823
Progress of Theoretical Physics, Vol. 94, No.5, November 1995
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
N aofusa HONZA W A
Atomic Energy Research Institute, College of Science and Technology
Nihon University, Tokyo 101
(Received July 20, 1995)
We investigate the mean square radius of the deuteron in the Bethe-Salpeter formalism with
Lorentz covariance. All possible relativistic effects are considered within the additive current
concerning the constituent nucleons. We argue the possibilities to explain the discrepancy in the
mean square radius of the deuteron between experiment and non-relativistic nuclear potential models.
It is found that the relativistic corrections cannot account for the discrepancy, since the relevant
relativistic corrections cancel each other. For the relativistic corrections to the non-relativistic
theory, it is necessary to have at least a consistent treatment of both the equation and the electromag·
netic current of the deuteron as a bound system. Furthermore, we estimate the contributions of the
off-shell effects for the photon-nucleon interaction vertex. These off-shell effects may be candidates
to explain the discrepancy.
§ 1.
Introduction
The mean square (MS) radius of the deuteron rm 2 is a characteristic quantity
derived from the deuteron wave function. Hence, the calculation of the MS radius
may be a good check of the nucleon-nucleon potential models. In non-relativistic
(NR) nuclear theory, rm 2 is given 1l as
1 ("'
rm 2 =4Jo r 2 [u(r) 2 +w(r) 2 ]dr,
(1·1)
where u(r) and w(r) are, respectively, the usual radial S-and D-wave functions of
the deuteron. In typical NR potential models, the theoretical value of rm 2 is given as
rm 2 =(1.95~1.98fm) 22 J, 3 J (the value depends somewhat on the model used). On the
other hand, the value of rm 2 (r~xpt) is experimentally derived from the observation of
the MS charge radius r2h; for there exists the following relation 4l between r2h and rm 2
with the proton charge radius rp=0.862(12) fm 5 l and the MS neutron charge radius rn 2
= -0.1192(18) fm 2 : 6l
(1·2)
where the final term represents the relativistic correction for Zitterbewegung, and we
have assumed that the isospin symmetry takes the mass m for both the proton and
neutron. For extremely low momentum transfer, it is possible to obtain the value of
~xpt from the experimental data with good confidence by measuring the ratio of the
cross section of the elastic electron-deuteron scattering to that of electron-proton
scattering. One of the most widely accepted values is given as dxpt =(1.9546(47)
fm) 2 _7l- 9 J Thus the discrepancy between NR potential models and experiment is
(1·3)
824
N. Honzawa
It has been reported 9 > that NR potential models are unable to explain this discrepancy
simultaneously with the other low energy parameters of the proton-neutron system.
The main efforts to explain this discrepancy have been made along two directions,
taking into account the relativistic effects, or the effects of meson-exchange current
(MEC). 10 > In order to make any meaningful quantitative prediction for electromagnetic processes, it is indispensable to have an effective form of the conserved current
for the deuteron system, and in doing so its covariant description is necessary. In a
previous work/ 0 we have investigated the magnetic and quadrupole moments of
deuteron in the BS formalism. We have shown that the well-known difficulty concerning the magnetic and quadrupole moments in the NR nuclear potential models
may be solved through a nonperturvative relativistic effect, the "normalization
effect". That is, if we assume an admixture of negative energy amplitudes with
magnitudes of a few hundredths of positive amplitudes, we can fit both the experimental values of the magnetic and quadrupole moments at the same time with a single
value of the D,wave state probability.
In this paper we investigate all possible relativistic effects on the MS radius of the
deuteron in the BS formalism with special attention on maintaining Lorentz covariance. We discuss the possibilities to explain the discrepancy in Eq. (1 · 3). Here, it
should be noted that some important parts out of the MEC effects such as "pair"
current in the NR nuclear theories are automatically included in our covariant
treatment.
This paper is organized as follows. In the next section, we review the description of the deuteron in the BS formalism so far as required for our problem. In § 3,
we present the qualitative analysis on the MS radius of deuteron and argue the
possibilities to explain the discrepancy. In § 4, the details of the relativistic corrections to the MS radius of the deuteron are studied by semi-quantitative analysis. The
off-shell effects in the photon-nucleon interaction vertex are also discussed. Concluding remarks will be given in the final section.
