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Quark exchange currents in nuclei
Nuclear Physics North-Holland
A526 (1991) 495-546
QUARK
EXCHANGE
CURRENTS
IN NUCLEI*
Effective operator description Yoshiaki
YAMAUCHI,
Alfons
BUCHMANN’
and Amand
lnnstitut.fiir Theoretische Physik, Universitiit Tiibingen, Aufder
FAESSLER
Morgenstelle 14, 7400 Tiibingen, Germany
Akito ARIMA Department of Physics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received
24 September
1990
Abstract: In this paper we introduce for the first time effective quark exchange current operators which can be used in nuclear structure calculations. To this end we first solve the equation of motion for a microscopic quark hamiltonian using the quark cluster model. We then construct the electromagnetic current operator on the quark level, i.e. the photon is coupled directly to the quarks. By eliminating the quark (internal) degrees of freedom, we derive an effective electromagnetic current operator on the nucleon level. A part of this effective current corresponds to the conventional impulse and meson exchange currents with vertex factors predicted by the quark model. In addition, this effective current contains new non-local and isospin-dependent terms which are generated by the Pauli principle on the quark level (quark exchange between nucleons). When we evaluate these quark exchange currents, we use harmonic-oscillator wave functions as nuclear wave functions including short-range correlations. We introduce these short-range correlations by solving the Bethe-Goldstone equation with our effective NN potential, which is derived from a microscopic quark hamiltonian. We investigate the role of these additional quark exchange currents in the magnetic moments and the elastic magnetic form factors of several closed-shell *l nuclei, such as 15N > “0 7and 39K.
1. Introduction In the last decade, several groups have searched for clear evidence of quark effects in nuclei in the framework of various quark models. Among these, the quark cluster model
seems to be the most appropriate,
because
it can successfully
describe
the
short-range part of the NN interaction, which is the fundamental input for treating any nuclear many-body problem. The main feature of the quark cluster model is that due to the Pauli principle on the quark level, various quark exchange processes between the three-quark clusters (nucleons) occur. For example, a gluon or a pion and a quark are simultaneously exchanged between two nucleons. It has been found that these gluon-quark lm6) and pion-quark 7-9) exchange processes produce a shortrange repulsion in NN scattering, thus providing a microscopic explanation of a long standing problem. l
Supported
by the DFG under contract number Department of Physics, Faculty
’ Present address:
Fa 67/10-5. of Science, University
of Tokyo,
113, Japan. 03759474/91/$03.50
@ 1991 - Elsevier
Science
Publishers
B.V. (North-Holland)
Bunkyo-ku,
Tokyo
496
Y. Yamauchi et al. / Quark exchange currents
Aside from NN scattering, electron-nucleus scattering seems to be an interesting subject for investigating quark effects ‘*-**), because electron scattering at high momentum
transfers
magnifies
the short-range
which in turn is related to the short-range
part of the nuclear
part of the NN interaction.
work 16-z*), we applied
the quark cluster model to electron-deuteron
to the Pauli
new electromagnetic
principle,
currents
wave function In our previous scattering.
Due
arise that are associated
with
the quark, gluon-quark, and pion-quark exchange processes, which generate the short-range repulsion between the nucleons. These currents are called quark exchange currents. They are not present in a conventional calculation of the electromagnetic properties of the deuteron. It has been shown that they appreciably influence the electromagnetic form factors of the deuteron ‘6-‘8*20) and the deuteron electrodisintegration cross section ‘9,20) at high momentum transfers (q 2 5 fm-‘). In the present work, we investigate the role of these quark exchange currents in the magnetic form factors of other light nuclei. Since a microscopic treatment in terms of quark degrees of freedom is not feasible for nuclei with A > 2, it is essential to devise a method which allows us to incorporate quark degrees of freedom into a conventional
nuclear
structure
calculation.
In nuclear
structure
calculations,
the
nucleus is usually regarded as an aggregate of nucleons interacting via a given NN interaction, and any physical quantity is expressed in terms of nucleon dynamical variables, i.e. the position, momentum, spin, and isospin of the nucleons, rather than in terms of the corresponding quark dynamical variables. A similar and more familiar method is usually adopted in the treatment of meson degrees of freedom in nuclei *‘). It is therefore instructive to recall this method first. It is well known that the long-range part of the NN interaction is transmitted by one-pion exchange, and the original lagrangian describing the nuclear system not only contains nucleon but also pion degrees of freedom. However, the usual nuclear physics prescription for including these pionic degrees of freedom has been to eliminate them explicitly and to express them implicitly in terms of an effective two-body potential (one-pion exchange potential) which depends only on nucleon dynamical variables. Similarly, if the nucleus is subjected to an external electromagnetic field the original
lagrangian
not only contains
the coupling
of the electromag-
netic field to the nucleon, but also to the pion field. The electromagnetic interaction of the external field with the pions is then ususally described in terms of an effective electromagnetic two-body operator (one-pion exchange current) acting in the nucleon space only. Due to the gauge invariance of the underlying field theory, the effective one-pion exchange current which implicitly describes pion degrees of freedom in the electromagnetic interaction, is linked to the effective one-pion exchange potential via the continuity equation for the electromagnetic current. The following is a graphic way of understanding the origin of the one-pion exchange current which bears some resemblance to the quark exchange currents to be discussed below. The one-pion exchange potential is an isospin-dependent charge exchange potential, i.e. by emitting a positive pion a proton can suddenly change
Y. Yamauchi et al. / Quark exchange currents
into a neutron.
