Off-shell effects in electromagnetic interaction with bound nucleons
Off-energy-shell effects in one- and two-nucleon photoabsorption are studied in the reaction γ³He → pd at intermediate energies. The calculations are carried out with the ³He wave functions for the Bonn and Paris potentials.
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Off-shell effects in electromagnetic interaction with bound nucleons / A.A. Belyaev, W. Glöckle, J. Golak, H. Kamada, V.V. Kotlyar, H. Witała // Вопросы атомной науки и техники. — 2001. — № 6. — С. 187-191. — Бібліогр.: 42 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859823193598984192 |
|---|---|
| author | Belyaev, A.A. Glöckle, W. Golak, J. Kamada, H. Kotlyar, V.V. Witała, H. |
| author_facet | Belyaev, A.A. Glöckle, W. Golak, J. Kamada, H. Kotlyar, V.V. Witała, H. |
| citation_txt | Off-shell effects in electromagnetic interaction with bound nucleons / A.A. Belyaev, W. Glöckle, J. Golak, H. Kamada, V.V. Kotlyar, H. Witała // Вопросы атомной науки и техники. — 2001. — № 6. — С. 187-191. — Бібліогр.: 42 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Off-energy-shell effects in one- and two-nucleon photoabsorption are studied in the reaction γ³He → pd at intermediate energies. The calculations are carried out with the ³He wave functions for the Bonn and Paris potentials.
|
| first_indexed | 2025-12-07T15:27:16Z |
| format | Article |
| fulltext |
OFF-SHELL EFFECTS IN ELECTROMAGNETIC INTERACTION
WITH BOUND NUCLEONS
A.A. Belyaev a, W. Glöckle b, J. Golak b, H. Kamada c, V.V. Kotlyar a, H. Witała d
a-National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
b-Institut fűr Theoretische Physik II, Ruhr Universität Bochum, 44780 Bochum, Germany
c-Department of Physics, Faculty of Engineering, Kyushu Institute of Technology,
1-1 Sensuicho, Tobata, Kitakyushu 804-8550, Japan
d-Institute of Physics, Jagellonian University, PL 30059 Cracow, Poland
Off-energy-shell effects in one- and two-nucleon photoabsorption are studied in the reaction pdHe →3γ at inter-
mediate energies. The calculations are carried out with the 3He wave functions for the Bonn and Paris potentials.
PACS: 25.10.+s, 25.20.-x, 27.10.+h
1. INTRODUCTION
Uncertainties due to the off-shell effects in the cross
section of the electron scattering off a bound nucleon
were discussed in 1,2. Large efforts were made to un-
derstand the role of the off-shell effects in the produc-
tion of electron-positron pairs in the virtual Compton
scattering on the proton 3, the proton-proton brem-
sstrahlung 4, the Compton scattering on the nucleon 5,
the deuteron photo- 6 and electrodisintegration 7-8, the
elastic scattering of electrons off 3He 9, the two-body
3He photodisintegration 10,11. Detailed discussion of
the problems in the theory of EM interaction with the
off-shell particles and references to previous work can
be found in 12,13.
Articles 14-15 contribute to the theory of EM
interaction with off-mass-shell particles. The origin of
the off-energy-shell (OES) effects in the framework of
the theory with the on-mass-shell particles can be
clarified within approach 16,17,18 based on Fukuda-
Sawada-Taketani-Okubo transformation. OES
modifications were introduced into the single-nucleon
current in 19,20,21-22. Two-body interaction currents
taking into account OES corrections were constructed in
23 using model 24 of the pion-exchange currents (πEC).
It was shown 25,26 that due to the OES effects the
relative role of the two-nucleon mechanisms generated
by the pion exchange could be visibly reduced in
pdHeγ3 → at MeV.200Eγ
Mechanisms of 3He photodisintegration and proton-
deuteron radiative capture are intensively studied both
below 27-28 and above 29,30,31,32,33-34 the pion
production threshold. A considerable progress has been
made, in particular, in the region of photon energies
MeV,140Eγ where the interaction currents and re-
scattering effects were treated simultaneously 35 -36.
The meson exchange currents (MEC) were demonstra-
ted to bring important contributions to the differential
cross sections and the polarization observables.