§ 2.
Bethe-Salpeter formalism
In the bound-state BS equation for the deuteron, the BS amplitude xa(P; P)
satisfies 12 >
(2 ·1)
where G(p- p') is the meson-nucleon interaction kernel, and the inverse of the nucleon
propagator (Dirac operator) for the nucleon momentum Pr,. is given as
(2·2)
where P,.=(PI + Pz),. and p,.=(1/2)(p~- Pz),. denote the total four momentum and the
relative one, respectively: Ma is the mass of a deuteron and P/=- Mi: a represents the other quantum numbers, such as angular momentum, of the deuteron state.
It is well-known that the BS equation can be reduced to the Shrodinger equation in the
non-relativistic limit, v~ c.
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
825
In general, the BS amplitude has sixteen components, because the constituent
nucleons are the Dirac particles. Here we carry out the partial wave decomposition
of the BS amplitude in the rest frame of deuteron in terms of the product of the free
Dirac spinor u~:(p;) of the constituent nucleon as follows:
(2·3)
where p<o>=(O, iMd), p=(pt, p2) and s=(st, s2); the index s;( = ±1/2) and p;( = ±1)
denote the spin and energy spin states, respectively, of the ith nucleon. In Eq. (2·3)
the component amplitude ( qJ/(p, Po)) is represented by a sum of the eight partial wave
component amplitudes with the spectroscopic number aP= 25 + 1L/''P',
(2·4)
where S, L and ] denote the total spin, orbital angular momentum, and the total
angular momentum, respectively. Here the reduction of the numbers from the initial
sixteen for x to eight for qJ is due to parity conservation. It should be noted that only
the first two amplitudes with 3 St ++ and 3 Dt ++ states correspond to the usual states in
the NR potential models. According to our previous work/ 1 > the positive energy
amplitudes for the ordinary one boson exchange (OBE) interaction picture are expected to be damped for the region of IPI and Po:
(2·5a)
and for the negative energy amplitude:
qJP(p, Po)__. 0
for
IPI2 /-lB ,
Po 2 /-lB
(2·5b)
,
where p = (-, - ), ( +, -) or (-, + ). Here f-lB stands for the mass of the exchange
boson in the meson-nucleon interactions, and M=Md/2.
We assume that the electromagnetic (EM) interaction of the deuteron is given as
a sum of the individual interactions of constituent nucleons. Due to the invariance of
the BS equation for the exchange of constituent nucleon one and nucleon two as the
isoscalar nature of the deuteron, the conserved EM current of the deuteron is given as
fp.(a',al(P', P)=- ieNd jd 4PXa'(p'; P')rp.(ll(q)SF( 2 )(
~ P- p
r
1
Xa(P; P)
I
(2·6)
where p'i=p+q/2, P'=P+q, Nd=1/(27r)\12Po2Po and f is the conjugate of x. It
should be noted that the effective EM current in Eq. (2·6) satisfies the current
conservation as q,.jp.<a',a>(P', P)=O. The photon-nucleon interaction vertex F~'- 0 >(q) is
given by the on-shell form
rp.(l>(q)= rp.< 1 >Ft(q 2 ) - ~ a1!JqvF2(q 2),
2
(2. 7)
where Gp.v=(rp.rv- rvrp.)/2i and Ft(q 2 ) (F2(q 2 )) represents the isoscalar Dirac (Pauli)
form factor of nucleons with Ft(O)=F2(0)=1 and K= Kp+ Kn (J-lp/2m=(1 + Kp)/2m and
f-lnl2m= Kn/21'!l stand for the proton and neutron magnetic moments, respectively).
We need the BS amplitude for deuteron in the general moving frame in order to
826
N. Honzawa
calculate the matrix element of the deuteron current in Eq. (2·6). For this purpose,
we describe the transformation property of the BS amplitude under the Lorentz
transformation A,..JJ, namely,
(2·8)
(2·9)
where cP> is the Pauli spin matrix of the ith nucleon, and E=/P 2 +Mi.