Subsequently,
the positive
pion
point
by a neutron
which
changes
in space
equation must
for the electromagnetic
flow between
charge exchange between
can be reabsorbed
That is, in the presence
the total nuclear
into
at some other
a proton.
The continuity
then implies that an electromagnetic
the sites of the two nucleons
reaction.
the nucleons
then
current
497
current
which
current
have participated
of an isospin-dependent is not simply
in the
interaction
the sum of the single-
nucleon currents (impulse approximation) but must be accompanied by an additional two-body current in order to satisfy charge conservation. Thus, the one-pion exchange current describes the additional electromagnetic interaction of the external field with the underlying
pion
degrees
of freedom
and represents
the pionic e&ct
in the electromagnetic properties of the nucleus. In order to incorporate quark degrees of freedom into the nuclear many-body problem, we proceed in a similar fashion. We start with a description of the nucleus and its electromagnetic interaction with an external field in terms of quark degrees of freedom by using the quark cluster model. We then eliminate the quarks explicitly and introduce effective operators in the nucleon space. The equation of motion for the nuclear system is then described by a Schrodinger equation with an effective NN interaction. Likewise, the electromagnetic interaction of the nucleus with an external field is then described by an effective electromagnetic current operator in the nucleon space. Due to the Pauli principle on the quark level these effective operators acquire non-local and isospin-dependent parts even if the original operator between the quarks (e.g. the color magnetic interaction) was local and isospin for the case of the effective independent. This is shown, for example, in refs. 3,536722) NN interaction. Analogously, in the case of the effective electromagnetic current, additional non-local and isospin dependent two-body currents arise. Again, this is a consequence of the underlying quark degrees of freedom, and in particular, of the antisymmetrization principle on the quark level. We will refer to these additional electromagnetic currents as quark exchange currents. These quark exchange currents represent the quark efict in the electromagnetic properties of the nucleus. Finally, we stress that the continuity equation for the electromagnetic provides an intimate connection between the effective NN interaction effective electromagnetic current. As mentioned before, the short-range between
two nucleons
comes from the non-local
and isospin-dependent
current and the repulsion part of the
effective NN interaction. The physical origin of this non-locality and isospin dependence is the underlying quark structure of the nucleon and the Pauli principle on the quark level, which leads to various quark exchange processes between the nucleons. Then, because of the non-locality and isospin dependence of the effective NN interaction, corresponding effective two-body currents are required in order to satisfy current conservation. These effective two-body currents are the quark exchange currents. Another more graphic explanation of the origin of the quark exchange currents is as follows. When a proton comes close to a neutron, for example, they can exchange an up and a down quark between them, thereby
Y. Yamauchi et al. / Quark exchange currents
498
transforming
the original
then implies
that there must be a corresponding
the nucleons.
Thus,
proton
in order
to satisfy
we start with an electromagnetic the hamiltonian
into a neutron current
current
which
and vice versa. Charge quark
exchange
conservation satisfies
conservation
current
on the nucleon
current
conservation
on the quark level, and define the effective electromagnetic
in a way consistent
with the effective
between level, with current
NN interaction.
In this paper, we will investigate the influence of these quark exchange currents on the magnetic form factors of various light nuclei, where the shell-model picture is well established. Because the quark exchange currents contribute only in the short-range region, the evaluation of the quark exchange currents in the shell model must be done by taking appropriate short-range correlations between the nucleons into account, otherwise no reliable conclusion about the role of quark degrees of freedom in the nucleus can be drawn. In our picture, the short-range repulsion itself comes from quark degrees of freedom and is described in terms of the aforementioned effective NN interaction. Therefore, we will introduce the short-range correlation consistently with the quark exchange currents by solving the Bethe-Goldstone equation 23-28) with our effective NN interaction. The paper is organized as follows. In sect. 2, we first rewrite the quark exchange currents in terms of nucleon dynamical variables and show that these quark exchange currents represent a genuine quark effect in the electromagnetic nucleus. In sect. 3, these quark exchange currents are evaluated calculation employing consistent short-range correlations. In sect. results, and in sect. 5, we conclude with the main findings of the
properties of the in a shell model 4, we discuss our present paper.
