy, since a modification of ΓπN leads to a
new coupling constant at the πN D13(1520) vertex. A reliable solution of this problem
mandates a complete analysis of πN scattering taking into account effects from ReΣ.
The results for the spectral shape of the D13(1520) presented in this work show a strong
similarity to the textbook case of a stable state, whose mass is below the multi-particle
threshold [56]. In the absence of interactions, all the spectral strength sits in the particle
peak. After the interactions have been turned on, however, strength appears at masses
above the threshold. For the spectral function to remain normalized, this implies that
strength has to be removed from the quasi-particle peak.
The large decay width of ΓNρ= 300 MeV for the P13(1720) given in [57] also leads to
problems when calculating ReΣ, requiring a very small cutoff value Λ. Since such a large
decay width seems questionable for a resonance close to the nominal Nρ threshold and
coupling in a p-wave to this channel, we instead use the PDG estimate ΓNρ= 110 MeV
for this resonance. See also the discussion in Chapter 3.
3
ρN Scattering and Experiment
Experimental information on the coupling of mesons to a baryon resonance enters into our
model via the coupling constants fM, which are adjusted to the experimentally observed
partial decay width of that resonance into the respective meson-nucleon channel. Whereas
for the Nπ and Nη final states those branching ratios are rather well known, some un-
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certainty prevails for the decay width into the Nρ channel. This is due to the principal
difficulties associated with the experimental identification of a ρ meson at energies below or
only slightly above the nominal threshold, as occurring in the decay of, e. g., the D13(1520)
or the P13(1720). In the following we discuss the experimental information about the decay
of the D13(1520) in some detail and also touch on some of the uncertainties concerning a
few other states.
3.1The D13Amplitude
A major part of the results of this work hinges on the coupling of the D13(1520) resonance
to the Nρ channel. According to the PDG [55], one has ΓNρ= 24 MeV. This is close to
the value suggested by Manley et al [57], where ΓNρ= 26 MeV is found. In this work we
adopt the value of [57] and give some motivation for our choice in the following paragraphs.
The experimental information on ΓNρis primarily derived from a partial wave analysis
of πN → ππN scattering presented in [54], where a very clear resonance structure is found
in the πN → ρN channel of the D13partial wave amplitude for energies around 1.5 GeV.
Similar results are reported in Herndon et al[58] and Dolbeau et al [59]. Only in the work of
Brody [60] no coupling of the D13(1520) to the Nρ channel is found, since below√s = 1.6
GeV the Nρ contribution is set to zero by hand. In subsequent analyses, the partial decay
width ΓNρhas been extracted from a resonance fit of this partial wave amplitude. Both
the work of Manley et al [57] and Vrana et al[61] achieve a reasonable fit by assigning a
relatively large value (in view of the available phase space) to ΓNρ. In [57] a partial width
of 26 MeV is found, whereas the analysis of [61] reports a value of 12 MeV.
Further support for a rather large value for ΓNρcomes from a complementary exper-
iment, where photoproduction of pion pairs on the nucleon has been studied [62]. The
two-π invariant mass spectra - measured at photon energies just below the nominal thresh-
old for ρ production - follow the expected phase space distribution in the isoscalar channel
(π0π0). In the isovector channel a systematic asymmetry favouring larger invariant masses
is reported. An appealing interpretation of this finding assigns this asymmetry to the ρ
meson, which does not couple to the isoscalar channel. In a subsequent theoretical analysis
[63], this conjecture has been put on a more solid basis. There, a coupling of the D13(1520)
to the Nρ channel in line with the PDG values is necessary for a successful description of
the data.
In [22, 21] meson-nucleon scattering is described in terms of 4-point interactions. After
iterating the interaction, the resonant structures seen in experiment emerge dynamically.
Fitting these structures with a Breit-Wigner type ansatz, width and mass can be compared
the results of other analyses. For ΓNρa value of about 6 MeV is found in [22], smaller
than the results from [57, 61]. This is claimed to be due to the fact that a direct fit of
π N → ρN data from [60] is performed, thus leaving the region around the D13(1520)
essentially unconstrained. In [21] an even smaller coupling is obtained after the inclusion
of photo-induced data to the coupled channel analysis. There the direct constraint of
photo data to the hadronic ρN vertex results from the assumption of strict vector meson
dominance. The ρ coupling strength is then determined rather indirectly by the isovector
part of the photon-coupling and not from hadronic data.
Although we cannot exclude the possibility of such a weak coupling to the Nρ channel,
we believe that there is enough experimental evidence supporting a rather large width ΓNρ
of the D13(1520). We base our calculations on ΓNρ= 26 MeV as suggested in [57]. In order
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to get some feeling for the sensitivity of our results on ΓNρwe present also calculations
using ΓNρ= 12 MeV reported in [61].
3.2Other Partial Waves
Uncertainties concerning resonance parameters not only exist for the D13(1520) state, but
for many of the high lying resonances included in this work. Fortunately, in most cases
it turns out that the results are not too sensitive to changes in the parameters. This is
not true, however, for the P13(1720) and the D33(1700) states, which have a large energy
overlap with the Nρ system.
For both resonances a large branching ratio into the 2πN final state is well established
in the literature, see for example [57, 61, 52, 64]. However, so far no agreement has been
reached both for the total width of the resonance and for the relative strength of Nρ and
∆π contributions. Note first, that the decay of the P13(1720) to N ρ is strongly suppressed
from phase space, if one takes into account that the coupling is p-wave. In this light, the
huge partial decay width assigned in [57] of about 300 MeV seems questionable. Therefore
we have opted to take the PDG value ΓNρ= 110 MeV [55] for this channel, which is in
agreement with the findings in [61]. For the D33(1700) in [57] a value of 46 MeV is found,
whereas the PDG suggests a value of 120 MeV. Here we follow the results of [57]. This way
we arrive at a conservative estimate concerning the influence of both states for in-medium
effects.
4Mesons and Baryons in the Nuclear Medium
Up to now we have discussed aspects of the meson-nucleon and baryon-nucleon interaction
in the vacuum. We are now in a position to consider these interactions in the nuclear
medium. The goal is to achieve a coupled-channel analysis of the in-medium properties of
pions, η and ρ mesons as well as baryon resonances. We therefore need to calculate the
spectral function of all particles under consideration:
Amed
ρmed(k) = −1
M(q) = −1
πIm
1
vac(q) − Π+
q2− m2
M− Π+
1
R− Σ+
M(q)
(18)
πIm
k2− m2
med(k)
,
where M stands for the meson under consideration. Here we denote the full in-medium
self energy of meson M by the sum Π+
of all processes that take place only in the medium. For resonances this splitting is not
reasonable within our model, since the vacuum and in-medium self energies are generated
by the same type of diagrams.
For the mesons we consider effects from resonant meson-nucleon scattering, where the
nucleon is provided by the surrounding nuclear medium. This leads to the excitation
of particle-hole pairs, see Fig. 7. In the baryon sector, the in-medium decay width of
resonances is affected by two mechanisms: Pauli blocking reduces the width, whereas
resonance-nucleon scattering leads to a broadening. We generate collisional broadening –
being due to the exchange of mesons – by replacing the vacuum meson propagator Dvac
its in-medium counterpart Dmed
M
in the resonance self energy Fig. 2. Applying Cutkosky’s
vac(q) + Π+
M(q), where Π+
Mcontains the contribution
Mby
15