Proton-deuteron radiative capture and two- and three-body photodisintegration of 3He at low energy are described using realistic hadronic dynamics and including the Coulomb force. The sensitivity of the observables to the relativistic corrections of the one-nucleon electromagnetic current operator is studied. Significant effects of the relativistic spin-orbit charge are found for the vector-analyzing powers in the proton-deuteron radiative capture and for the beam-target parallel-antiparallel spin asymmetry in the three-body photodisintegration of 3He

Deltuva, A.

Fonseca, A.C.

Sauer, P.U.

English

Institutional Repository of Leibniz Universität Hannover

PHYSICAL REVIEW C 80, 064004 (2009)
Relativistic corrections of one-nucleon current in low-energy three-nucleon photonuclear reactions
A. Deltuva,1 A. C. Fonseca,1 and P. U. Sauer2
1Centro de Fı´sica Nuclear da Universidade de Lisboa, P-1649-003 Lisbon, Portugal
2Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, D-30167 Hannover, Germany
(Received 19 November 2009; published 22 December 2009)
Proton-deuteron radiative capture and two- and three-body photodisintegration of 3He at low energy are
described using realistic hadronic dynamics and including the Coulomb force. The sensitivity of the observables
to the relativistic corrections of the one-nucleon electromagnetic current operator is studied. Significant effects
of the relativistic spin-orbit charge are found for the vector-analyzing powers in the proton-deuteron radiative
capture and for the beam-target parallel-antiparallel spin asymmetry in the three-body photodisintegration of 3He.
DOI: 10.1103/PhysRevC.80.064004 PACS number(s): 25.20.−x, 21.45.−v, 24.70.+s, 25.40.Lw
I. INTRODUCTION
Electromagnetic (e.m.) reactions in the three-nucleon (3N )
system have been extensively studied in the past with the aim
of testing various models of the hadronic two-nucleon (2N )
and 3N interaction and of the nuclear e.m. current [1–6].
In the nucleon-deuteron (Nd) radiative capture at higher
energies [4], we observed quite a significant effect of the
relativistic spin-orbit correction of the one-nucleon (1N )
charge operator. This is consistent with the findings of
Refs. [7–9] for the photodisintegration of the deuteron.
However, quite surprisingly, we found a sizable effect of the
same relativistic spin-orbit correction also for the proton (p)
and deuteron (d) vector-analyzing powers in the low-energy
pd radiative capture [10]. The aim of this article is to calculate
the observables of the low-energy 3N photonuclear reactions.
For pd radiative capture, we compare with existing data.
For two- and three-body photodisintegration of 3He, we
make predictions for experiments that are in the process of
being performed at the High Intensity Gamma-ray Source
(HIGS) facility [11]. We study the sensitivity of observables
to relativistic corrections of the 1N charge and spatial current.
In Sec. II, we describe the chosen model for the hadronic
interaction and e.m. current. Selected results for the pd
radiative capture and photodisintegration of 3He are presented
in Sec. III. We summarize in Sec. IV.
II. DYNAMICS
The hadronic dynamics is based on the realistic two-baryon
coupled-channel potential CD Bonn +  [12] that allows for a
virtual excitation of a nucleon to a  isobar and thereby yields
effective many-nucleon forces [13]. The nonrelativistic part of
the e.m. current is taken over from Ref. [4] and has one-baryon
and two-baryon pieces. Besides the standard purely nucleonic
part, there are additional parts involving the  isobar, which
then make effective 2N and 3N contributions to the e.m.
current that are consistent with the effective 2N and 3N forces.