2. Quark exchange currents In this section, we derive the quark exchange currents as effective two-body operators on the nucleon level using the quark cluster model based on the resonating group method (RGM). As mentioned in the introduction, these effective electromagnetic currents and the effective NN interaction are intimately related by the requirement of current conservation. Thus, the definition of the quark exchange currents as effective
operators
must
be performed
carefully
in a way consistent
with the
derivation of the NN interaction. Therefore, we first present the derivation of the effective NN interaction in the two-nucleon system in sect. 2.1. We start with a microscopic description of the NN system as a six-quark system in which the quarks interact via a given microscopic hamiltonian. We then obtain the Schrodinger equation with the effective NN potential by eliminating the quark degrees of freedom (or by integrating over the internal coordinates) and interpreting the renormalized wave function as the NN wave function. The effective NN potential is defined in such a way that it gives the same result as solving the equation of motion in the six-quark system directly. In deriving the effective NN interaction from the underlying quark dynamics, we pay special attention to the relation between the wave
499
Y. Yamauchi et al. / Quark exchange currents
function
in the two-nucleon
we turn
to the derivation
exchange current
current.
two-nucleon
and that in the six-quark
system.
In sect. 2.2,
of the effective
electromagnetic
current
or the quark
We start from the quark
electromagnetic
currents
which satisfy
conservation
electromagnetic
system
with the given quark hamiltonian.
current
by using
the relation
system and that in the six-quark
Then we derive the effective
between
the wave function
in the
system. The effective electromagnetic
current is defined in such a way that its matrix elements between NN wave functions gives the same result as the corresponding matrix elements of the microscopic quark currents taken between six-quark wave functions. Due to the Pauli principle on the quark level the effective electromagnetic current contains a non-local part, the quark exchange current. Finally, in sect. 2.3 we show that as long as we restrict ourselves to effective two-body operators, the description of the nucleus as an A-nucleon system using the effective NN interaction and effective electromagnetic current obtained in the two-nucleon system is equivalent to a microscopic description of the nucleus as a 3A-quark system using the microscopic quark hamiltonian and quark
electromagnetic
2.1. THE
EFFECTIVE
current.
NN
INTERACTION
Let us start with the total hamiltonian
for the six-quark
system given by (2.la)
Here mq = irn,is the quark mass. The center-of-mass (c.m.) motion of the six-quark system is eliminated via the second term where PC is the center-of-momentum interaction Vy) consists of the coordinate and MG = 6m,. The quark-quark confinement term one-pion exchange vy”f = -a,(X;
Vrnf, the one-gluon (OPE) term VyPEP:
exchange
(OGE)
term
V$‘GEP, and
the
(2.lb)
. Aj)& ,
(2.lc) -__
___A2 A”-m’,’
where fmq= gfrNNand fiNN/47r=0.08.
(2.ld)
The cut-off parameter A in the VoPEP is taken to be 4.2 fm-‘, which is estimated from the size of the pion ‘). The strong coupling constant (Y, is determined from the experimental N-A mass splitting and
500
Y. Yamauchi
the strength nucleon
of the confinement
mass against
parameter)
variations
b,. Here the nucleon
use the parameter of this hamiltonian The RGM
et al. / Quark exchange currents
potential
a, from
of the nucleon-size
the stability parameter
condition
is in a (0~)~ harmonic-oscillator
configuration.
set b, = 0.5 fm, LY,= 0.36, and a, = 67.27 MeV * fm-‘. and its parameters
wave function
of the
(harmonic-oscillator We
More details
are given in refs. 1s,‘9).
for the six-quark
system is then written
as
(2.2) where Qi$(&,
iji.J is the intrinsic
wave function
of the ith cluster. For these intrinsic
wave functions we choose only the nucleon (OS)’ harmonic-oscillator configuration. The single quark coordinates p,, . . . , p6 are then transformed into two internal coordinates for each cluster, &, and &2, and one c.m. coordinate for each cluster, ri (i = 1,2). Since the c.m. motion of the six-quark system as a whole is eliminated, the wave function ~(‘12’ does not depend on the c.m. coordinate R,2 = t( rl + rJ, but only on the intercluster (relative) coordinate r12 = r, - r2. Due to the Pauli principle between quarks, the six-quark wave function !P(rJ must be totally antisymmetric under the exchange of any two quarks. Since the intrinsic wave functions are antisymmetric, the antisymmetry of the six-quark wave function through the product of antisymmetrizers &2Q(c’“ster) and dCq)
can be achieved
%d(c’“ster) = v$( 1 - P,4P25P36) )
(2.3a) (2.3b)
acting on the wave function of eq. (2.2). Since the operator P,4P25P36 represents just the exchange of the two clusters as a whole, ~~~~~~~~~~ is the antisymmetrizer for the clusters. In eq. (2.2), this is already incorporated by choosing the relative wave function Ik ( r12) to be antisymmetric with respect to the exchange of the clusters as a whole. Thus the Pauli principle between nucleons comes from the antisymmetrizer &‘c’uster), when we introduce
the nucleon
space. Note that in addition
to the obvious
two-nucleon component where there is no overlap between the clusters, the RGM wave function also includes the six-quark component where the quarks are not confined to their respective clusters. To a large extent, this coexistence of NN and six-quark components in the RGM wave function follows from the antisymmetrization principle on the quark level. Once the total hamiltonian and the intrinsic wave function of the clusters are given, the equation of motion which determines the relative wave function P(r,,) is given by the RGM equation of motion:
1(ii1@$(Si.l,$i2))(H’q’--E)ly(6q)(pl,. .. ~6) dgr,r.. . d&.2=0. 3
(2.4)
501
Y. Yamauchi et al. / Quark exchange currents
Integration
over the internal
coordinates
leads to an integro-differential
equation
of the form
J
H(r,z,
The hamiltonian
kernel
r{JP(rL)
drL= E
J
N(r12, 42) p(&)
H (r, r’) and norm kernel
d42 .