The BS amplitude should be normalized so that the total charge of the deuteron,
given by the EM current Eq. (2·6), becomes correctly ed=e. By using the formula
e=limq-o]o< 1' 1>(P', P) and Eq. (2·3), after some calculations, we can write the normalization condition of the BS amplitude as
with the definition of "pseudoprobability"
(2·11)
where ¢/(IPI, Po) represents the radial BS amplitude, and we have used the simplified
notation Icts-:-1 =I Cs +I representing the pseudoprobability of the 3 St ++ state of the
deuteron, and in the same manner IC:ls~I=ICs-I.IC:fo~I=ICo±l, IC:fp';I=IC:Jpl and ICrp';l=
IC!PI. Thus we see that the inclusion of negative energy states into the amplitude
necessarily reinforces the contribution of the positive energy states. In our previous
work, 11 > we have shown by using the ordinary OBE interaction kernel that the
pseudoprobability for 3 s~-- partial states is dominant over the other negative energy
states as p_ :=::::I Cs -1 2 :=: : 0.015. Therefore, for the MS radius, it is expected that the
contributions of the negative energy states, except for the 3 s~-- partial state, are much
smaller than the discrepancy. In the following analysis, we are exclusively concerned with the contribution of the 3 s~-- partial state.
§ 3.
Qualitative analysis of the MS radius of deuteron
In this section we shall make a general analysis of the MS radius of deuteron,
taking into account the qualitative behavior of the component amplitude in Eq. (2·5).
The MS charge radius r~h in the BS formalism is given by the following
formula: 13 >
a ]o
-e1 lj:rl aq2
2
r~h=
0 1
' >
(q/2, - q/2),
(3·1)
where ]o< 1•1>(q/2, -q/2) is the charge density in the Breit frame deriving from the EM
current in Eq. (2·6). In Eq. (3·1), we take the polarization of deuteron as a'=a=1
(with the third axis as the direction of quantization), reflecting our physical situation.
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
827
The Breit frame is defined by p~<B>=(q/2, iPo) and P~8 >=( -q/2, iPo). By substituting
Eq. (2·8) into Eq. (2·6), the EM current in the Breit frame is represented in terms of
the BS amplitude in the rest frame as follows:
fp.(p'< 8 >, p<Bl)=.fp.(q/2, -q/2)
= - ieNd jd 4PX(P';
r
~ p<o>_ p X(P; p<o>)
1
p<O>)f,..(q)SF(2)(
(3·2)
with f,.. 10 (q)=S 10 (A)S< 2>(A)r,..< 0 (q)S 0 >(A)S 12 >(J1), and the relative momentum p~ is
related by the boost transformation p'=A 2P+(1/2)Aq. Furthermore, by using the
decomposition of the BS amplitude in Eq. (2·3), we obtain the final expression of the
EM current of deuteron as
4
1
J,..(q/2,- q/2)=- ieNd L: L: (d
1 P&~:(p', Po)f~>:.:t(p', p, q)f/J/(p, Po)S~;>(pz)p',s' p,s}
(3·3)
with
f~>:.:t(p', p, q)= iu~{(p)®u~H- p)T,.. 10 (q)r4< 2 >u~i(p)®u~~(- p),
(3·4)
where s~;>(pz)- 1 =pzEp-M+Po and Ep=Jp 2 +m2 • In most of the conventional
relativistic models, the process of resorting to the Breit frame is missing, and the
effective current in the rest frame, aside from the conservation of the space momentum, has been directly applied for the static charge and current densities.
By substituting Eq. (3 · 3) into Eq. (3 ·1), the MS charge radius r~h in the BS
formalism is given as
(3·5)
where
(3·6a)
(3·6b)
(3·6c)
Here, <rA 2 ), <rv 2 ) and <dv> include various relativistic effects in our covariant
treatments. <rA 2>and <r/> correspond to the MS radius in the NR formula Eq. (1·1)
and rch- rm 2 , respectively. We obtain the following form for <rA 2
):
(3·7)
where
828
N. Honzawa
2
<rA )NP=
< rA 2>ret =
~- Mi
1
r=
iNd~~d 4P[!