Since the underlying potential CD Bonn +  is based on the
exchange of standard isovector mesons π and ρ and isoscalar
mesons ω and σ , the same meson exchanges are contained in
the effective nucleonic forces and currents. The initial and final
3N bound and scattering states are described in the isospin
formalism by the exact Faddeev and Alt, Grassberger, and
Sandhas three-particle equations [14], respectively, that are
solved in momentum space using partial wave decomposition
[15]. The Coulomb interaction between charged baryons
is important in the considered low-energy reactions and is
included using the method of screening and renormalization
[16–18]. The current is expanded in electric and magnetic
multipoles. The magnetic multipoles are calculated from the
one- and two-baryon parts of the spatial current. The electric
multipoles use the Siegert form of the current without the long-
wavelength approximation; assuming current conservation,
the dominant parts of the one-baryon convection current and
the diagonal π - and ρ-exchange current are taken into account
implicitly in the Siegert part of the electric multipoles by
the Coulomb multipoles of the charge density; the remaining
non-Siegert part of the electric multipoles not accounted for
by the charge density is calculated using explicit one- and
two-baryon spatial currents. Although the continuity equation
is not exactly fulfilled for the nonlocal potential CD Bonn + 
or when the relativistic 1N current corrections are included, the
employed calculational scheme, based on the Siegert form of
the current, effectively corrects the current nonconservation, as
argued in Ref. [19], and is therefore reliable for the observables
of low-energy photoreactions considered in this article. We
also note that the use of the nonrelativistic hadronic dynamics,
together with the relativistic current corrections, is somehow
inconsistent. However, in the low-energy reactions, it is natural
to expect the effect of the relativistic hadronic dynamics to
be small; this expectation is consistent with the results of
Ref. [20].
To isolate the effects of various relativistic current correc-
tions, we perform several calculations (all using CD Bonn +
 as the hadronic potential and including Coulomb) with a
different choice of e.m. current:
(i) The nonrelativistic one- and two-baryon e.m. currents
are used as in the standard calculations of Ref. [4],
except for the parameters of the single nucleon-
transition current, which are taken from Ref. [21]. The
corresponding results in the figures are given by the
dash-dotted curves.
(ii) The relativistic spin-orbit correction to the 1N charge,
as given by Eq. (A11a) of Ref. [4], is included in
0556-2813/2009/80(6)/064004(5) 064004-1 ©2009 The American Physical Society
A. DELTUVA, A. C. FONSECA, AND P. U. SAUER PHYSICAL REVIEW C 80, 064004 (2009)
-0.1
0.0
0 60 120 180
A y
(d)
Θc.m.  (deg)
Ed = 10 MeV
-0.1
0.0
0 60 120 180
Θc.m.  (deg)
Ed = 17.5 MeV
-0.1
0.0
0 60 120 18
Θc.m.  (deg)
Ed = 29 MeV
FIG. 1. (Color online) Deuteron vector-analyzing power Ay(d) for pd radiative capture at Ed = 10, 17.5, and 29 MeV as a function of
the c.m. d-3He scattering angle. Results obtained with nonrelativistic e.m. current (dash-dotted curves), including relativistic spin-orbit charge
(solid curves), and including, in addition, relativistic corrections to the spatial current (dotted curves), are compared with the experimental data
from Refs. [25–27].
addition to (i); it contributes to the Siegert term. The
corresponding results in the figures are given by the
solid curves, consistently with Ref. [10].
(iii) With respect to the relativistic Darwin-Foldy correction
to the 1N charge, there is a controversy in the literature:
according to Ref. [22], it vanishes identically for the
reactions with real photons (γ ), whereas it yields a
finite contribution according to Ref. [23]. We include
the latter in addition to (ii); however, in all studied low-
energy photoreactions, its effect turns out to be entirely
negligible and will therefore not be shown separately.
(iv) In addition to (iii), relativistic corrections to the 1N spa-
tial current from Ref. [23] are included; they contribute
to the magnetic multipoles and to the non-Siegert part
of the electric multipoles. The corresponding results in
the figures are given by the dotted curves.
(v) In addition to (iv), the nucleon- transition charge that
is of the relativistic order, as given by Eq. (A11b) of
Ref. [4], is included. However, the corresponding
results are not shown separately because the effect is
entirely negligible.
The technical details of our calculations are explained in Refs.
[4,10,24].
0.0
0.2
0.4
0 60 120 180
A y
(p)
Θc.m.  (deg)
FIG. 2. (Color online) Proton vector-analyzing power Ay(p) for
pd radiative capture at Ep = 5 MeV as a function of the c.m. p-γ
scattering angle. Curves are as in Fig. 1, and the experimental data
are from Ref. [26].
III. RESULTS
We start with the pd radiative capture, where significant
relativistic effects were predicted in Ref. [10]. In Fig. 1, we
show the deuteron vector-analyzing power Ay(d) at deuteron
laboratory energies Ed = 10, 17.5, and 29 MeV as a function
of the center-of-mass (c.m.) d-3He (or p-γ ) scattering angle.
