Vector- and Pseudoscalar-baryon coupled channel systems

Abstract

In this manuscript, I will report the details of our recent work on the
vector meson-baryon (VB) interaction, which we studied with the motivation of
finding dynamical generation of resonances in the corresponding systems. We
started our study by building a formalism based on the hidden local symmetry
and calculating the leading order contributions to the scattering equations by
summing the diagrams with: (a) a vector meson exchange in the t-channel (b) an
octet baryon exchange in the s-, u-channels and (c) a contact interaction
arising from the part of the vector meson-baryon Lagrangian which is related to
the anomalous magnetic moment of the baryons. We find the contribution from all
these sources, except the s-channel, to be important. The amplitudes obtained
by solving the coupled channel Bethe-Salpeter equations for the systems with
total strangeness zero, show generation of one isospin 3/2, spin 1/2 resonance
and three isospin 1/2 resonances: two with spin 3/2 and one with spin 1/2. We
identify these resonances with $\Delta$ (1900) $S_{31}$, $N^*$(2080) $D_{13}$,
$N^*$(1700) $D_{13}$, and $N^*$(2090) $S_{11}$, respectively.
We have further extended our study by including pseudoscalar meson-baryon
(PB) as the coupled channels of VB systems. For this, we obtain the PB
$\rightarrow$ VB amplitudes by using the Kroll-Ruddermann term where,
considering the vector meson dominance phenomena, the photon is replaced by a
vector meson. The calculations done within this formalism reveal a very strong
coupling of the VB channels to the low-lying resonances like $\Lambda$(1405)
and $\Lambda$(1670), which can have important implications on certain reactions
producing them. In addition to this, we find that the effect of coupling the
higher mass states to the lighter channels is not restricted to increasing the
width of those states, it can be far more strong.

Full text

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arXiv:1202.3840v1 [nucl-th] 17 Feb 2012
Vector- and Pseudoscalar-baryon coupled channel
systems
K. P. Khemchandani1†, A. Mart´ ınez Torres2, H. Kaneko1,
H. Nagahiro3, A. Hosaka1
1Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047, Japan.
2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan.
3Department of Physics, Nara Women’s University, Nara 630-8506, Japan.
E-mail:
†kanchan@rcnp.osaka-u.ac.jp
Abstract.
meson-baryon (VB) interaction, which we studied with the motivation of finding dynamical
generation of resonances in the corresponding systems. We started our study by building a
formalism based on the hidden local symmetry and calculating the leading order contributions
to the scattering equations by summing the diagrams with: (a) a vector meson exchange in
the t-channel (b) an octet baryon exchange in the s-, u-channels and (c) a contact interaction
arising from the part of the vector meson-baryon Lagrangian which is related to the anomalous
magnetic moment of the baryons. We find the contribution from all these sources, except the
s-channel, to be important. The amplitudes obtained by solving the coupled channel Bethe-
Salpeter equations for the systems with total strangeness zero, show generation of one isospin
3/2, spin 1/2 resonance and three isospin 1/2 resonances: two with spin 3/2 and one with
spin 1/2. We identify these resonances with ∆ (1900) S31, N∗(2080) D13, N∗(1700) D13, and
N∗(2090) S11, respectively.
We have further extended our study by including pseudoscalar meson-baryon (PB) as the
coupled channels of VB systems.For this, we obtain the PB → VB amplitudes by using
the Kroll-Ruddermann term where, considering the vector meson dominance phenomena, the
photon is replaced by a vector meson. The calculations done within this formalism reveal a very
strong coupling of the VB channels to the low-lying resonances like Λ(1405) and Λ(1670), which
can have important implications on certain reactions producing them. In addition to this, we
find that the effect of coupling the higher mass states to the lighter channels is not restricted
to increasing the width of those states, it can be far more strong.
In this manuscript, I will report the details of our recent work on the vector
1. Introduction
The vector mesons, which are understood as (q¯ q)l=1composite systems, have attracted a lot of
attention historically, beginning from the observation of the universality of the ρ-meson couplings
to the light hadrons, which can be understood in terms of the fluctuation of a photon to a vector
meson that subsequently interact with the hadrons. Further findings, like the dominance of the
the vector mesons in explaining the electromagnetic form factors of the hadrons [1] stimulated
the formulation of a theory where the vector mesons are treated as the dynamical gauge bosons
of the hidden local symmetry of the non-linear sigma model [2, 3, 4]. The resulting theory has
been widely used in hadron physics involving the vector mesons (see Refs. [5, 6, 7, 8, 9, 10, 11]
as examples).