The interaction currents are included implicitly in
37-38 via the extended Siegert theorem and explicitly in
39,40,41,42,43,44,45-46. While rescattering in the pd-
system has been taken into account in 47-48, investi-
gations 49,50,51,52,53-54 have been performed in the
plane-wave approximation. The techniques to treat the
MEC in calculations with the exact solutions of the Fad-
deev-like equations for the initial and final states have
been elaborated in 55. Approach 56 is based on the
multipole and partial-wave decompositions. A distingui-
shing feature of 57,58,59,60,61 is the use of vector va-
riables following 62,63.
Aim of the present report is to improve and to detail mo-
del of πEC 64 that embodies the OES corrections. Other
purpose of the paper is to study influence of the OES ef-
fects in one- and two-nucleon photoabsorption on obser-
vables in the reaction pdHe →3γ taking advantage on
the precise numerical solutions of the Faddeev equa-
tions for the 3N bound state obtained in 65-66 with the
realistic NN potentials.
2. MODEL OF THE NUCLEAR CURRENT
In the present calculations we take into account one-
and two-nucleon contributions to the nuclear current
),()()( ]2[
μ
]1[
μμ xJxJxJ
+= where
∑∑
<
==
βαα
α βα ).;()(and);()( ]2[]1[ xJxJxJxJ
Various approaches are used to derive a construction
for the one-nucleon current );( αxJ
(for discussion see,
e.g., 67,68,69,70-71. The current of interest can be obtai-
ned from the expression for the matrix element
),()'(');0(' μμ pupupJp Γ=
α (1)
where μp and μ'p are momenta of the nucleon with the
label α, for instance, ( )pEp p
,μ = . The spin indices are
omitted for brevity. For the nucleons on the mass shell,
i.e., ,2222
μ Np MpEp =−= and 22
μ NMp =′ the γNN
vertex function μΓ has the form 72-73
( )
( ) ,)'()'(
2
1
γ)'(),'(
ν
μν2
1
μ2
1
μ
ppippF
M
ppFpp
N
−−+
+−=Γ
σλ
λ
(2)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 187-191. 187
where μνσ denotes the commutator of the -γ matrices
and NM is the nucleon mass. The vertex function de-
pends on the Dirac and Pauli form factors (FFs), 1F and
2F , with the arguments belonging to the space-like re-
gion since for the on-mass-shell nucleons 0)'( 2
μ ≤− pp .
Taking into account that the spinors )'(' pu and
)( pu obey the Dirac equation one can get a representa-
tion of current (1),(2) that is convenient for the use in
the calculations with the nuclear wave functions expres-
sed in terms of the two-component Pauli spinors.
Following 74-75 we keep the arguments of the EM
FFs as .)'()()'( 22
'
2
μ ppEEpp pp
−−−=− As far as
three-momentum is conserved in the intermediate states
within the framework of the theory with the nucleons on
the mass shell, one has .)()'( 22
'
2
μ QEEpp pp
−−=−
The vector μ)'( pp − cannot be replaced by the four-
momentum transfer ),(μ QQ
ω= in the general case
since .' ω≠− pp EE
The current is expanded in powers of )./( NMp For
example, the lowest order in )/( NMp for the Fourier
transform of the charge density reads
).)(()(
)(
22
1
]1[
QEEFQpp
pQp
pQp
−−−+′=
=′
+
δ
ρ
(3)
For the next terms in the )/( NMp expansion of the
one-body current we refer, e.g., to 76,77-78.
One can decompose the arguments of the FFs
),))2/()'((1()'( 22
μ
++⋅−−=− NMppnQpp (4)
where the unit vector .||/ QQn
= Nevertheless, we
retain the arguments in the form of Eq. (4) to derive the
expressions for the two-body currents.
Approximating 22
μ)'( Qpp
−≈− we get the prescrip-
tion of Ref. 79, where the nonrelativistic limit of the
Dirac current was suggested to include the FFs ).( 2QF i
−
It is worth noting that constructions for the one-body
current 80 that are obtained with the help of the Foldy-
Wouthuysen transformation applied to the Dirac equation
for a nucleon in an external EM field contain )( 2
µQF i .