The effect of the 1N spin-orbit charge is sizable and clearly
improves the description of the experimental data, whereas the
relativistic corrections of the 1N spatial current contributing
to the magnetic multipoles and to the non-Siegert part of the
electric multipoles yield only minor changes. Furthermore,
it is interesting to note that the spin-orbit charge effect
increases with decreasing energy. A strong and beneficial
effect is seen also in the proton analyzing power Ay(p) at
Ep = 5 MeV proton laboratory energy in Fig. 2. In contrast,
the differential cross section and deuteron tensor analyzing
powers at low energies remain almost unaffected by the
spin-orbit charge, as can be seen from our previous results [10]
at Ed = 6 MeV; those results are not documented in this
article. The spin-orbit charge effect shows up in the deuteron
tensor analyzing powers as the energy increases, especially
at forward and backward angles, where the differential cross
section is small. In Fig. 3, we show Ayy at Ed = 45 MeV;
0.0
0.1
0 60 120 180
A y
y
Θc.m.  (deg)
FIG. 3. (Color online) Deuteron tensor analyzing power Ayy for
pd radiative capture at Ed = 45 MeV as a function of the c.m. d-3He
scattering angle. Curves are as in Fig. 1, and the experimental data
are from Ref. [27].
064004-2
RELATIVISTIC CORRECTIONS OF ONE-NUCLEON . . . PHYSICAL REVIEW C 80, 064004 (2009)
0.0
0.1
0.2
0.3
0 60 120 180
dσ
/d
Ω
 
 
(µb
/s
r)
Θc.m.  (deg)
-0.2
0.0
0 60 120 180
A y
(p)
Θc.m.  (deg)
-0.1
0.0
0.1
0 60 120 180
A y
(d)
Θc.m.  (deg)
FIG. 4. (Color online) Observables of pd radiative capture at Ep = 100 MeV as functions of the c.m. p-γ scattering angle. Curves are as
in Fig. 1, and the experimental data are from Refs. [30] (circles) and [31] (squares).
further examples can be found in Refs. [10,27]. Finally, we
note that our nonrelativistic results, as documented by the
dash-dotted curves, are consistent with the results of Ref. [6]
derived from the different hadronic interactions and e.m.
currents.
One may conjecture that large spin-orbit charge effect
in low-energy Ay(p) and Ay(d) is a result of the high
momentum components present in the 3He wave function.
We therefore performed additional calculations with AV18
[28] and effective field theory N3LO [29] potentials, which,
although having quite comparable 3He binding energies—
6.923 MeV and 7.128 MeV, respectively—differ significantly
in the 3He D-state probability, 8.47% and 6.32%, and in the ex-
pectation value for the 3He internal kinetic energy, 45.68 MeV
and 33.78 MeV. However, the spin-orbit charge effect turns out
to be the same in both cases, thereby ruling out the preceding
conjecture. In fact, at low energy, only the matrix elements
〈3He|E1|pd(4PJ )〉 with J = 1/2 and 3/2, that is, the E1
transitions between the 4PJ pd scattering states and the 3He
bound state, calculated using either the Siegert form (as in
our standard procedure) or the explicit one- and two-baryon
spatial currents, are significantly affected by the relativistic
corrections. The nonrelativistic (spin-independent) charge
contribution for these matrix elements is suppressed compared
to the 〈3He|E1|pd(2PJ )〉 due to the total spin of the 3He
bound state being predominantly S = 1/2. As a consequence,
the spin-orbit charge becomes important, especially for the
〈3He|E1|pd(4P3/2)〉, which is responsible for about 90% of
the observed effect in the vector-analyzing powers, which are
very sensitive to this particular matrix element.
For curiosity, we show also in Fig. 4 the results at
considerably higher energies, that is, Ep = 100 MeV or Ed =
200 MeV, where not only the spin-orbit charge, but also the
relativistic corrections of the 1N spatial current, yield visible
effects for the differential cross section and spin observables.
The description of the experimental data is quite satisfactory,
although in this energy regime one can expect further changes
due to relativistic hadronic dynamics.