:;,2 ifJl(p, Po) J(1)/(p, Po)S~2,>(p2)- 1 ,
4
iNd"i}.
p,s}(d
• P[ fifJ/(p, Po)] (1)/(p,
{< -
(3·8a)
Po)S~~>(p2)- 1 ,
(3·8b)
;p 3(M- -Po ) apo
a +p2ap
a2 2
Po + Po2) ap20
2M-
+p·_2_[1-2(MPo) _LJ}
op
oPo '
(3·8c)
by using the formulae
.
hm
q~O
a2 P~
_ . M- Po
aQj2 -_ __j!J__
M 28p.j z M 2 op.4 .
d
(3·9)
d
By carrying out the angular integral and transforming the spherical Bessel function
jL(Pr) as
(3·10)
we obtain the following Fourier conjugate spatiotemporal expression of
Eq. (3·8a) for 3 StP and 3 DtP states:
2
<rA )NP=-
iNi~!
£""1:drdroiaP(r, ro)r 1/Jl(r, ro)(
2
Pt! p
2
m-
M),
2
<rA )NP
is
(3·11)
where we have used the non-relativistic approximation Er-> m and Po---> 0 for S~~>(p2)- 1
and the isoscalar nature of deuteron. It is clear that formula (3·11) with p=( +, +)
corresponds to the non-relativistic Eq. (1·1). For the contribution from the 3 s~-­
state, it is expected to be
<rA 2 >_
~
NP~-
<-_2>-p
_ <r>- p-rm 2~
r
-----2-
rm
m" ) p-rm 2~
-2p-rm 2 ,
~- 10
2
(
~--
mp
(3 ·12)
where m" and mp are the mass of the pion and rho-meson, respectively. Thus the
contribution for negative energy state becomes negligible in regard to the discrepancy. As a result, <rA 2)NP may be estimated as
(rA 2 )Np=(ri)r!ip+(rA 2 )NP~ rm 2(P+ -10- 2p_)=rm 2(1 + p_ -10- 2 P-)~ rm 2(1 + p_).
(3·13)
Thus, if we have an admixture of negative energy amplitudes with pseudoprobability
p_ into positive ones, the MS radius rm 2 in Eq. (1·1) is modified through the normalization condition in Eq. (2·10). This effect is called the normalization effect. It is
essential for the normalization effect that the contributions of the negative energy
states themselves are negligible to the extra contribution through the normalization
condition.
The second term <ri>rel in Eq. (3·7) and <r~v> in Eq. (3·5) represent the Lorentz
deformation effects, such as the relativistic effects for the relative motion of constituent nucleons, the Lorentz contraction and retardation, depending on the details of the
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
829
BS amplitude. These terms may cancel each other (see the discussion to be given in
§ 4), and thus may be negligible in comparison to the discrepancy, although quite
interesting theoretically. Thus we neglect these terms.
For the <rl> term (see Appendix A) we get the following formula for the case of
3
3
StP and DtP states:
(3 ·14)
It may be worth while to note that the relativistic correction for the Zitterbewegung
is deduced from this term. By substituting Eqs. (3·13) and (3·14) into Eq. (3·5), we
obtain the final result of the charge radius of the deuteron in the BS formalism
corresponding to Eq. (1· 2) as
aris=rm 2 P-+
4 ~zP-+ 2 ~zP-+- 8 ~ 2 (1+2K)
3
m~M
(P++P-).
(3 ·15)
This formula is identical with the usual one in Eq. (1·2); by putting P+=1 (P-=0) and
then 81is=(3/8m2 )(1+2K)(m-M)/M:=::::l0- 3 /m 2 ), which is negligible in regard to the
discrepancy. However, from our previous analysis 11 ) as p_::::::lcs-1 2 ::::::0.015, there is
the contribution out of the normalization effect, and then we numerically obtain the
final result as follows:
(3·16)
where <rm 2)NR and <rm 2)ss are the redefined MS radii in the NR nuclear theory and the
BS formalism, respectively. From Eq. (3 ·15), these quantities are related as
(3 ·17)
Summarizing the results of the above analysis in this section, we conclude that
most of the relativistic corrections to the NR MS radius, except for the normalization
effect, are numerically too small to explain the discrepancy. For the pseudoprobability p_ :=::::I Cs -1 2 :=: : 0.015, the order of magnitude of the extra contribution from
the normalization effect is favorable, but its correction has the wrong sign. It seems
that the situation in the NR potential models does not make a change for the better.