Two-nucleon interaction currents satisfy the conti-
nuity equation ,],[ ]2[]1[ JQV ⋅=
ρ where V is the nuc-
leon-nucleon potential. We restrict our treatment by the
contributions to the current ]2[J generated by the pion
exchange. The respective part of the potential is given
by the one-pion-exchange interaction. For the matrix
elements one has
),);(()()( 2
1 α βδα β βαβαβαβα kkVkkppVpp −+=′′
where
),()()()()();( βτατβσασα β πβα
⋅⋅⋅−= kvkkkV (5)
),(1
2
1)( 22
222
2
2 kF
kmm
fkv NN
NN
π
ππ
π
π π +
= (6)
αp is the momentum of the α-th nucleon, ,ααα ppk
−′=
ααα ppk
−′= , NNfπ is the pseudovector πN coupling
constant, πm is the pion mass, ))(()( ατασ
denotes the
Pauli vector in the spin (isospin) space of particle α. For
the πNN FF )( 2
πNN kF we use the parametrization in
the monopole form ),/()()( 22222
πNN kmkF +Λ−Λ= πππ
where πΛ is the cut-off parameter.
To meat the requirements imposed by the continuity
equation, when the charge density is chosen in the form of
Eq. (3) and NN interaction is given by Eq. (5), pion exchan-
ge currents should consist 81 of the following terms
),;();();();( αβQJαβQJαβQJαβQJ isiv'iv
++=
where );( α βQJ
is the Fourier transform of the current
),;( α βxJ
the isovector currents ~);( αβQJ iv
,])()(i[~ zβτατ
× )(),(~);( βτατ zz
iv' αβQJ
and the
isoscalar current ).()(~);( βτατ
⋅αβQJ is For the cur-
rent
),;,,,()(
);(
α βδ
α β
βαβαβα
βαβα
ppppJQkk
ppQJpp
′′−+=
=′′
the leading )/( NMp order can be derived form the
corresponding expressions 82.
In the case when the approximation 22
μ)'( Qpp
−≈− is
used for the arguments of the EM FFs, model 83 yields
);();();( α βα βα β π QJQJQJ s
+= , (7)
with
);,();,,,( ,, α βα β βα
π
βαβα
π kkJppppJ ss
=′′ (8)
like the original currents. The seagull and pion-in-flight
currents are
))()()()()()((
)(])()(i[);,( 2v
1z
απαβπβ
βα
ασβσβσασ
βτατα β
kvkkvk
QFkkJ s
⋅+⋅−×
×−×=
(9)
and
)./())()((
)()()(
)(])()(i[);,(
22
2v
1z
βααπβπ
βαβα
βα
π
βσασ
βτατα β
kkkvkv
kkkk
QFkkJ
−−×
×⋅⋅−×
×−×=
(10)
An extension of model (7), (9), (10) can be cast into
the form
).;();();(
);(
,,, α βα βα β
α β
ππ QJQJQJ
QJ
md
OS
mi
OS
mis
OS
OS
++=
=
(11)
The currents
)),()()()(
)()()()((
])()(i[);( z
,
απα
βπβ
βασβσ
αβσασ
βτατα β
kvFk
kvFk
QJ mis
OS
⋅+
+⋅−
××=
(12)
188
)/())()()()((
)()()(
])()(i[);(
22
z
,
βααπβπ
βαβα
π
βα
βσασ
βτατα β
kkkvFkvF
kkkk
QJ mi
OS
−−×
×⋅⋅−×
××=
(13)
satisfy the continuity equation with the charge density (3)
and NN interaction (5). According to 84 one can reckon
these currents among “model independent” ones since the
structure of the currents is determined by the requirement
of current conservation entirely. The EM FFs appear in
the expressions for the currents in the superposition
)),)((
))((()(
22
1
22
12
1
QEEF
QEEFF
pQp
V
Qpp
V
−−+
+−−=
+
−′′
αα
αα
α
where VF1 is the isovector FF of the nucleon.
The transverse current
)/())()())(()((
)()()(
])()(i[);(
22
2
1
z
,
βααπβπ
βαβα
π
βα
βσασ
βτατα β
kkkvkvFF
kkkk
QJ
transv
md
OS
−+−×
×⋅⋅−×
××=
(14)
is not constrained by the continuity equation and is in a
certain sense “model dependent”. The r.h.s. of Eq. (14)
contains the transverse component of the vector defined as
]].[[ nAnAtransv
××= One has 0);(, =α βπ QJ md
OS
when
the arguments of the EM FFs are taken to be 2Q
− or .2
µQ
Current (14) is introduced with the aim to compen-
sate the singularity in transv
mi
OS QJ ));(( , α βπ
that springs
from the factor )/(1 22
βα kk − . As a result one has
),/())()((
)()()(
))()((])()(i[
));();((
22
2
1
z
,,
βααπβπ
βαβα
ππ
βσασ
βαβτατ
α βα β
kkkvkv
kkkk
FF
QJQJ
transv
transv
md
OS
mi
OS
−−×
×⋅⋅−×
×+×=
=+
(15)
where )).((2)( ααβα knnkkk transv
⋅−=− Expression
(15) is free from singularities similar to Eq. (10).
