The Baryon octet-vector meson
interaction and dynamically generated
resonances in the S=0 sector
Bao-Xi SUN (孙宝玺)
Beijing University of Technology
Hirschegg 2014
Hadrons from Quarks and Gluons,
Jan. 12-18, 2014
Content
1. Vector meson-baryon decuplet interaction
S. Sarkar, B. -X. Sun, E. Oset, M. J. Vicente
Vacas, EPJA 44: 431 (2010).
2. Vector meson-baryon octet interaction
E. Oset, A. Ramos, EPJA 44: 445 (2010).
3. Vector meson-baryon octet interaction
K. P. Khemchandani, H. Kaneko, H. Nagahiro, A.
Hosaka, PRD 83:114041 (2011)
4. My present work on Vector meson-baryon octet
interaction
Lippman-Schwinger Eqution
With a kernel of effective interaction, we can solve the LS equation.
The amplitude satisfies the unitary relation exactly, and the
properties of hadronic resonances generated dynamically can be
obtained. These resonances do not appear in the effective
Lagrangian density.
T  V  V GT  [1  VG]1V
Hidden-gauge symmetry
• In order to construct the vector meson-baryon
octet interaction Largrangian density, we
consider the SU(3) flavor local gauge
symmetry neglecting the mass term, and then
we obtain

L   g B   V  , B   B  B V 

Vector-vector Interaction
Vector octet-baryon decuplet
interaction
• Now I will discuss the interaction
between vector mesons and baryon
decuplet in the chiral unitary
approach. Because the interaction
Lagrangian is not known, we will try
to obtain the vector octet –baryon
decuplet interaction potentials by
comparing the pseudoscalar meson
– baryon decuplet interaction.
Interaction Vertex
•
When the momentum transfer is far less than the mass of the
vector meson in the propagator, we can neglect the square of the
momentum in the propagator. Therefore the t-channel interaction
between vector meson and baryon is obtained:
u-channel and s-channel
• We neglected the contribution from schannel and u-channel interaction since
we thought their effects are trivial at that
time.
Vector meson – baryon loop function in the
dimensional regularization scheme
Vector meson-baryon loop function accounting for
the width of the intermediate states
S. Sarkar, B. X. Sun, E.Oset et al., EPJA 44, 431 (2010)
S. Sarkar, B. X. Sun, E.Oset et al., EPJA 44, 431 (2010)
•
Ten resonances in the different
strangeness and isospin channels.
Degenerate in JP=1/2-, 3/2-, 5/2-.
Vector-baryon octet interaction
• E. Oset and A. Ramos,
EPJA 44, 445 (2010).
Tensor coupling between vector
meson and baryon octet
•
K. P. Khemchandani, H. Kaneko, H. Nagahiro and A. Hosaka,
PRD 83, 114041 (2011).
In this article, a tensor coupling term between vector meson and baryon
octet is added, which is relevant to the magnetic moments of the baryons,
and is also gauge invariant.
K. P. Khemchandani et al.,PRD 83, 114041 (2011)
K. P. Khemchandani et al.,PRD 83, 114041 (2011)
• In addition to t-channel, the s-channel,u-channel
and contact interaction are taken into account,
and a non-relativistic interaction potential
between vector meson and baryon octet with
strangeness S=0 is obtained.
• When u-channel and s-channel are taken into
account, the hadronic resonances generated
dynamically with different spins are not
degenerate again.
Results for Strangeness S=0
• S=0 K. P. Khemchandani et al.,PRD 83, 114041 (2011)
• I=1/2, S=0, E. Oset and A. Ramos,
EPJA 44, 445 (2010).
SU(3) vector –baryon interaction
• The SU(3) generalized form of the vector
meson baryon interaction Lagrangian can
be written as
Mixing of the vector octet and
singlet
• In order to obtain the right couplings for
the physical omega meson and phi
meson, we need to consider the mixing of
their octet and singlet components.
Vector meson-baryon-baryon vertices
• The Lagrangian related to the s-, u-, tchannel interactions is explicitly written as
• Moreover, the contact term between the
vector meson and the baryon can be
obtained from
t- channel interaction
• If the momentum of the initial meson is
similar to that of the final vector meson,
i.e., q_2~q_1, the momentum transfer
k=q_2-q_1 is trivial null approximately, and
then the t- channel interaction can be
written as
BBV vertex
The vertex of two baryons and a vector meson
and the coupling constants g_1 and g_2 are different for different baryons
and meson, these coupling constant can be obtained from SU(3) symmetry.
t-channel supplement
• However, if the difference between q_2
and q_1 is taken into account, an
additional term should be supplemented in
the t- channel of the baryon and meson
interaction.
u- , s- channel interactions
• The u- channel and s- channel interactions
between the baryon and the vector meson
can be written as
Contact term
• The contact interaction between baryons
and mesons is written as
• Of course, the coupling constant can be
obtained from SU(3) symmetry.
Total potentials
• the total kernel of the vector meson- baryon
octet interaction is a summation of the t-, s-, uchannels and contact term, which is functions of
the total energy in the center of mass system
and the scattering angleθ.
• Moreover, we should emphasize that we have
not made any approximation to obtain of the
vector meson-baryon octet potential.
Lippman-Schwinger Equation
• The relativistic coupling-channel LippmanSchwinger equation is written as
with a loop function in the on-shell
factorization approach
S-wave approxiamtion
• the amplitude T can be expanded in a
series of Legendre polynomials
with the coefficient
• In the S-wave approximation, only the T_0
is calculated for different channels.
I=1/2,S=0,J=1/2 channels
Pole positions and couplings
I=1/2,S=0,J=3/2,J_z=1/2 channels
Pole positions and couplings
I=3/2, S=0, J=1/2 Channels
Pole positions and couplings
I=3/2, S=0, J=3/2, J_z=1/2
channels
Summary 1
• 1. 14 poles are found in the complex total
energy plane for the channel with
strangeness zero S=0, and the couplings
to baryons and vector mesons are also
calculated. Their PDG counterparts are
discussed.
• 2. The spin of the dynamically generated
resonances are determined, and the spin
of the resonance is not degenerate again.
• 3. The contact term plays an important
role.
Scattering angle θ=0
• The forward scattering amplitude are the
same as each other when the orientation
of the total spin takes inverse values,
which is easy to understand since the spin
is conserved when θ is zero.
Total spin is not conserved
• However, the amplitudes at non-zero
scattering angle θ are different when the
orientation of the total spin takes different
values, which implies the total spin is not
conserved in the S-wave approximation in
the relativistic framework.
Partial wave analysis
• In the partial wave analysis, I calculate the
amplitudes with different spins:
• <1,0; ½, ½|T|1,0; ½, ½>,
• <1,1; ½, ½|T|1,1; ½, ½>,
• <1,-1; ½, ½|T|1,-1; ½, ½>,
• <1,1; ½, -½|T|1,0; ½, ½>,
• <1,0; ½, -½|T|1,-1; ½, ½>.
S=0 and I=1/2 channel
Partial wave analysis
• Since the spins are not conserved in the
interaction process, for example, the
amplitudes <1,1; ½, ½|T|1,1; ½, ½>
• and <1,-1; ½, -½|T|1,-1; ½, -½> take
different values at a fixed total energy in
the center of mass system, the results
might depend on the choice of the spins of
the initial and final states. It means we can
only obtain some approximate results.
•
•
Thanks