§ 4.
4.1.
The possibilities to explain the discrepancy
Semi-quantitative estimate to the relativistic corrections
In the previous section we do not take into account the contributions of both
In order to estimate to their contributions, we need a concrete
expression of the BS amplitude depending on the interaction kernel. It is difficult,
however, to analytically solve the BS equation, even in the case of the OBE interaction kernel. In this subsection we shall make a semiquantitative analysis to estimate
their contributions, reproducing the low energy behavior of deuteron. It is well<rA 2)rei and <r~v>.
830
N. Honzawa
known that the BS equation is reduced to the Salpeter equation 14> by making the
instantaneous approximation as Po- P'o=O for the interaction kernel G(p- p').
Furthermore, if we neglect the negative energy states and make the NR approximation Ep=/p2 +m2 ~m+p2 /2m, the Salpeter equation coincides with the Shrodinger
equation.
From this fact, we assume, so far as an analysis of the MS radius, that the
Salpeter amplitude lf!s ++(p) for the positive energy states is approximately equal to
the NR wave function of deuteron (u(p), w(p)). Furthermore, we neglect the contri·
butions of the negative energy states, except for the normalization effect, from the
analysis in the previous section. The relationship between the BS amplitude
(j)s ++(p, Po) and the Salpeter amplitude lfls ++(p) for the positive energy states is given
as
(j)s ++(p, Po)=[Sp] lfls ++(p), [Sp]=
;i
(Ep- ;;)-:--~
02_ ie ·
(4 ·1)
By substituting Eq. (4·1) into Eq. (3·5) and carrying out the integral with respect to
the relative energy Po, we obtain the following formula:
(4 ·2)
where <rm 2 >s corresponds to rm 2 in Eq. (1·1). arA 2 , adel, arh and arv 2 denote the
relativistic corrections for <rA 2 >NP, <rA 2 >rel, <rh> and <r/)-3/4m2 in Eqs. (3·5) and
(3·7), respectively. These explicit forms are given in Appendix B. We here tenta·
tively use the wave function in the Paris potential, 3 > which is widely accepted in the
NR potential model, to the Salpeter amplitude lJ!++(p). By using the formulae (B·2)
~(B·6) and carrying out numerical calculations, we obtain as follows in the case p_
~ ICs -~ 2 ~0.015,
<rm 2 >s=<rm 2 >Parls(l + P-)=(1.986 fm) 2
8r;el=0.01666 fm 2 ,
8r~v=
'
-0.01867 fm 2
arA 2 = -1.006 X lo-s fm 2 '
,
Br/=5.061 X 10- 4 fm 2 ,
(4·3)
where <rm 2 )Paris represents the MS radius in the Paris potential model. The relativistic corrections for Bde1 and Brh have magnitudes on the order of the discrepancy, but
their terms apparently cancel each other due to their opposite sign. This result
indicates that the calculation of the relativistic corrections to a NR theory is quite
delicate. As a result, ariotal=arA 2 +ar;ei+Bdv+Brv 2 =-1.504Xl0- 3 fm 2 , leading to a
negligible result with the correct sign for the discrepancy, and we obtain a result
similar to that in the previous section:
(4 ·4)
In this may, we conclude that the relativistic corrections are not able to explain the
discrepancy concerning the MS radius of the deuteron in the NR nuclear potential
models. Another possibility may be the MEC effect. However, it has been
reported 9 >,IO> that the MEC contribution has the wrong sign and is several times larger
than the discrepancy, although the value depends strongly on the model used.
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
831
The off-shell effects of the photon-nucleon interaction vertex
In this subsection, we should like to argue the possibility to explain the discrepancy of the MS radius by taking into account the off-shell effect of the photon-nucleon
interaction vertex for the bound system. So far we have assumed the on-shell form
for the photon-nucleon interaction vertex in Eq. (2 · 7). This assumption is introduced
mainly to avoid the complexity of the calculations. In the case of the bound system,
the photon-nucleon interaction vertex has to modify the on-shell form in Eq. (2 · 7) as
4.2.