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Using this effective field theory based on the hidden local symmetry, we have focussed on
understanding the low-energy interaction of vector mesons with baryons, with the idea of finding
dynamical generation of baryon resonances, motivated by the large branching ratios of some of
the excited baryons to the vector meson-baryon decay channels. Generation of resonances in
vector meson-baryon systems has earlier been reported in several phenomenological studies also
[12, 13, 14, 15, 16, 17]. We proceeded with our study by reminding ourselves that in case of the
vector meson-baryon interaction, we cannot rely on the low energy theorems which have worked
well in case of the pseudoscalar mesons (see, for example, Refs. [18, 19, 20, 21, 22, 23, 24] where
several baryon resonances have been found to get generated in the pseudoscalar-baryon or even
in two/three pseudoscalar-baryon systems). Keeping this in mind, we built a formalism [25]
on the basis of the hidden local symmetry and calculated the four diagrams shown in Fig. 1,
which correspond to: (a) a vector meson exchange in the t-channel, (b) a contact interaction
demanded by the gauge invariance (corresponding to the hidden local symmetry (HLS)) of the
vector meson-baryon interaction Lagrangian, which can be written for the SU(2) case as:
LρN= −g¯ N
?
F1γµρµ+F2
4Mσµνρµν
?
N,(1)
and (c, d) the s- and u-channel octet baryon exchange.
P2
P1
P1
P2
(a)
(b)
(c)
(d)
K1
K2
P1
P2
q
K1
K2
P1
P2
K1
K2
K1
K2
Figure 1. Different diagrams contributing to the vector meson-baryon interactions (see details
in the text).
Writing the HLS gauge invariant Lagrangian given by Eq. (1), explicitly, we get [25]
L = −g¯ N
?
γµρµ+
κρ
4M[∂µρν− ∂νρµ+ ig (ρµρν− ρνρµ)]σµν?
N,(2)

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where the nucleon field transforms as N → h(x)N with h(x) being an element of the hidden
local symmetry, F1(q2= 0) = 1, F2(q2= 0) = κρ∼ 3.22 is the anomalous magnetic coupling for
the ρNN vertex, M is the mass of the nucleon and
ρµ=1
2? τ · ? ρµ=1
2
?
ρ0
µ
√2ρ+
−ρ0
µ
√2ρ−
µ
µ
?
.(3)
Using Eq. (2), we can write Lagrangians for the vertices involving one and two meson fields,
respectively, as
LρNN
LρρNN
=−g¯ N
−ig2¯ N (ρµρν− ρνρµ)σµνN,
?
γµρµ+
κρ
4M(∂µρν− ∂νρµ)σµν?
N(4)
=(5)
where the former is used to obtain the amplitudes for the s- and u-channel diagrams [25]
Vs
ρN
=
g2
4
g2
4
?
?
1 −κρm
1 +κρm
2M
?2
?2
1
m + 2M
1
m − 2M
?1 − 2? tρ·? tN
?1 + 2? tρ·? tN
?(1 − 2? sρ· ? sN),
?(1 + 2? sρ· ? sN),
(6)
Vu
ρN
=
2M
(7)
and latter one gives a contact interaction with the amplitude given by
VCT
ρN=g2κρ
M
?tρ·? tN? sρ· ? sN. (8)
It should be emphasized here that this contact interaction arises from the [ρµ,ρν] term of the
vector tensor, which is essentially required to obtain the HLS gauge invariance of the Lagrangian
given by Eq. (2).
To calculate the t-channel diagrams, together with Eq. (4), we also need
L3ρ= −2igTr (∂µVνVµVν− ∂µVνVνVµ), (9)
which comes from the non-Abelian kinetic term of the ρ-meson.
amplitude has the form [25]
The resulting t-channel
Vt
ρN=−
1
2f2
1
4f2
π
?K0
?K0
1+ K0
2
?? ǫ1· ? ǫ2
?? ǫ1· ? ǫ2
for I = 1/2(10)
=
π
1+ K0
2
for I = 3/2,(11)
which is spin independent in nature. However the contact, s- and u-channel interactions lead
to spin dependent amplitudes. A comparison of Eqs. (6), (7), (8), (10) and (11) shows that
the contribution from all the four diagrams is equally important (it can be easily seen by using
the KSRF relation: g = mρ/√2fπ). We have calculated the amplitudes for other VB channels
using a more general Lagrangian [25], which we will give in the next section. In the subsequent
section, we will review the results of the full coupled channel scattering equations for the total
strangeness zero VB systems.
Having built a reliable formalism for the vector-meson baryon interactions, we extended
our study by including pseudoscalar-baryon systems as coupled channels [26]. Since several
resonances decay with large branching ratios to both PB and VB channels, it is possible, then,
that a coupled PB-VB dynamics can play important role in explanation of the properties of

