A. Haga1
Y. Horikawa2
Osaka Univ. Jpn.
Y. Tanaka3 1.RCNP,
2.Juntendo Univ. Jpn.
3.Nagoya Inst. Tech. Jpn
H. Toki1
・History and Motivations
・What is Nuclear Polarization?
Lamb shift, which is well known as the energy difference between the Dirac theory and experiment
in the bound electron states have been explained by the quantum electrodynamics (QED).
In the
evaluation of energy levels of exotic atoms like muonic atom, however, it is also required to
take into account the effect that bound particle distorts the nucleus because of its small radius.
The level shift due to these effects is traditionally called “nuclear-polarization ” effect.
‹First calculation of nuclear polarization.
R. Cole, Jr., Phys. Rev. 177, 164(1969)
‹Discovery of anomaly in Δ2p splitting energy of muonic 208Pb.
(At Los Alamos) Y. Yamazaki, Phys. Rev. Lett. 42, 1470(1979)
‹Existence of the same kind of discrepancies in other levels.
(At PSI)
P. Bergem et. al. Phys. Rev. C 37, (1988)
‹Large transverse contribution in hydrogen-like heavy atoms.
N. Yamanaka, A. Haga et. al. Phys. Rev. A 63 062502(2001)
We establish the calculation method of
gauge-invariant nuclear polarization.
Fig.1 Nucleus is polarized by field of bound particle
・Feynman Diagram
Success of precise calculation will give us
new probe for study of nuclear structure.
The reduction formula for gauge-invariant polarization propagator of virtual photon:
Feynman diagrams for
nuclear polarization
in lowest order.
If we employ the relativistic system, the ladder and cross diagrams
are considered with negative-energy nucleon states.
But…
Lepton
Nucleus Lepton
We compute
nuclear excitations
more than
1000!!
Nucleus
Fig.2(a) Ladder and cross diagrams
If the nucleon is treated as
the non-relativistic particle,
we must include the Seagull
diagram to achieve the gauge
invariance.
Not
polarized !
Lepton
or, if non-relativistic system is employed,
Nucleus
Fig.2(b) Seagull diagram
is also required to be gauge invariant.
Phys. Rev. A65, 052509(2002)
Phys. Rev. A66, 034501(2002)
We could explain about a half of the anomaly in the Δ2p splitting,
while a quarter of the anomaly in the Δ3p splitting.
Within the nuclear-polarization calculation, there are
ambiguities, e.g., relativistic effect of nucleon system:
z The contribution of anti-nucleon states
z The effective mass effect
z The renormalization effect of Dirac sea
Transverse interaction including the
seagull contribution gives rise to nuclear
polarization with different muon-spin
dependence from Coulomb interaction.
Δ2p
Transverse contribution in electric dipole
states of nucleus is sensitive for choice
of the gauge.
Nuclear polarization of
electronic 208Pb has serious gauge
dependence if one calculates it with only
the ladder and cross diagrams.
The seagull diagram is quite important in
restoring the gauge invariance.
On the other hand, the inclusion of all
produces results very similar to the
calculations with Coulomb interaction
only (denoted as CNP).
Fig4. Relation between relativistic and non-relativistic nuclear models
Open circles are shifted
to closed ones by the effect
of transverse interaction.
The experimental allowable
region of nuclear polarization.
Δ3p
Fig.3 Allowed nuclear-polarization energies
The contribution from the anti-nucleon states in the relativistic model
corresponds to the seagull diagram in the non-relativistic nuclear model.
On the other hand, the relativistic result is enhanced over non-relativistic
one due to the effective mass effect.
Now we study the renormalization effect in the nucleon system.
1. What is Nuclear Polarization?
Lamb shift, which is well known as the energy difference between
the Dirac theory and experiment in the bound electron states have
been explained by the quantum electrodynamics (QED).
In the
evaluation of energy levels of exotic atoms like muonic atom,
however, it is also required to take into account the effect that
bound particle distorts the nucleus because of its small radius.
The level shift due to these effects is traditionally called “nuclearpolarization ” effect.
Fig.1 Nucleus is polarized by field of bound particle
2. History and Motivations
‹First calculation of nuclear polarization.
R. Cole, Jr., Phys. Rev. 177, 164(1969)
‹Discovery of anomaly in Δ2p splitting energy of muonic 208Pb.
(At Los Alamos) Y. Yamazaki, Phys. Rev. Lett. 42, 1470(1979)
‹Existence of the same kind of discrepancies in other levels.
(At PSI)
P. Bergem et. al. Phys. Rev. C 37, (1988)
‹Large transverse contribution in hydrogen-like heavy atoms.
N. Yamanaka, A. Haga et. al. Phys. Rev. A 63 062502(2001)
We establish the calculation method of
gauge-invariant nuclear polarization.
Success of precise calculation will give us
new probe for study of nuclear structure.
3. Feynman Diagram
Feynman diagrams for
nuclear polarization
in lowest order.
Lepton
But…
Nucleus Lepton
Nucleus
Fig.2(a) Ladder and cross diagrams
If the nucleon is treated as
the non-relativistic particle,
we must include the Seagull
diagram to achieve the gauge
invariance.
Not
polarized !
Lepton
Nucleus
Fig.2(b) Seagull diagram
The reduction formula for gauge-invariant polarization
propagator of virtual photon:
The first and second terms of the right-hand side contribute
to the nuclear polarization.
The second term corresponds to the seagull diagram.
Formalism
For the nuclear polarization due to ladder and cross
diagrams depicted in Fig.2(a), we obtain
If we employ the relativistic nuclear model, the antinucleon states contribute to nuclear polarization in
dependence on the sign of excited energy.
We compute the nuclear excitations more than 1000
by random-phase approximation in finite system.
For non-relativistic system, we must calculate the seagull
contribution to be gauge invariance.
Results
Phys. Rev. A65, 052509(2002)
Transverse contribution in electric dipole
states of nucleus is sensitive for choice of the
gauge. Nuclear polarization of electronic 208Pb
has serious gauge dependence if one calculates
it with only the ladder and cross diagrams. The
seagull diagram is quite important in restoring
the gauge invariance.
On the other hand, the inclusion of all
produces results very similar to the calculations
with Coulomb interaction only (denoted as
CNP).
Results
Transverse interaction
including the seagull
contribution gives rise to
nuclear polarization
with different muonspin dependence from
Coulomb interaction.
Open circles are shifted
to closed ones by the effect
of transverse interaction.
Δ2p
The experimental
allowable
region of nuclear
polarization.
Fig.3 Allowed nuclear-polarization energies
Δ3p
Summary
We could explain about a half of the anomaly in the Δ2p splitting,
while a quarter of the anomaly in the Δ3p splitting.
Within the nuclear-polarization calculation, there are
ambiguities, e.g., relativistic effect of nucleon system:
z The contribution of anti-nucleon states
z The effective mass effect
z The renormalization effect of Dirac sea
Fig4. Relation between relativistic and non-relativistic nuclear models
The contribution from the anti-nucleon states in the relativistic model
corresponds to the seagull diagram in the non-relativistic nuclear model.
On the other hand, the relativistic result is enhanced over non-relativistic
one due to the effective mass effect.
Now we study the renormalization effect in the nucleon system.