(4 ·5)
where A~'ou(p!, PL) represents the off-shell vertex, and it vanishes for the on-shell case
p?=- m 2 and p/=- m 2 • It is obtained by calculating the higher order corrections of
photon- and meson-nucleon interactions and by renormalizing. Generally, it is
difficult to obtain its explicit form since the nuclear interaction is strong. Then, we
phenomenologically assume the relevant form for A~'ou(p!, PL) and estimate its correction to the MS radius of deuteron. Since A/rr(p!, P1) is the Lorentz invariant, and it
vanishes for the on-shell case, we can expect its explicit form to be
A~'off(Pi,
P1)= C(PI~+ m
2
)r~' 0 >+ Cz(zPivr) 1
>+ m)r~'<L>
+ ···,
(L)
+ Ca( z·p'1vYv (1)+ m ) 6,vqv
(4·6)
where C; (i=l, 2, 3, ···)is a function of Pi~ and P~~'· Furthermore, due to the conservation of deuteron current, the form in Eq. (4·6) is restricted by the following WardTakahashi identity:
(4·7)
with
-z
ir/l)pll' + m+ L:R(PL) '
(4 ·8)
where L:R(PL) represents the renormalized self-energy, which vanishes for the on-shell
case. Then the meson-nucleon interaction kernel is modified, and the BS equation in
Eq. (2 ·1) is generally changed to the renormalized one. It is interesting that the third
term in Eq. (4·6) automatically satisfies the current conservation without the change
of the BS equation and the normalization condition. However, its correction to the
MS radius is negligible in comparison to the discrepancy, since the leading q 2 -dependence becomes ~ Ca(m- M)q 2 •
We assume the most simple form for A/rr(pi, P1) as
(4·9)
with a constant C. This equation satisfies current conservation without a change of
the nucleon propagator, although the normalization condition slightly modifies the
order of the binding energy. It is worth while to note that Eq. (4·9) has a form
similar to that we have obtained by assuming the relativistic separable interaction
kernel to the BS equation. 15 > By using Eq. (4·9), we have
832
N. Honzawa
3
-- 2cm z 4m2.
<rorrz >~~-23 c-
(4·10)
This off-shell effect has the possibility of explaining the discrepancy of the MS radius,
It should be noted that the contribution of this off -shell effect in
Eq. (4·9) to the magnetic and quadrupole moments is negligible. This off-shell effect
may play a more important role for high-energy nuclear physics. More detailed
analysis on this off-shell effect will be given separately.
if Cm 2 ~ 1 and C > 0.
§ 5.
Concluding remarks
In this paper we have investigated the MS radius of the deuteron in the BS
formalism with special attention on maintaining the Lorentz covariance, starting
directly from the expression in the Breit frame of the conserved effective deuteron
current and argued the possibilities to explain the discrepancy in Eq. (1· 3). In our
investigation we have intended to extract the results as generally as possible, being
independent of dynamical details. It should be noted that some important parts out
of the MEC effects such as "pair" current in the NR nuclear theories are automatically
included in our covariant treatment. One characteristic feature of the BS formalism
is the existence of the negative energy states. By neglecting the negative energy
states, the BS amplitude has only two states CS1++ and 3 D1++) which correspond to
them in usual NR nuclear potential models. From this point of view, we can classify
the contributions to the MS radius of the deuteron as the following three parts: the
main part comes from the positive energy states including many kinds of relativistic
effects such as the relative motions of constituent nucleons and boost effects: the
second one comes from the negative energy states themselves: the third one is the
normalization effect, which is non-perturbative and appears through the normaliza·
tion condition. It should be noted that the second and third contributions may be
related to each other (see the discussion in § 3).