Next we consider the two-body photodisintegration of 3He
that is related to the pd radiative capture by time reversal.
In Fig. 5, we present results for the linear photon asymmetry
, the target analyzing power Ay(3He), and the beam-target
parallel-antiparallel spin asymmetry AP-A. The importance
of the spin-orbit charge is different for these observables:
 remains almost unaffected; Ay(3He) shows moderate effect
around the minimum; and AP-A is strongly affected, but
only at forward and backward angles, where the differential
cross section is small. The different effects are due to the
different sensitivity of the observables to the crucial matrix
elements 〈3He|E1|pd(4PJ )〉, as already encountered in the
radiative capture. The effect of the relativistic corrections
of the 1N spatial current is negligible for all considered
3He photodisintegration observables and therefore will be not
shown for the three-body breakup.
0.0
0.4
0.8
0 60 120 180
Σ
Θlab  (deg)
0.0
0.2
0 60 120 180
A y
(3 H
e)
Θlab  (deg)
-0.4
-0.2
0.0
0 60 120 180
A P
-A
Θlab  (deg)
FIG. 5. (Color online) Observables of 3He two-body photodisintegration at Eγ = 15.2 MeV as functions of the laboratory p-γ scattering
angle. Curves are as in Fig. 1.
064004-3
A. DELTUVA, A. C. FONSECA, AND P. U. SAUER PHYSICAL REVIEW C 80, 064004 (2009)
0
10
20
30
40
0 2 4d
3 σ
/d
Ω
N
 
dE
N
 
 
(µb
 M
eV
-
1 s
r-
1 )
EN  (MeV)
0.0
0.4
0.8
0 2 4
Σ
EN  (MeV)
-0.1
0.0
0.1
0 2 4
A y
(3 H
e)
EN  (MeV)
FIG. 6. (Color online) Observables of 3He three-body photodisintegration at Eγ = 15.2 MeV as functions of the detected nucleon laboratory
energy for the nucleon laboratory scattering angle N = 30◦. Results obtained with nonrelativistic e.m. current are given by thick (thin)
dash-dotted curves for the detected proton (neutron), whereas the results including relativistic spin-orbit charge are given by thick (thin) solid
curves for the detected proton (neutron). Solid and dash-dotted curves lie almost on top of each other.
Finally, in Figs. 6 and 7, we show our predictions
for the semi-inclusive observables of the 3He three-body
photodisintegration. We note in passing that the inclusion
of the Coulomb interaction for this reaction is—among all
reactions considered in this article—most important; that
fact is also explicitly documented in Refs. [24,32]. The
inclusion of the spin-orbit charge yields no visible changes
in the semi-inclusive differential cross section and in the
spin observables  and Ay(3He) but leads to a significant
effect for the asymmetry AP-A, which, in contrast to the
two-body photodisintegration, is seen at all scattering angles
of the detected proton or neutron. Note that the corre-
sponding observable in the deuteron photodisintegration is
affected by the relativistic 1N current corrections in a similar
way [9].
IV. SUMMARY
We performed calculations of 3N photonuclear reactions
at low energies and studied the effects of various relativistic
corrections to the 1N e.m. current operator. The spin-orbit
charge is the only important correction when the Siegert form
of the electric multipoles is used. The realistic coupled-channel
potential CD Bonn +  with Coulomb and the corresponding
e.m. current, including relativistic 1N corrections, are quite
successful in accounting for the existing pd radiative capture
data. Sizable and beneficial effects of the spin-orbit charge
were found for proton and deuteron vector-analyzing powers,
even at low energies, where the corrections owing to relativistic
hadronic dynamics are expected to be small. With increasing
energy, the effect of the relativistic 1N current corrections
becomes visible for other observables, as well.
0.0
0.1
A P
-A
Eγ = 11.4 MeV
0.0
0.1
Eγ = 15.2 MeV
ΘN = 30
o
0.0
0.1 Eγ = 20.6 MeV
0.0
0.1
0 2
A P
-A
EN  (MeV)
0.0
0.1
0 2 4
EN  (MeV)
ΘN = 90
o
0.0
0.1
0 2 4 6 8
EN  (MeV)
FIG. 7. (Color online) Parallel-antiparallel spin asymmetry of 3He three-body photodisintegration at Eγ = 11.4, 15.2, and 20.6 MeV as a
function of the detected nucleon laboratory energy for (top) N = 30◦ and (bottom) N = 90◦. Curves are as in Fig. 6.