N( r, r’) are defined
(2Sa) as
We are concerned only with the orbital part of the wave function and have suppressed the spin, isospin, and color dependence here. By dividing the antisymmetrizer (1 -9P3J into the identity operator part (1) and the permutation part (-9P3,J, these kernels are decomposed into two terms, a direct term and an exchange term. For example, the norm kernel is decomposed as follows: N(r,r’)=
N’D’(r)S(r-r’)+N~EX~(r,r’)=S(r-r’)-90~~fff)[P~~rc~]~(r,r’),
(2.6a) (2.6b)
where Ol;*‘[ P:FC) ] 1s . the effective spin-isospin operator which gives the same result as the quark exchange operator P$s6Tc’ in the spin-isospin and color space of the quarks. We will give an explicit expression for this effective spin-isospin operator later. Eqs. (2.6) are derived using familiar techniques from nuclear cluster theory *‘). Details of this derivation are given in the appendix. These RGM expressions clearly show that the operators resulting after integration over the internal coordinates become non-local due to the quark exchange process (Pauli principle). While the direct term contributes everywhere, the exchange term is only effective from eqs. (2.6). the norm kernel the cluster radius
in the overlap region of the clusters which can be understood easily (The function C?(r, r’), i.e. the radial part of the exchange term of NCEX’, decreases rapidly as the intercluster distance r or r’ exceeds and is practically zero for twice the cluster radius.) The hamiltonian
kernel can be further decomposed into different types. For the one-body operator (kinetic energy) there is one direct term (fig. la) and two different exchange terms (fig. lb, c). For the two-body operator (potential) one must consider two different direct terms (fig. Id, e) and five different exchange terms (fig. If-j). By solving eq. (2Sa) or eq. (2.4), we get the six-quark RGM wave function as defined by eq. (2.2). As mentioned before, it is fairly cumbersome to extend this microscopic treatment on the quark level to nuclei with A > 2. Therefore, we will introduce the nucleon space and use a Schrodinger equation with an effective NN
502
Y. Yamauchi
123
et al. / Quark exchange
654
123
654
123
(a)
1 123
Cd)
I
654
(b)
654 (e>
currents
123
N
123
654
Cc)
654 (f)
1
123
654 (9)
1 123
654
Fig. 1. The different types of RGM kernels in the quark cluster model. For a one-body quark operator: (a) direct term; (b), (c) exchange terms. For a two-body quark operator: (d) direct (intracluster) term, (e) direct (intercluster) term, (f-j) exchange terms. The solid lines represent quarks and the boxes quark operators.
potential written in terms of nucleon dynamical variables. This effective description must then yield the same result as solving the six-quark wave function from eq. (2.4) directly. One obvious candidate for an appropriate NN wave function and an effective Schrodinger equation is the relative wave function ?P(r,,) and eq. (2.5a). There is, however, an important difference between ly(r12) in eq. (2Sa) and the conventional NN wave function. The solution of eq. (2.5a) !P(r,J does not satisfy the usual orthonormality condition and Schrodinger equation because of the appearance of the norm kernel in eq. (2.5a). Therefore the solution of eq. (2.5a) cannot be interpreted as a NN wave function. In order to define a proper NN wave function,
Y. Yamauchi et al. / Quark exchange currents
the renormalized
wave function G(r)
is introduced
503
where
N”’
J
N-“‘(r,
with I!? = E - 2Eint, where
J
N”2(r,
IS . the square
consequence, the renormalized for a non-local potential
V,,(r,r’)=
=
r’)V(r’)
root operator
wave function
r”)H(r”, E,,, defines
dr’,
(2.7) of the norm
kernel.
P satisfies the Schrodinger
rrn)N-“2(r”‘,
r’)dr”di~~-~-2~i,,,,
the total internal
energy
As a
equation
(2.9)
of one cluster.
Next, we turn to the discussion of the spin-isospin and color degrees of freedom, which we have omitted for simplicity up to now. In order to rewrite the spin-isospin and color part of the six-quark matrix elements, we introduce effective spin-isospin operators in the NN space. These effective operators OCeff) must then satisfy the condition that for any state they yield the same matrix elements as the corresponding spin-isospin and color operators OCq) when evaluated on the quark level. Due to the quark exchange property of the permutation operator the resulting effective NN operators will in general be two-body operators acting on the spin-isospin coordinates of nucleons 1 and 2 (xsf~~O~;*“~x%& = (*~~~‘c’(o(q)lx~~~,c,)
(2.10)
for (ST,), (S,T’J=(OO), (Ol), (lo), (11). As an example, consider the effective of quarks 3 and 6 operator corresponding to P$zTc), which acts on the coordinates belonging to the first and second cluster Oie:)[ P ~SQTC)]=~[(1+~1IN).1/N))+~(1+~7(1N).71N))aiN).a~N)], which can be easily checked
using eq. (2.10). This equation
(2.11)
is also given in ref. 22).