First, we have shown by the qualitative argument that all the relativistic correc·
tions in the positive energy states, except for the Zitterbewegung term 3/4m 2 , to the
MS radius of deuteron are relatively small compared to the discrepancy, so that we
obtained the ordinary formula for the MS radius in Eq. (1·2). Secondly, we have
investigated the details of the relativistic corrections by carrying out the appropriate
three dimensional reduction, corresponding to the instantaneous approximation, so
that it reproduces the low energy behavior of deuteron. Here, we must make the
three dimensional reduction after the calculation of the matrix elements concerning
the MS radius. We have shown that some relativistic corrections have the same
order of magnitude as the discrepancy, but their terms apparently cancel each other
due to their opposite sign. We have found that the calculation of the relativistic
corrections to the NR theory is sensitive and that it is important to have at least a
consistent treatment for both the equation and the electromagnetic current of the
deuteron as a bound system. We have shown also, by qualitative argument, that the
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
833
contributions of the negative energy states themselves are negligible compared to the
discrepancy. For the normalization effect, if we take P-=0.015, its contribution has
the correct order of magnitude with the wrong sign. As a result, since the total
relativistic corrections tend to make the discrepancy larger, we may conclude that it
is impossible to explain the discrepancy concerning the MS radius of deuteron in the
NR nuclear potential model including the relativistic corrections within the additive
current for the consitiuent nucleons.
Finally, we have phenomenologically estimated the contributions of the off-shell
effects for photon-nucleon interaction vertex. These off-shell effects may be candidates to explain the discrepancy.
Acknowledgements
The author would like to thank Professor S. Ishida for his encouragement and
useful comments. He also thanks Professor M. Wada, Professor 0. Shito, Professor
M. Imoto and other members of laboratory of Physics at Nihon University for helpful
discussions and their support. He is grateful to Dr. M. Nakamura for a reading of
this manuscript and useful comments.
Appendix A
In this appendix, we describe the calculations for the term of
Eq. (3·4), we get the formula for the case of 3 5/ and 3 D1P states as
<rl>.
From
P( ,
)
. a2 r-(l)p'
11m---:12
0 s',s P , p, q
q~o
uq
(A·1)
where <·>s•,s denotes the expectation value for the spin states, and R;(p) (i=1~3) is
defined by
R/"(P)
4~~k:~2~)+8,::EP ( 2 + ;:,~)(m~M)-Bm(E~+m) ( 1 -
;:,)
x[7-3p2+(1- ;:r(1+pz)]+ Bm(E~+m)(1- ;:r
(A·2a)
834
N. Honzawa
(A·2b)
(A·2c)
Therefore, we obtain the following formula:
3
<r/>= -6 '::Ia 2 FI(q)lq=o+<rR 2 >+<rr 2 >- M 2 +lim t 2 Nd
uq
4 d
q~O uq
(A·3)
with
(A·4a)
and
<rr 2 >=- iNd p~sjd 4 Pfb~,(p, Po)R3(p) ( a 0 >• a< 2>
X fP/(p,
(a<I>.p)\a<2>.p))
p
s',s
Po)S~~>(p2)- 1 •
(A·4b)
Furthermore, by using limq~o(o 2 Nd/oq 2 )=3/4Mi and -6(oFI(q)/oq 2 )iq=o= rp 2 + rn 2
-3K/2m 2 , we obtain the formula
<rl>=rp2 +rn 2 +
4!2P++
2~2P-+ s!2 (1+2K)m;M (P++P-),
(A·5)
which is derived by <rr 2 )::::::0 by virtue of R3(p)::::::O and
3
1 + P2 +-3-M- Po
R I P'(p)~-3_
~ 4m2
2
8m2
M P2 ' Rl'(p):::::: 4 /:z2 ( 1 +p1 + M~Po
PI).
(A·6)
Here, we neglect the terms much smaller than the discrepancy from the qualitative
behavior of the BS amplitudes in Eq. (2·5) and the non-relativistic approximation Ep
:::::om and Po::::::O.