064004-4
RELATIVISTIC CORRECTIONS OF ONE-NUCLEON . . . PHYSICAL REVIEW C 80, 064004 (2009)
Among the observables of the two- and three-body pho-
todisintegration of 3He, the beam-target parallel-antiparallel
spin asymmetry AP-A appears to be most sensitive to the
relativistic 1N current corrections. Polarization data do not yet
exist for these reactions, but the corresponding experiments
are under way at the HIGS facility. The upcoming data
will provide crucial tests for the chosen hadronic and e.m.
dynamics.
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064004-5


Relativistic corrections of one-nucleon current in low-energy three-nucleon photonuclear reactions

Institutionelles Repositorium der Leibniz Universität Hannover

PHYSICAL REVIEW C 80, 064004 (2009)Relativistic corrections of one-nucleon current in low-energy three-nucleon photonuclear reactionsA. Deltuva,1 A. C. Fonseca,1 and P. U. Sauer21Centro de Fı´sica Nuclear da Universidade de Lisboa, P-1649-003 Lisbon, Portugal2Institut fu¨r Theoretische Physik, Leibniz Universita¨t Hannover, D-30167 Hannover, Germany(Received 19 November 2009; published 22 December 2009)Proton-deuteron radiative capture and two- and three-body photodisintegration of 3He at low energy aredescribed using realistic hadronic dynamics and including the Coulomb force. The sensitivity of the observablesto the relativistic corrections of the one-nucleon electromagnetic current operator is studied. Significant effectsof the relativistic spin-orbit charge are found for the vector-analyzing powers in the proton-deuteron radiativecapture and for the beam-target parallel-antiparallel spin asymmetry in the three-body photodisintegration of 3He.DOI: 10.1103/PhysRevC.80.064004 PACS number(s): 25.20.−x, 21.45.−v, 24.70.+s, 25.40.LwI. INTRODUCTIONElectromagnetic (e.m.) reactions in the three-nucleon (3N )system have been extensively studied in the past with the aimof testing various models of the hadronic two-nucleon (2N )and 3N interaction and of the nuclear e.m. current [1–6].In the nucleon-deuteron (Nd) radiative capture at higherenergies [4], we observed quite a significant effect of therelativistic spin-orbit correction of the one-nucleon (1N )charge operator. This is consistent with the findings ofRefs. [7–9] for the photodisintegration of the deuteron.However, quite surprisingly, we found a sizable effect of thesame relativistic spin-orbit correction also for the proton (p)and deuteron (d) vector-analyzing powers in the low-energypd radiative capture [10]. The aim of this article is to calculatethe observables of the low-energy 3N photonuclear reactions.For pd radiative capture, we compare with existing data.For two- and three-body photodisintegration of 3He, wemake predictions for experiments that are in the process ofbeing performed at the High Intensity Gamma-ray Source(HIGS) facility [11]. We study the sensitivity of observablesto relativistic corrections of the 1N charge and spatial current.In Sec. II, we describe the chosen model for the hadronicinteraction and e.m. current. Selected results for the pdradiative capture and photodisintegration of 3He are presentedin Sec. III. We summarize in Sec. IV.II. DYNAMICSThe hadronic dynamics is based on the realistic two-baryoncoupled-channel potential CD Bonn +  [12] that allows for avirtual excitation of a nucleon to a  isobar and thereby yieldseffective many-nucleon forces [13]. The nonrelativistic part ofthe e.m. current is taken over from Ref. [4] and has one-baryonand two-baryon pieces. Besides the standard purely nucleonicpart, there are additional parts involving the  isobar, whichthen make effective 2N and 3N contributions to the e.m.current that are consistent with the effective 2N and 3N forces.Since the underlying potential CD Bonn +  is based on theexchange of standard isovector mesons π and ρ and isoscalarmesons ω and σ , the same meson exchanges are contained inthe effective nucleonic forces and currents. The initial and final3N bound and scattering states are described in the isospinformalism by the exact Faddeev and Alt, Grassberger, andSandhas three-particle equations [14], respectively, that aresolved in momentum space using partial wave decomposition[15]. The Coulomb