Eq. (2.11) clearly exhibits the charge exchange character (i.e. the isospin dependence (N) * 7iN)) of the effective operators even when the initial quark operators are 71 isospin-independent. In this way, we can systematically rewrite all quark operators in terms of effective NN operators, and from this derivation it is evident that the effective Schrodinger equation in the two-nucleon space given by eq. (2.8) gives the same result as solving the equation of motion for the six-quark system given by eq. (2.4). Although our effective NN potential V,, is written in terms of nuclear dynamical variables, it represents the effect of the quark degrees of freedom in the same way as a microscopic calculation would. We have chosen the effective description mainly because it is more convenient for practical applications to nuclei with A > 2. After the relation between the six-quark wave function and the effective NN wave function has been
Y. Yamauchi et al. / Quark exchange currents
504
clarified,
the derivation
in a way consistent
of the effective
with the definition
quark
exchange
of the effective
currents
can be performed
NN interaction.
This will be
done in the next subsection. However, effective
before
ending
NN interaction
hamiltonian
kernel
this subsection,
let us investigate
given by eq. (2.9) in greater
into a direct H(r,
and an exchange
evaluated
the
r’).
(2.12a)
and is given by Pf*
PD’(
of the
term
r’)=HcD)(r)G(r-r’)+HcEX)(r,
The direct term can be readily
the properties
detail. First we decompose
Vp2p”‘( r,J ,
r12)= 2Ei”t + ~+
(2.12b)
where V I”
=
$
$“)
. $‘),,\N)
. v,~$N”’
.
v,
j-
n
dk e-tk-exp
(-{btk’)
2~~ k’+mf,
A. (2.12c)
Note that Eint comes from the intracluster
terms represented
by fig. la for the kinetic
energy and fig. Id for the potential. The direct potential VypP results from the intercluster term represented by fig. le. The integration over the internal coordinates results in the factor exp (-ibtk2). It is easily understood that the direct potential is the same as the conventional vertex factor F,,(k) =exp confinement and one-gluon
one-pion exchange (-~6’,k’)(A’/(k’+A’))‘/ exchange potentials,
potential, which contains the TN twice. In the case of the there is no corresponding inter-
cluster term (fig. le) because of the color operator hi * A,, which represents color confinement. Using eqs. (2.12), we can rewrite the non-local potential of eq. (2.9) as follows V,,( r, r’) = Vp2pEp(r) . 6( r - r’) + V?Fp( r, r’) . Here, the first term is the conventional (2.12c), and the effective quark-exchange
(2.13a)
one-pion exchange potential given in eq. potential VpFp consists of two terms
V?zp( r, r’) = V?i’( r, r’) + VP,““( r, r’) . The quark-interchange as V?i’:‘( r, r’) =
J
potential
V $” and the renormalization
N-‘/2( r, r”)H(EX)( #‘, r’rr)N-“2(
V FTN(r, r’) = J ~-‘/2(~, ry~(D)(r9~-W(
(2.13b) term Vy2ENare defined
rnr, r’) dr” dr”’
,
r”, r’) dr”-H(D)(r)G(r-r’).
The quark-exchange potential is only effective in the region overlap or at short relative distances between the two nucleons.
(2.13~)
(2.13d)
where the clusters Note that N-‘j2 is
Y. Yamauchi et al. / Quark exchange currents
different (2.6)),
from the 6 function
only in the short-range
so that the renormalization
repulsion
this short-range
in NN scattering.
repulsion
or overlap
term gives an appreciable
this region. Previous work has shown that the quark-exchange a short-range
505
is attributable
region
(see eqs.
contribution
only in
Vp,“‘( r, r’) produces
potential
It is well understood to the Pauli
principle,
that the origin
of
as well as to the
spin-spin term of the OGEP and OPEP between quarks, which also create the N-A mass splitting in the three-quark system. Since our effective potential so far lacks enough intermediate-range attraction, we simply add a phenomenological c exchange
V” to our effective
potential V”(r,J
= -$
V,, in eq. (2.8).
