Extended Siegert Theorem in the Relativistic Investigation of the Deuteron Photodisintegration Reaction

0.16 MB·English

by  S. G. Bondarenko

EXTENDED SIEGERT THEOREM IN THE RELATIVISTIC
INVESTIGATION OF THE DEUTERON PHOTODISINTEGRATION
REACTION
S.G. Bondarenko∗ and V.V. Burov†
BLTP, Joint Institute for Nuclear Research, 141980 Dubna, Russia
K.Yu. Kazakov‡ and D.V. Shulga§
LTNP, Far Eastern National University, Sukhanov Str. 8, 690095 Vladivostok, Russia
Thecontributionofthetwo-bodyexchangecurrentisinvestigatedforthereactionofthedeuteron
photodisintegration in the framework of the Bethe-Salpeter formalism and with using extended
Siegerttheorem. Thistheoremallowtoexpressthereactionamplitudeintermsofextendedelectric
9 and magnetic dipole moments of the system. The resultant analytical expression is faultless with
0 respecttobothtranslationandgaugeinvariance. Itpermitstoperformcalculationsofthedeuteron
0 photodisintegration cross section and polarization observables taking into account two-body ex-
2 changecurrent implicitly.

n Keywords: deuteronphotodisintegrarion,Siegerttheorem,Bethe-Salpeter formalism.
a
J