As the representation
))()()()((
)()(||/
])()(i[));(( z
,
απβπ
βα
π
βα
βσασ
βτατα β
kvFkvF
kkQn
QJ long
mi
OS
−×
×⋅⋅×
××=
indicates, the longitudinal component of current (13)
has no singularities either. For any vector one has
).( AnnAlong
⋅=
Currents (10) and (15) have much in common. On
the other hand, feature (8) is lost in model of πEC (11)-
(15) consistent with charge density (3). If the arguments
of the EM FFs in (11)-(15) are substituted by 2
µQ ,
constructions 85 for the πEC will be recovered.
3. RELATIVE ROLE OF THE OFF-ENERGY-
SHELL EFFECTS IN THE 3He TWO-BODY
PHOTODISINTEGRATION
To study the role of the OES effects in the reaction
dpHeγ 3 +→+ the construction of convection and spin
currents (CC and SC) from 86,87 and model (7)-(10) of
πEC are used. For the nucleon EM FFs the dipole pa-
rametrization and the scaling rule 88,89 are taken. The
calculations have been performed with the Bochum –
Cracow wave functions (WFs) for 3N bound state ob-
tained 90-91 for the Bonn and Paris potentials. The re-
scattering effects in the final pd-state are neglected. The
reaction amplitudes are computed within approach 92,
93,94 that allows us do not apply any multipole decom-
positions for the nuclear current.
As is demonstrated in Fig. 1, the OES corrections in
the nuclear current affect the differential cross section σ
at MeV,175γ >E compensating contributions of the
πEC and decreasing the σ values. At MeV,200~γE
variations of the cross section due to OES effects and
changes caused by increase of the cut-off parameter πΛ
from 750 MeV/c up to 1.2 GeV/c are of the same order in
magnitude. The observed dependencies indicate impor-
tance of treatment of the OES effects.
150 175 200 225 250 275 300
10-1
100
Elab
γ
, MeV
Λ
π
=1200 MeV/c
Λ
π
=
=750 MeV/c
θ cm
p
=90o
σ [CC;π EC]+σ [SC;SOC],
WFs for Paris potential
σ,
µ
b/
sr
Fig. 1. Energy dependences of the differential
cross section for .3 pdHe →γ Influence of the off-
energy-shell effects is displayed by the dash-dotted
and dash-dot-dotted curves. The experimental points
", C and � are taken from 95,96 and 97, respectively
The detailed analysis reveals that the substitution
2
γ
2
μ Ep)(p' −→− in the arguments of the nucleon EM FFs is
not a very rough approximation. Nevertheless, The sub-
stitution is not well justified at forward and backward
angles pθ of proton emission. It should be stressed, that
in studying the processes with real photons the EM FFs
of nucleons are usually taken at point .02 =µQ
The approximated expressions can be factorized ta-
king the EM FF out from the nuclear overlap integrals.
This transformation allows one to explain an observation
that the cross section asymmetry coefficient Σ for the
reaction dpHe +→+ 3γ
with linearly polarized photons
is insensitive to the OES effects at least under the condi-
tions when the πEC play an important role. Really, since
the polarization observables, in particular, the photon
asymmetry, are given by a ratio of quadratic forms of the
amplitudes, they do not depend on the OES modifications
189
of the nuclear current introduced in the factorized form
both in the numerator and in the denominator.
To assess the relative role of the OES effects we de-
monstrate in Fig. 2 dependence of the differential cross
section and the beam asymmetry on the number αN of
the partial-wave component of the 3N bound state wave
function included in the calculations and on the choice
of the nucleon-nucleon potential.