NN potential
l _ e-m,,‘-l2- e -2u sinh (m,r,,)
--&
,2 1 [cash (2~) - l] eC”-‘lr
r,,G2R0
r12) 2&,
(2.14)
with cy = m,R,,, m, = 520.0 MeV and R,, = 1.34 b, [ref. ‘)I. The aN coupling constant g, is determined in order to reproduce the binding energy of the deuteron: gt/4m = w exchange between 2.55 [refs. 18,‘9)]. Even if we had started with a microscopic quarks, we would still end up with an effective NN potential with essentially the same characteristics as the one in eq. (2.14). This is due to the fact that the effects resulting from quark antisymmetrization are negligible for a spin-isospin independent central potential like the (T potential 30). In our formalism this means that the corresponding quark exchange potential V,,QEP is negligible because of the cancellation between the quark-interchange V$” term and the renormalization term Vr:N; thus, only the direct term of the sigma potential survives. Finally, we give a summary of the procedure needed to derive the effective operators in the nucleon space from the corresponding operators in the quark space. There are two steps. The first step is to integrate over the internal coordinates and to create the RGM kernel as shown in eq. (2.5b). A part of this RGM kernel corresponds to the conventional one-pion exchange potential with vertex factors predicted by the quark model. In addition, due to the Pauli principle on the quark level, this RGM
kernel
acquires
a non-local
part,
which then leads to the quark-
term as given in eq. (2.13~). The second step consists of the renormalizagiven in eq. (2.9). This procedure then leads to the renormalization term given in eq. (2.13d). interchange
tion procedure
2.2. DEFINITION
OF THE
QUARK
EXCHANGE
CURRENTS
After the six-quark wave function is obtained by solving eq. (2.4) or (2.5a), the electromagnetic interaction of the six-quark system with an external field is described in terms of the six-quark elements of the quark charge and current operators ( p(6q)lJ$)( p(6q)), where JF’ IS . the quark electromagnetic current given as JzLq)= ; JIpi’“P’( pi, 4) + I=1
; i
{Jz:+“)(
pi, pi, 4) +$=‘“o”)(
pi, pi, 4)) _
(2.15a)
506
Y. Yamauchi et al. / Quark exchange currents
Here, brevity
the three-momentum
of the photon
we write the quark
order to satisfy current
charge
conservation
we introduce
not only the quark
also the pion
JzLpion) and gluon
is denoted
and current
by q. For convenience
as a four-vector
with the quark hamiltonian impulse
current
quark
current)
currents
(two-body
Jt’imp),
q)
=Z& tq
(2.15b) ajq)
x
but
currents):
&q:imp)( pi, q) = ejq) eir.p, , fq:imp)(pi,
In
given in eq. (2.la)
(one-body
JjLq’g’“on)exchange
and
current.
eiw,
+2
eiw,
v,)
,
(2.1%)
4
Jpio”ypi,
pj, q) = 0 )
(2.15d)
J(q:pion)(pi, pj, q) = J(q:pion-pair)(pi, pj, q) +J(q:pionic)(pi,
.P:pio-ypi,
pj, q) =
pj, q) ,
(2.15e) -k,
eff$! ($)
XT?)),
eiq.P, ,p),y)
.
vp,,( !?$z_T$)
2 xj5&T+(‘++d
(2.15f)
> r
fq:pionid(Pi,
pi,
q)
=
-
je&
(@)x
p)pp
. vp,ajq) * v,
T
ei(rl-k).
k2+m; A2 kz+nz
Jpq’uo”ypi, pj, q) = 0 )
A2 (q-k)2+A2
P,
(q-k)‘+mZ, l/2 ’
(2.1W (2.15h)
where
the isospin structure of the quark charge operator eiq) is given by ei”‘= ei(l + 37g)). In ref. I’), we also considered other two-body currents, (e.g. the isoscular pion-pair current) which are of higher order in (l/ mq). These higher order terms in the charge and current operators are the counterparts of the corresponding higher order terms in the kinetic energy, OPEP and OGEP as required by current conservation. Since we do not consider these higher order terms in the hamiltonian, we also do not consider the corresponding higher order terms in the electromagnetic current in order to satisfy current conservation. For the same reason, we do not choose the gluon-pair current, which we used in refs. 18*19),but the gauge invariant gluon current which satisfies current conservation with (the LS force of) the OGEP. The gauge invariant gluon current differs from the gluon-pair current by a factor t. More details of the one-gluon exchange current are given in ref. 20).
Y. Yamauchi et al. / Quark exchange currents
Having
defined
wave function
the description
of eq. (2.7), which
(2.2), we now introduce terms of nucleon description six-quark
of the two-nucleon is related
by the renormalized wave function
currents,
of eq.
which are written
in
and which yield exactly the same result as a microscopic
of the electromagnetic wave function.
system
to the six-quark
the effective electromagnetic
variables
507
interaction
Thus the effective
in terms
of quark
electromagnetic
currents
current
Jt”’
and the is defined
by ly@l)) .
($jJ1”“‘1 I@ = (l@Q)j#$j We follow
the same procedure
as in the definition
(2.16)
of the effective
NN potential.
First, we integrate over the internal coordinates and rewrite the six-quark matrix element in eq. (2.16) as a matrix element which contains only the relative wave function: (Pq)(J;)J?Pq))
)
= ( P(J;b)+JFX’J?P)
(2.17a)
with
I?QZ(Ei.l,li.,)}J:)
J:“‘( rn, Rn, 4) = J { X
i=l
1I?@C(Ej.l,5j.2)I
dtl.1 . * . 42.2
(2.17b)
3
j=l
JFx)(r,
r’,
R,2, q) =
I? J-t
@;Vnt(gi.l?
gi.2)
i=l
6(r12-
r)Jjp)(-9p36)
I
* * . G2
~(r12-r")&l
dr12,
(2.17~)
j=l
where we decomposed the RGM kernel into the direct term JLD’ and the exchange term JF"' . The direct term can be readily evaluated and will be shown to be the same as the conventional electromagnetic current, that is, the impulse current JE”‘“’ and meson exchange current JIMEC) (pion-pair and pionic current) with the one-nucleon vertex factors calculated in the quark model:
JlD)(r12,RI*, q)= $ JE”“(ri, q)+JIMEC)(rl,
r,, q).