5

] I. INTRODUCTION
h
t
-
l Oneofthemostimportantmethodofthenuclen-nucleoninteractioninvestigationisthereactionofthe
c
electronsandphotonsscatteringonthedeuteron. Inthepreviousworks[1],[2],[3]thereactionofdeuteron
u
photodisintegration has been studied on basis of the Bethe-Salpeter formalism within the framework of
n
[ one-bodyapproximation. Finalstateinteractionhasbeenconsideredbesides. Howeveritwasinsufficient
for complete description of experimental data in the wide range of energy. With increasing of photon
1
energy the significant contribution to differential cross section comes from meson exchange currents. An
v
appropriate method of description of such effects is the extended Siegert theorem [4], [5]. Let’s consider
5
it briefly.
2
4
0
. II. THE EXTENDED SIEGERT’S THEOREM
1
0
9 Generalization of Siegert’s theorem is made without conventional decomposition of the EM current
0 intotwoparts–theconvectioncurrentassociatedwiththe motionofnucleusasawholeandtheintrinsic
v: current [6].
i Firstonneedstoconstructthematrixelementfortheabsorptionofarealphotonwiththree-momentum
X
q, while the nuclear system makes a transition from an internal state P i to a final internal state
i
r P =P +qf . | i
a | f i i
The required matrix element is given by
Sγ = d4x P f j (x)P i 0 Aµ(x)1 . (1)
if Z h f | µ | i ih γ| | γi
Applying the invariance with respect to four translation we may write
j (x)=eıP·xj (0)e−ıP·x, (2)
µ µ
where P is the operator of total four momentum of the system, i.e.
P P i =P P i , P P f =P P f . (3)
i i i f f f
| i | i | i | i
Introducing Eq. (2) into Eq. (1) we find
Sγ = P f j (0) P i 0 Aµ(P P )1 . (4)
if h f | µ | i ih γ| f − i | γi
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected]
2
with
1
Aµ(q)= εµ(q) a (q)+a+(q) eıωt, (5)
√2ω s s s
s=X0,1,2,3 (cid:0) (cid:1)
where a (q) and a+(q) are the destruction and creation operators, respectively, for a photon of three-
s s
momentum q and unit four-polarizationvector ε. The transversality condition is ε q =0.
·
Now we will separate the matrix element into two parts using the identity
1
εe−ıq·x = dλ ∇ ε xe−ıλq·x +ıωλx ε′e−ıλq·x , (6)
x
Z · ×
(cid:8) (cid:0) (cid:1) (cid:9)
0
whereεandε′ =q ε/ω representtheunitelectricandmagneticpolarizationvectors,respectively. This
×
identity is equivalent to a gauge transformation of the EM potential in QED
AT(x) A (x)=AT(x) ∂ Λ(x), (7)
µ → µ µ − µ
where the superscript T on A indicates it is in the transverse gauge,i.e. AT =(0,εTe−ıq·x), and Λ(x) is
such an function of x that the current has the Siegert limit. Suppressing the time dependence it means
that
lim Λ(x)=ε x. (8)
ω→0 ·
Thus the identity (6) is a gauge transformation with the following choice for Λ(x) (Foldy gauge [7])
1
Λ(x)=ε x dλe−ıλq·x =ε x 1 1ıq x 1(q x)2+O(q3) . (9)
· Z ∼ · − 2 · − 6 ·
(cid:0) (cid:1)
0
In principle, it should make no difference for the final result at low energies. However, in approximate
many-body calculations with j not necessarily conserved, the choice of gauge does become important.
µ
The Foldy gauge has the good theoretical property that A(x) projects out from j(x) only the magnetic
part, i.e. all magnetic effects are contained in j (x). It means that the knowledge of the nonrelativistic
m
one-body nucleon charge density ρ is optimized in the Foldy gauge.
(1)
Substituting Eq. (6) into Eq. (4), one has
P f j (0)P i =ı(E E )D (q) ıq M (q), (10)
f µ i f i if if
h | | i − − ×
where q = P P , E and E is the total energy of the nuclear system in the initial and final states.
f i i f
−
The quantities D (q) and M (q) has the form
fi fi
1
D (q) = dλ dx x ρ (x;P ) eıλq·x, (11)
if if i
Z Z
0
1
M (q) = dλ dx λ x j (x;P ) eıλq·x. (12)
if Z Z × if i
(cid:2) (cid:3)
0
with
hPf f|j(0)|Piii = Z dx eı(Pf−Pi)·xjif(x;Pi), (13)
Pf f ρ(0)Pii = dx eı(Pf−Pi)·xρif(x;Pi). (14)
h | | i Z
Reversing Eqs. (14) and (13), we verify that
1
j (x;P ) = dp e−ıp·x p+P f j(0)P i , (15)
if i (2π)3 Z h i | | i i
1
ρ (x;P ) = dp e−ıp·x p+P f ρ(0)P i . (16)
if i (2π)3 Z h i | | i i
3
From Eqs. (11) and (14) we immediately find
ıq D (q)= P f ρ(0)P i P f ρ(0)P i . (17)
if f i i i
· h | | i−h | | i
Atthis pointwearereadyto writetheS-matrixelementinterms ofthe matrixelementsoperatorsD(q)
and M(q). Substituting Eq. (10) into Eq. (4) and using the relation (17) we obtain
Sγ = P f ρ(0)P i 0 φ(q)1 +ı[q 0 φ(q)1 ω 0 A(q)1 ] D (q)
if h i | | i ih γ| | γi h γ| | γi− h γ| | γi · if
+ ı[q M (q)] 0 A(q)1 . (18)
if γ γ
× ·h | | i
Finally introducing the strength of the electric and magnetic fields (in momentum space)
E(q) = ıωA(q) ıq φ(q), (19)