Appendix B
We present the formula of the MS radius of deuteron in the instantaneous
approximation as,
(B·1)
with
(B·2)
The Mean Square Radius of Deuteron in the Bethe-Salpeter Formalism
~
urA
0
2_
Nd fd3
=---z7r
1
835
a 17r++*(
)] 1Tr++( )
·ap
~
p ~
p
[
p 4Ep(Ep- M) p
- ~;o fd p{- 8Ep(ip- M) + 8Ep 3 (~:- M) + 8Ep
3
2( : : -
M) 2 }11Jf++(p)l
2
(B·3)
1 (
M )(
-8M2 1 - Ep 3
--b +
)
PL1 [p· a:
2
0
P
2
m~p ]( 1 -- ;: )
2m(E:+ m)
8~Ep ( 2+ ;:
4MEp 3
2
(B·4)
'
2 M~~P211Jf++(p)l }
2
Ep(E:+ m) ( 1 - ;;
22
- 4m(E:+ m) ( 1 - ; : )[ 2
)(
Y}'
:q)
1-
+( 1 +:: YJ
2
-
p
1
+-KEp(Ep+m) mEp
- [ Ep(E:+ m) +
X{
2
Ep )[ 3
1 - 2M
4MEP
1Jf++*(p)] 1Jf++(p)+
- Nd rp2dp .1_11/JD ++(p)l2{
271"
(
4MEp(~p- M)2 ]}lw++(p)l
4MEp 2 r;P_ M)
orlv= ~~o jd 3
2
1
(
m
Ep )[
( 1-M
Ep(Ep+m) 1+ 4Ep 2
m
3
+ 4Ep
2
3
)
1 ( 2 m 2 )]
+ Ep
+ 8MEP
+ 8m(E:+m) ( 1-;: Y[ 1+( 1+;: YJ
+~[ 2 + m-M
4mEp
M
1 ( 1_ ___!!!___)] 3
EP
_K_
4mEp
(2 +___!!!___)( 1Ep
E_! )
M
(B· 5)
'
N. Honzawa
836
(B·6)
where w++(p) represents the Salpeter amplitude for S- and D-wave states, and
cPD ++(p) represents the radial Salpeter amplitude for D-wave state corresponding to
the w(p) in the NR treatment. Equation (B·2) coincides with the Fourier conjugate
expression in Eq. (1·1) by identifying w++(p) with the NR wave function.
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
See, for example, ]. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (John Wiley &
Sons, New York, 1952).
T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962), 382.
R. V. Reid, Ann. of Phys. 50 (1968), 411.
R. A. Bryan and A. Gersten, Phys. Rev. D6 (1972), 341.
R. de Tourreil, B. Reuben and D. W. L. Sprung, Nucl. Phys. A242 (1975), 445.
M. Lacombe et a!., Phys. Rev. C21 (1980), 861; Phys. Lett. 101B (1981), 139.
S. Klarsfeld, ]. Martorell and D. W. L. Sprung, ]. of Phys. GlO (1984), 165.
G. G. Simon et a!., Nucl. Phys. A333 (1980), 381.
V. E. Krohn and G. R. Ringo, Phys. Rev. DB (1973), 1305.
L. Koester, W. Nistler and W. Waschkowski, Phys. Rev. Lett. 36 (1976), 1021.
R. W. Berard et a!., Phy Lett. 47B (1973), 355.
G. G. Simon, CH. Schmitt and V. H. Walther, Nucl. Phys. A364 (1981), 285.
L. ]. Allen et a!., ]. of Phys. G7 (1981), 1367.
]. P. McTavish, ]. of Phys. G8 (1982), 911.
S. Klarsfeld et a!., Nucl. Phys. A456 (1986), 373.
M. Kohno, ]. of Phys. G9 (1983), L85.
R. G. Arnold, C. E. Carlson and F. Gross, Phys. Rev. C21 (1980), 1426.
M. ]. Zuilhof and ]. A. Tjon, Phys. Rev. C24 (1981), 736.
N. Honzawa and S. Ishida, Phys. Rev. C45 (1992), 45.
Y. Nambu, Prog. Theor. Phys. 5 (1950), 614.
H. A. Bethe and E. E. Salpeter, Phys. Rev. 82 (1951), 309.
E. E. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232.
See, for the review, N. Nakanishi, Prog. Theor. Phys. Suppl. No. 43 (1969), 1.
D. Lurie, Particle and Fields (Interscience, New York, 1968).
See, for example, S. Ishida, in Proc. 13th International Conference on Few Body Problems in
Physics, Adelaide, Australia, FIAS-R-216 (1992), p. 248; Ref. 11).
N. Honzawa, S. Ishida and M. Oda, Prog. Theor. Phys. 74 (1985), 939.
E. E. Salpeter, Phys. Rev. 87 (1952), 328.
H. Ito, W. W. Buck and F. Gross, Phys. Rev. C43 (1991), 2483.