(2.18)
i=*
The conventional
impulse
current
is given as
Jgmp)(ri, q) = e(GL(q’)+
TizG”,(q2)) ei4’c ,
(2.19a)
JCimp)( ri, q) = &{(G^,(q2)+~i,GK(q2))qxajN’ei*.~ N
+(G”,(q2)+
7,,GL(q2))2
ei9“, V,} ,
(2.19b)
508
Y. Yamauchi et al. / Quark exchange currents
with Gsh,(q2)4$mP)(q2)+
,3$&'on)(q2),
Q’(+
G$duon)(q2)+
G$"P'($)+
(2.19~) G$Pion)(q2),
Gg( q’) = G’,( q2) = Gsimp)( q2) = i exp (-ab’,q”) G zmp)( q’) = 1 exp (-ibis’)
(2.19d) ,
(2.19e)
.
(2.19f)
The conventional impulse current comes from the direct term of the quark impulse current represented by fig. la and from the intracluster terms of the quark-pion and quark-gluon currents represented by fig. Id. The contribution from the quark-pion and quark-gluon intracluster terms shown in fig. Id is included in eqs. (2.19a, b) through the corresponding corrections to the one-nucleon form factors, GCg’“““)(q2) and GCpio”)(q2) g’iven in eqs. (2.19c, d). Since the explicit expressions for Gcg’“‘“)(q2) and GCpion)(q2) were already given in refs. ‘83’9), we will not repeat them here. The conventional meson exchange current is given as JLMEC)(r,, rj, q) = 0 ,
(2.20a)
J’MEC’( ri, PI, q) = J(pio”-payJ.i, rj, q) +J(pio”ic)(& #pion-pair)(ri, ‘;, q) =!&!$E
J(pionyri,
I - ------
X
,-‘k.‘a,
27r2 k2+mt
Pi, q) = -i,*
I
(2.20b)
(TjN) x TIN))Z ei4’rLajN)aJN) . v,, dk
X
rj, q) )
F,N(k2)F,N,(k
($J) x $Q$J) 71
q)+(i++j)
. Vp/N’
(2.2Oc)
. v,,
$(2k-q)&$
x Fmv(k2)F,,((q
7T (2.20d)
- k)2) ,
with ,
(2.20e)
(2.20f) The conventional meson exchange current comes from the intercluster term represented by fig. le. The integration over the internal coordinates results in the vertex are factors F-N and F,,,,N g iven in eqs. (2.20e, f). More details of this derivation given in refs. ‘*,19). In the case of the gluon current, there is no corresponding intercluster term fig. le because of the color operator Ai. Ai, which represents color confinement.
Y. Yamauchi et al. / Quark exchange currents
Now, let us turn to the exchange
terms. Here we evaluate
which comes from the quark impulse for comparison. evaluated
The exchange
in the same
coordinate
of the
unchanged
under
only the exchange
and also the corresponding
six-quark
term
direct term
terms which come from other quark currents
manner.
can be
We start
with the spatial component. The c.m. symmetric and remains R,2 is completely
system
the application
on R,2 can be factored
current
509
of the antisymmetrizer.
out from the integral
Therefore
the dependence
in eq. (2.17~). Thus,
we first rewrite
the quark impulse current by transforming the single quark coordinates p, , . . . , p6 to a c.m. coordinate, R,, and the coordinates relative to the c.m. frame of the six quarks
(pi - R,J as follows
&+“P)(p
) q)
= s{3q
x ,id
eiq.(pc-R)
eiwR+2
. 3 eiq.(p,-R)(vpc
-iv,)
eir.R
N +eiq-(p,--R)
e iq.R
V,} .
(2.21)
We then insert eq. (2.21) into eq. (2.17~) and evaluate the integrals over the internal and the relative coordinate r,2. We need to evaluate the
coordinates &,.r . . . & following integrals:
ir &X&1.1, S,.J S(h2 - r) i=, I eiq.(~3-R,2) 3e’q”(P’-RlZ)(Vp, -iVR,2) X eiq.(PZ-RI2) Px6 3eiq”p~-R1z’(Vp,-~VR,~)P36 i
e4+--R
121 P36
3eiq’(p3-R12)(Vpj - iVR,,) Ps6 1
x
S(r12- r’) d&
. * . dh2 dr12
eiiq.rv,6cr
The operators
H’, P* can be derived
H+(
r, r’;
k)
=
LP( r, r’)
ry
P+(r,
r’;
$4)
P-( r,
r’;
jq)
using familiar
theory 29). Since we show the details only the final result here
_
techniques
of this derivation
.