−
H(q) = ıq A(q), (20)
×
we cast the S-matrix element in the manifestly gauge independent form
S = 0 E(q)1 D (q) 0 H(q)1 M (q)+ P f ρ(0)P i 0 φ(q)1 . (21)
iγ→f γ γ if γ γ if i i γ γ
−h | | i· −h | | i· h | | ih | | i
The scattering amplitude for the photon absorption is written in gauge independent form
Tγ = 0 E(0)1 D (q) 0 H(0)1 M (q) (22)
if −h γ| | γi· fi −h γ| | γi· fi
with
1
0 E(0)1 = ωε(q) qε (q) , (23)
γ γ 0
h | | i √2ω −
(cid:0) (cid:1)
1
0 H(0)1 = q ε(q). (24)
γ γ
h | | i √2ω ×
The last term in Eq. (21) is equal to zero, since it is proportional to the matrix element of the total
charge between orthogonal states. This equation is consistent with requirements of gauge invariance as
well as invariance with respect to translations.
III. APPLICATION TO RELATIVISTIC FORMALISM
Now we shall write down T-matrix in Siegert’s form within Bethe-Salpeter formalism (21)
1 1
Tγ = ωε(q) qε (q) D (q) [q ε(q)] M (q). (25)
if −√2ω − 0 · if − √2ω × · if
(cid:0) (cid:1)
The operators D (q) and M (q) could be obtained from Eqs. (11) and (12) with the use of Eqs. (15)
fi fi
and (16)
1
dλ
D (q) = ı λq+P f ρ(0)P i , (26)
if q i i
− Z λ ∇ h | | i
0
1
M (q) = ı dλ λq+P f j(0)P i . (27)
if q i i
− Z ∇ ×h | | i
0
Substituting expression for the matrix element of the EM current between two relativistic states
dsdk
P f jµ(0)P i = ı χ¯ (s) Λµ(s,k;P ,P ) χ (k), (28)
h f | | i i Z (2π)8 Pf f i Pi
into Eqs. (26) and (27) we obtain
1
dλ d4sd4s′
D (q) = χ¯ (s) Λ0(s,s′;P +λk,P ) χ (s′), (29)
if Z λ ∇qZ (2π)8 Pi+λq i i Pi
0
1
d4sd4s′
M (q) = dλ χ¯ (s) Λ(s,s′;P +λq,P ) χ (s′). (30)
if Z ∇q×Z (2π)8 Pi+λq i i Pi
0
4
Using the transverse gauge in calculations, i.e. εT =0 and εT q=0, one finds that
0 ·
Tγ = ω εT D (q) 1 εT′ M (q), ρ= 1. (31)
if −r2 ρ · if − √2 ρ · if ±
One concludes that this matrix elements gives response of the nuclear system in two transverse perpen-
dicular directions (defined by the three-vectors ε and ε′) with respect to the photon three-momentum
q.
IV. RESULTS AND DISCUSSIONS
The results of our calculations are depicted in the Fig. 1 and Fig. 2. The calculations has performed
in nonrelativistic (Shrodinger equation)and relativistic (Bethe-Salpeter equation) models. In both cases
weuseGrazIIpotentialofnucleon-nucleoninteractionandcarryoutthecomputationswithandwithout
including two-bodycurrenteffectivelyviaextendedSiegerttheorem. We haven’ttakenintoaccountfinal
state interaction in this work. Notations are explained in the figure’s captions.
It is seen in the first plot of Fig. 1 that two-body effects give a large contribution to the differential
cross section even at the photon energy equal to 20 MeV. One-body approximation is 30% less than
experimental data. Calculations with using extended Siegert theorem allow to agree with experimental
data.
Relativistic effects give larger contribution to two-body current than to one body current. We can see
that practically for all the plots at the figures, particularly for T at 200 MeV. For T we can see large
22 20
contribution from relativistic effects even at photon energy equal to 20 Mev.
[1] K.Yu.Kazakov and S.Eh. Shirmovsky,Phys. Rev.C. 2001. V.63. P. 014002.
[2] K. Yu.Kazakov and D.V. Shulga, Phys.Rev.C. 2002. V. 65. P.064002.
[3] S.G.Bondarenko,V.V.Burov,K.Yu.Kazakov,D.V.Shulga.Phys.Part.Nucl.Lett.2004. V.1.N.4(121).
P. 17.
[4] A.F. Siegert, Phys. Rev.52 (1937) 787.
[5] J.L. Friar and S.Fallieros, Phys. Rev.C29 (1984) 1645.
[6] A.V. Shebeko,J. Nucl. Phys. 49 (1989) 1.
[7] L.L. Foldy,Phys.Rev. 92 (1953) 178.
[8] H. Arenho¨vel and M. Sanzone, Few Body Syst.Suppl.3 (1991) 1.
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(cid:10)(cid:11)(cid:6)(cid:16)(cid:17)(cid:1)(cid:18)(cid:1)(cid:11)(cid:19)
(cid:1)(cid:3)(cid:4)(cid:5)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) (cid:11)(cid:6)(cid:12)(cid:10)(cid:1)(cid:6)(cid:13)(cid:12)(cid:14)(cid:15)
(cid:2) (cid:11)(cid:6)(cid:16)(cid:17)(cid:1)(cid:18)(cid:1)(cid:11)(cid:19)
(cid:20)(cid:21)(cid:22)(cid:23) (cid:17)
(cid:5) (cid:23)(cid:5) (cid:21)(cid:5)(cid:5) (cid:21)(cid:23)(cid:5) (cid:18) (cid:18)(cid:22)(cid:17) (cid:18)(cid:22)(cid:23) (cid:18)(cid:22)(cid:24) (cid:18)(cid:22)(cid:25) (cid:18)(cid:22)(cid:26)
(cid:24)(cid:25)(cid:6)(cid:26)(cid:8)(cid:27) (cid:19)(cid:14)(cid:3)(cid:11)(cid:20)(cid:21)
FIG. 2: Tensor target asymmetry T22 and total cross section. Notation of the curves: nonrelativistic one-body
current (long-dashed); nonrelativistic two-body current (dash-dotted); relativistic one-body current (dotted);
relativistic two-body current (full).