interaction between charged baryonsis important in the considered low-energy reactions and isincluded using the method of screening and renormalization[16–18]. The current is expanded in electric and magneticmultipoles. The magnetic multipoles are calculated from theone- and two-baryon parts of the spatial current. The electricmultipoles use the Siegert form of the current without the long-wavelength approximation; assuming current conservation,the dominant parts of the one-baryon convection current andthe diagonal π - and ρ-exchange current are taken into accountimplicitly in the Siegert part of the electric multipoles bythe Coulomb multipoles of the charge density; the remainingnon-Siegert part of the electric multipoles not accounted forby the charge density is calculated using explicit one- andtwo-baryon spatial currents. Although the continuity equationis not exactly fulfilled for the nonlocal potential CD Bonn + or when the relativistic 1N current corrections are included, theemployed calculational scheme, based on the Siegert form ofthe current, effectively corrects the current nonconservation, asargued in Ref. [19], and is therefore reliable for the observablesof low-energy photoreactions considered in this article. Wealso note that the use of the nonrelativistic hadronic dynamics,together with the relativistic current corrections, is somehowinconsistent. However, in the low-energy reactions, it is naturalto expect the effect of the relativistic hadronic dynamics tobe small; this expectation is consistent with the results ofRef. [20].To isolate the effects of various relativistic current correc-tions, we perform several calculations (all using CD Bonn + as the hadronic potential and including Coulomb) with adifferent choice of e.m. current:(i) The nonrelativistic one- and two-baryon e.m. currentsare used as in the standard calculations of Ref. [4],except for the parameters of the single nucleon-transition current, which are taken from Ref. [21]. Thecorresponding results in the figures are given by thedash-dotted curves.(ii) The relativistic spin-orbit correction to the 1N charge,as given by Eq. (A11a) of Ref. [4], is included in0556-2813/2009/80(6)/064004(5) 064004-1 ©2009 The American Physical SocietyA. DELTUVA, A. C. FONSECA, AND P. U. SAUER PHYSICAL REVIEW C 80, 064004 (2009)-0.10.00 60 120 180A y(d)Θc.m.  (deg)Ed = 10 MeV-0.10.00 60 120 180Θc.m.  (deg)Ed = 17.5 MeV-0.10.00 60 120 18Θc.m.  (deg)Ed = 29 MeVFIG. 1. (Color online) Deuteron vector-analyzing power Ay(d) for pd radiative capture at Ed = 10, 17.5, and 29 MeV as a function ofthe c.m. d-3He scattering angle. Results obtained with nonrelativistic e.m. current (dash-dotted curves), including relativistic spin-orbit charge(solid curves), and including, in addition, relativistic corrections to the spatial current (dotted curves), are compared with the experimental datafrom Refs. [25–27].addition to (i); it contributes to the Siegert term. Thecorresponding results in the figures are given by thesolid curves, consistently with Ref. [10].(iii) With respect to the relativistic Darwin-Foldy correctionto the 1N charge, there is a controversy in the literature:according to Ref. [22], it vanishes identically for thereactions with real photons (γ ), whereas it yields afinite contribution according to Ref. [23]. We includethe latter in addition to (ii); however, in all studied low-energy photoreactions, its effect turns out to be entirelynegligible and will therefore not be shown separately.(iv) In addition to (iii), relativistic corrections to the 1N spa-tial current from Ref. [23] are included; they contributeto the magnetic multipoles and to the non-Siegert partof the electric multipoles. The corresponding results inthe figures are given by the dotted curves.(v) In addition to (iv), the nucleon- transition charge thatis of the relativistic order, as given by Eq. (A11b) ofRef. [4], is included. However, the correspondingresults are not shown separately because the effect isentirely negligible.The technical details of our calculations are explained in Refs.[4,10,24].0.00.20.40 60 120 180A y(p)Θc.m.  (deg)FIG. 2. (Color online) Proton vector-analyzing power Ay(p) forpd radiative capture at Ep = 5 MeV as a function of the c.m. p-γscattering angle. Curves are as in Fig. 1, and the experimental dataare from Ref. [26].III. RESULTSWe start with the pd radiative capture, where significantrelativistic effects were predicted in Ref. [10]. In Fig. 1, weshow the deuteron vector-analyzing power Ay(d) at deuteronlaboratory energies Ed = 10, 17.5, and 29 MeV as a functionof the center-of-mass (c.m.) d-3He (or p-γ ) scattering angle.The effect of the 1N spin-orbit charge is sizable and clearlyimproves the description of the experimental data, whereas therelativistic corrections of the 1N spatial current contributingto the magnetic multipoles and to the non-Siegert part of theelectric multipoles yield only minor changes. Furthermore,it is interesting to note that the spin-orbit charge effectincreases with decreasing energy. A strong and beneficialeffect is seen also in the proton analyzing power Ay(p) atEp = 5 MeV proton laboratory energy in Fig. 2. In contrast,the differential cross section and deuteron tensor analyzingpowers at low energies remain almost unaffected by thespin-orbit charge, as can be seen from our previous results [10]at Ed = 6 MeV; those results are not documented in thisarticle. The spin-orbit charge effect shows up in the deuterontensor analyzing powers as the energy increases, especiallyat forward and backward angles, where the differential crosssection is small. In Fig. 3, we show Ayy at Ed = 45 MeV;0.00.10 60 120 180A yyΘc.m.  (deg)FIG. 3. (Color online) Deuteron tensor analyzing power Ayy forpd radiative capture at Ed = 45 MeV as a function of the c.m. d-3Hescattering angle. Curves are as in Fig. 1, and the experimental dataare from Ref. [27].064004-2RELATIVISTIC CORRECTIONS OF ONE-NUCLEON . . . PHYSICAL REVIEW C 80, 064004 (2009)0.00.10.20.30 60 120 180dσ/dΩ  (µb/sr)Θc.m.  (deg)-0.20.00 60 120 180A y(p)Θc.m.  (deg)-0.10.00.10 60 120 180A y(d)Θc.m.  (deg)FIG. 4. (Color online) Observables of pd radiative capture at Ep = 100 MeV as functions of the c.m. p-γ scattering angle. Curves are asin Fig. 1, and the experimental data are from Refs. [30] (circles) and [31] (squares).further examples can be found in Refs. [10,27]. Finally, wenote that our nonrelativistic results, as documented by thedash-dotted curves, are consistent with the results of Ref. [6]derived from the different hadronic interactions and e.m.currents.One may conjecture that large spin-orbit charge effectin low-energy Ay(p) and Ay(d) is a result of the highmomentum components present in the 3He wave function.We therefore performed additional calculations with AV18[28] and effective field theory N3LO [29] potentials, which,although having quite comparable 3He binding energies—6.923 MeV and 7.128 MeV, respectively—differ significantlyin the 3He D-state probability, 8.47% and 6.32%, and in the ex-pectation value for the 3He internal kinetic energy, 45.68 MeVand 33.78 MeV. However, the spin-orbit charge effect turns outto be the same in both cases, thereby ruling out the precedingconjecture. In fact, at low energy, only the matrix elements〈3He|E1|pd(4PJ )〉 with J = 1/2 and 3/2, that is, the E1transitions between the 4PJ pd scattering states and the 3Hebound state, calculated using either the Siegert form (as inour standard procedure) or the explicit one- and two-baryonspatial currents, are significantly affected by the relativisticcorrections. The nonrelativistic (spin-independent) chargecontribution for these matrix elements is suppressed comparedto the 〈3He|E1|pd(2PJ )〉 due to the total spin of the 3Hebound state being predominantly S = 1/2. As a consequence,the spin-orbit charge becomes important, especially for the〈3He|E1|pd(4P3/2)〉, which is responsible for about 90% ofthe observed effect in the vector-analyzing powers, which arevery sensitive to this particular matrix element.For curiosity, we show also in Fig. 4 the results atconsiderably higher energies, that is, Ep = 100 MeV or Ed =200 MeV, where not only the spin-orbit charge, but also therelativistic corrections of the 1N spatial current, yield visibleeffects for the differential cross section and spin observables.The description of the experimental data is quite satisfactory,although in this energy regime one can expect further changesdue to