0 50 100 150 200 250 300
-1,0
-0,5
0,0
0,5
1,0
Σ [CC;SC]
Σ [CC;π EC;SC]
Σ
Elab
γ
, MeV
10-3
10-2
10-1
100
101
102
θ cm
p
=90o
σ [CC;π EC]+σ [SC]
σ [CC]+σ [SC]
Paris
N
α
=2
N
α
=5
N
α
=10
N
α
=34
σ,
µ
b/
sr
0 50 100 150 200 250 300
-1,0
-0,5
0,0
0,5
1,0
Σ [CC;SC]
Σ [CC;π EC;SC]
Σ
Elab
γ
, MeV
10-1
100
101
102
θ cm
p
=90o
N
α
=34
Bonn
Paris
σ [CC;π EC]+σ [SC]
σ [CC]+σ [SC]
σ,
µ
b/
sr
Fig. 2. Energy dependences of the differential cross section for pdHe →3γ and the asymmetry coefficient for
the reaction with linearly polarized photons. Results of the calculations with the Bochum-Cracow WFs for the
Bonn and Paris potential are shown. The experimental points , , , , and are taken from 98,99,100, 101,102-
103 and 104, respectively
As is seen from the Fig. 2, the differential cross sec-
tion ][];[ SCECCC σπσ + is not strongly affected by the
WF components when 2>αN , i.e., by the P-, D-, etc.
partial waves. The sensitivity of the cross section to the
choice of the WFs substantially decreases when ECπ
are taken into account. In this situation the variation of
σ due to OES effects should not be neglected. To reduce
the discrepancies between the result of the calculations
and the data, especially for the beam asymmetry Σ, the
three-nucleon mechanisms of photoabsorption and the
rescattering in the final state are to be included.
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192
A.A. Belyaev a, W. Glöckle b, J. Golak b, H. Kamada c, V.V. Kotlyar a, H. Witała d
a-National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
b-Institut fűr Theoretische Physik II, Ruhr Universität Bochum, 44780 Bochum, Germany
c-Department of Physics, Faculty of Engineering, Kyushu Institute of Technology,
1-1 Sensuicho, Tobata, Kitakyushu 804-8550, Japan
d-Institute of Physics, Jagellonian University, PL 30059 Cracow, Poland
PACS: 25.10.+s, 25.20.-x, 27.10.+h
1. INTRODUCTION
3. RELATIVE ROLE OF THE OFF-ENERGY-SHELL EFFECTS IN THE 3He TWO-BODY PHOTODISINTEGRATION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79484 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:27:16Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Belyaev, A.A. Glöckle, W. Golak, J. Kamada, H. Kotlyar, V.V. Witała, H. 2015-04-02T15:38:27Z 2015-04-02T15:38:27Z 2001 Off-shell effects in electromagnetic interaction with bound nucleons / A.A. Belyaev, W. Glöckle, J. Golak, H. Kamada, V.V. Kotlyar, H. Witała // Вопросы атомной науки и техники. — 2001. — № 6. — С. 187-191. — Бібліогр.: 42 назв. — англ. 1562-6016 PACS: 25.10.+s, 25.20.-x, 27.10.+h https://nasplib.isofts.kiev.ua/handle/123456789/79484 Off-energy-shell effects in one- and two-nucleon photoabsorption are studied in the reaction γ³He → pd at intermediate energies. The calculations are carried out with the ³He wave functions for the Bonn and Paris potentials. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Electrodynamics of high energies in matter and strong fields Off-shell effects in electromagnetic interaction with bound nucleons Эффекты схода с энергетической оболочки в электромагнитном взаимодействии со связанными нуклонами Article published earlier |
| spellingShingle | Off-shell effects in electromagnetic interaction with bound nucleons Belyaev, A.A. Glöckle, W. Golak, J. Kamada, H. Kotlyar, V.V. Witała, H. Electrodynamics of high energies in matter and strong fields |
| title | Off-shell effects in electromagnetic interaction with bound nucleons |
| title_alt | Эффекты схода с энергетической оболочки в электромагнитном взаимодействии со связанными нуклонами |
| title_full | Off-shell effects in electromagnetic interaction with bound nucleons |
| title_fullStr | Off-shell effects in electromagnetic interaction with bound nucleons |
| title_full_unstemmed | Off-shell effects in electromagnetic interaction with bound nucleons |
| title_short | Off-shell effects in electromagnetic interaction with bound nucleons |
| title_sort | off-shell effects in electromagnetic interaction with bound nucleons |
| topic | Electrodynamics of high energies in matter and strong fields |
| topic_facet | Electrodynamics of high energies in matter and strong fields |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79484 |
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