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some of such resonances. However, since most PB and VB channels lie well separated on the
energy scale, we did not know if we were going to get important information by coupling them.
Hence, we decided to concentrate on studying the low-lying resonances alone since those are
better understood. This also allowed us to rely on the t-channel calculations of PB → PB and
VB → VB vertices, which simplified the formalism. However, as we will discuss in Section 3,
it turned out that the coupling between the PB and VB channels can result in very useful and
significant findings, implying the requirement of a more detailed calculations.
2. Formalism
2.1. Vector-baryon interaction
To calculate the VB transition amplitudes, we generalized the SU(2) vector meson-baryon
Lagrangian of Eq. (2) to SU(3) to get
LV BB
=−g
?
?¯Bγµ[Vµ
8,B]? +
1
4M
?
F?¯Bσµν[∂µVν
?
8− ∂νVµ
8,B]?
+D?¯Bσµν{∂µVν
C0
4M?¯BσµνVµν
8− ∂νVµ
?
8,B}?+ ?¯BγµB??Vµ
0?
+
0B?,(12)
where the constants D = 2.4 and F = 0.82. These values were found to well reproduce the
magnetic moments of the baryons in Ref. [27]. Further, in our normalization scheme,
V =1
2






ρ0+ ω
√2ρ+
√2K∗+
√2ρ−
−ρ0+ ω
√2¯K∗0
√2K∗0
√2K∗−
√2φ






(13)
and
B =








1
√6Λ +
1
√2Σ0
Σ+
p
Σ−
1
√6Λ −
1
√2Σ0
n
Ξ−
Ξ0
−
?
2
3Λ








,(14)
which give the octet (V8), and the singlet (V0) fields of the vector mesons by considering the
ω − φ mixing into account. For this, we write the wave functions of the ω- and the φ-meson,
under the ideal mixing assumption, as
ω=
?1
?2
3ω8+
?2
?1
3ω0
φ=−
3φ8+
3φ0. (15)
With these inputs, we obtained the amplitudes for different processes involving all the VB
channels. All these amplitudes are listed in all details in Ref. [25].

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2.2. Coupling pseudoscalar- and vector meson-baryon systems
The new information required to carry out a coupled channel study of the PB-VB systems is to
write the PB ↔ VB vertex consistently. This is done by using the Kroll-Ruderman (KR) theorem
to write the Lagrangian for the γN → πN process and by replacing the γ by a vector meson via
the notion of the vector meson dominance. More concretely, we write the πN Lagrangian from
the Gell-Mann-Levi’s linear sigma model,
LπN=¯ψ [iγµ∂µ− gπNN(σ + i? τ.? πγ5)]ψ,(16)
and introduce a vector meson field as a gauge boson of the hidden local symmetry
i/ ∂ −→ i/ ∂ − g/ ρ,(17)
to obtain the Lagrangian for the ρN → πN vertex (see Ref [26] for more details)
LπNρN
=−i
−iggA
2fπ
g
2fπ
¯ N [π,ρµ]γµγ5N
→
¯ N [π,ρµ]γµγ5N,(18)
where π = ? τ ·? π, ρ = ? τ ·? ρ
The SU(3) Lagrangian corresponding to Eq. (18) can be written as
2and fπ= 93 MeV is the pion decay constant.
LPBV B=−ig
2fπ
?˜F?¯Bγµγ5[[P,Vµ],B]? +˜D?¯Bγµγ5{[P,Vµ],B}?
?
, (19)
where the trace ?...? has to be calculated in the flavor space and˜F = 0.46,˜D = 0.8 such that
˜F +˜D ≃ gA= 1.26. The ratio˜D/(˜F +˜D) ∼ 0.63 here is close to the quark model value of 0.6,
and the empirical values of˜F and˜D can be found, for example, in Ref. [28]. Using Eq. (19),
amplitudes for all the processes within SU(3) have been obtained and are given in Ref. [26].
In the next section, we will discuss some of the most important results of our work on this
topic.
3. Results and discussions
3.1. Strangeness 0 resonances found in the VB systems
Having calculated the amplitudes for different VB coupled channels with total strangeness 0,
which are ρN, ωN, φN, K∗Σ and K∗Λ, we solve the Bethe-Salpeter equation
T = V + V GT,(20)
where the kernels V (= Vt+VCT+Vu+Vs) are effectively treated as contact interactions, which,
in turn, can be factorized out of the loop integral involved in Eq. (20). Such a treatment, which
is suitable for studies involving low-energy interactions, reduces solving of integral Eq. (20) to
an algebraic one.
Since we are looking for dynamical generation of resonances, which implies working with
low-energy dynamics, we calculate all the amplitudes in the s-wave. This means that the states
found in our work can have Jπ= 1/2−,3/2−. Further, the total isospin of the VB system can be
1/2 or 3/2. Let us first discuss the results obtained for the total isospin 1/2 case. In Fig. 2, we
show the squared amplitude for the ρN → ρN process. We find it very illustrative to show the
results obtained by considering the t-channel exchange only for the VB interaction and compare
them with those obtained by considering all the four diagrams of Fig. 1.