relativistic hadronic dynamics.Next we consider the two-body photodisintegration of 3Hethat is related to the pd radiative capture by time reversal.In Fig. 5, we present results for the linear photon asymmetry, the target analyzing power Ay(3He), and the beam-targetparallel-antiparallel spin asymmetry AP-A. The importanceof the spin-orbit charge is different for these observables: remains almost unaffected; Ay(3He) shows moderate effectaround the minimum; and AP-A is strongly affected, butonly at forward and backward angles, where the differentialcross section is small. The different effects are due to thedifferent sensitivity of the observables to the crucial matrixelements 〈3He|E1|pd(4PJ )〉, as already encountered in theradiative capture. The effect of the relativistic correctionsof the 1N spatial current is negligible for all considered3He photodisintegration observables and therefore will be notshown for the three-body breakup.0.00.40.80 60 120 180ΣΘlab  (deg)0.00.20 60 120 180A y(3 He)Θlab  (deg)-0.4-0.20.00 60 120 180A P-AΘlab  (deg)FIG. 5. (Color online) Observables of 3He two-body photodisintegration at Eγ = 15.2 MeV as functions of the laboratory p-γ scatteringangle. Curves are as in Fig. 1.064004-3A. DELTUVA, A. C. FONSECA, AND P. U. SAUER PHYSICAL REVIEW C 80, 064004 (2009)0102030400 2 4d3 σ/dΩN dEN  (µb MeV-1 sr-1 )EN  (MeV)0.00.40.80 2 4ΣEN  (MeV)-0.10.00.10 2 4A y(3 He)EN  (MeV)FIG. 6. (Color online) Observables of 3He three-body photodisintegration at Eγ = 15.2 MeV as functions of the detected nucleon laboratoryenergy for the nucleon laboratory scattering angle N = 30◦. Results obtained with nonrelativistic e.m. current are given by thick (thin)dash-dotted curves for the detected proton (neutron), whereas the results including relativistic spin-orbit charge are given by thick (thin) solidcurves for the detected proton (neutron). Solid and dash-dotted curves lie almost on top of each other.Finally, in Figs. 6 and 7, we show our predictionsfor the semi-inclusive observables of the 3He three-bodyphotodisintegration. We note in passing that the inclusionof the Coulomb interaction for this reaction is—among allreactions considered in this article—most important; thatfact is also explicitly documented in Refs. [24,32]. Theinclusion of the spin-orbit charge yields no visible changesin the semi-inclusive differential cross section and in thespin observables  and Ay(3He) but leads to a significanteffect for the asymmetry AP-A, which, in contrast to thetwo-body photodisintegration, is seen at all scattering anglesof the detected proton or neutron. Note that the corre-sponding observable in the deuteron photodisintegration isaffected by the relativistic 1N current corrections in a similarway [9].IV. SUMMARYWe performed calculations of 3N photonuclear reactionsat low energies and studied the effects of various relativisticcorrections to the 1N e.m. current operator. The spin-orbitcharge is the only important correction when the Siegert formof the electric multipoles is used. The realistic coupled-channelpotential CD Bonn +  with Coulomb and the correspondinge.m. current, including relativistic 1N corrections, are quitesuccessful in accounting for the existing pd radiative capturedata. Sizable and beneficial effects of the spin-orbit chargewere found for proton and deuteron vector-analyzing powers,even at low energies, where the corrections owing to relativistichadronic dynamics are expected to be small. With increasingenergy, the effect of the relativistic 1N current correctionsbecomes visible for other observables, as well.0.00.1A P-AEγ = 11.4 MeV0.00.1Eγ = 15.2 MeVΘN = 30o0.00.1 Eγ = 20.6 MeV0.00.10 2A P-AEN  (MeV)0.00.10 2 4EN  (MeV)ΘN = 90o0.00.10 2 4 6 8EN  (MeV)FIG. 7. (Color online) Parallel-antiparallel spin asymmetry of 3He three-body photodisintegration at Eγ = 11.4, 15.2, and 20.6 MeV as afunction of the detected nucleon laboratory energy for (top) N = 30◦ and (bottom) N = 90◦. Curves are as in Fig. 6.064004-4RELATIVISTIC CORRECTIONS OF ONE-NUCLEON . . . 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Relativistic corrections of one-nucleon current in low-energy three-nucleon photonuclear reactions

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