(2.22a)
1 from nuclear
in the appendix,
exp [ isk( r + r’)] exp (&bik’)
,
cluster we give
(2.22b)
H-(r,
r’;
k) = Y(r, r’) exp [ i$k(r - r’)] exp (fbzk*) ,
P’(r,
r’;
k) = -i{e,H+(r,
r’) - H+(r, r’)v,.,} ,
(2.22d)
P-(r,
r’;
k) = -~{~,H-(r,
r’)+H-(r,
(2.22e)
r’)?,,},
(2.22c)
where LP(r, r’) is the radial part of the exchange term of the norm kernel given in eq. (2.6b). The derivative operators, 9, and d,, operate only on the final and initial
510
wave
Y. Yamauchi et al. / Quark exchange currents
functions,
evaluate
respectively.
Using
the direct and exchange
the operators
defined
in eq. (2.22),
terms which come from the quark impulse
we can current.
They are given as J(D.imP)(r, RI23 4) = JgAl, + Jg!w:R, + J~2”:,, #EX.imp)(,., ,.f, R
12, (I) = Jt:&!, + JE%z,
(2.23b)
+ JtcEoXn?.,~ ,
Jt$inI = -?2m,i
e @R,, q X o(lN)(;+$&))
J[Fpy& = -_?2m,i
e iq.R,z q x 20~*)[~:“)(t+~~:))(-9)P~~~~)]
+ _eiq~R,, e 2mNi
(2.23a)
,
e-ib$7’ eifr.r+
,
(l-2)
q x o~)[~~)(~+~~~))(_~)~~~TC)]
(2.23~)
epibiq2 ~+(r,
e-t@* H-(,.,
e J(D) =-eiq.R,, [conv:r]
(;+$&J))
e-~b~q2eilr.r~,+(lt,2),
(2.23e)
mNi
=_?_
[conv:r]
,J; iq) (2.23d)
+(1*2),
J(=)
,J; tq)
eiq’% 20~*)[(~+~~~))(_9)p~~~~)]
epk’$q* pi-(,,
,J; &)
mNi +e eWR12 o’,e;“‘[(++$7~‘)(-9)P$~Tc)] mNi
e-fbiq’ P-(*, r’;
$4)
(2.23f)
+(I-2)) J(D)
= -
J(=)
= -
[conv:Rl
e
2mNi
[conv:Rl
e
2m,i
e’q’R,zVR,,(~+~~(l~))
e-6‘bZq* q eifq”+(lH2),
eiq.% v,,,~o’,“,“)[(~+~~~)“!)(-~)P~~~~)]
(2.238) e-fbiq2 ~+(r,
e + -eiq'R,,VR,,0(leff)[(~+~~~))(-9)p~Tc)]e-~b~q2 2mNi
lt(l++2)
H-(r,r';iq)
(2.23h)
.
The effective spin-isospin cated. For example
operators
appearing
in eqs. (2.23) are rather
-~(l+~a(lN).aiN’)i(lrN’X?:N’)Z. there
are some
operators,
such as
useful
relations
20~~fi)[~~)p~sd_c) ] + o$$T
compli-
* cJy))(Ty)+TyyZ
O(le2R’[7~~)P~TC)]=~(l+$~iN)
However,
r’; $4)
between
$l)p$;w]=
2O&ff)[ rE’p$sQTC)] + O(lezff)[ @pit”‘]
these
(2.24a) effective
ay’O~;fi)[P:~=C’] = 7~~‘o$f’[p$~‘“‘]
spin-isospin
)
(2.24b)
.
(2.24~)
Y. Yamauchi et al. / Quark exchange currents
Now we consider that the direct
the properties
term given
as the conventional
of these direct and exchange
in eq. (2.23a)
impulse
511
current
together
terms. We can verify
with eq. (2.23c, e, g) is the same
given in eq. (2.19b) by transforming
nates r12 and RI2 to rl and r,. As can be seen from eq. (2.23a), the spatial of the impulse
current
J[sDdinI, the convection
can be decomposed current
the coordicomponent
into three types; the spin-magnetic
with the derivative
operating
current
on the c.m. motion
J(D) Fconv:Rl,
and the convection current with the derivative operating on the relative motion Jf,Dd,,:,, . This form is useful when we introduce the short-range correlations between nucleons in sect. 3. The exchange term can be decomposed in the same
(EX) and J[,,,,:,I (EX) correspond Jt:$&, J[conv:R~,
to corrections to Jfs”p!lnl,J~~\v:R1, and These exchange terms contain the operators H’ and P’ which correspond to corrections to eifq” and eifq” V,, respectively, as is easily seen from eqs. (2.22). way.
J(D) [conv:r]~
The contributions from fig. lb correspond to the parts which are expressed by the operators Ht and Pt and those from fig. lc are the parts which are expressed by the operators H- and P-. These exchange terms, given in eqs. (2.23), contain the radial part of the exchange term of the norm kernel 9 given in eq. (2.6b) through the operators H’ and P’. Therefore these exchange terms are effective only at short distances. The time component of the quark exchange current can also be expressed by the operators Jrx.imp)(r,
H” and is given as ,.I, R,,, 4) = eiq.R,, 20’,~~)ff’[(~+t~~=))(-9)p~S,~~)] e-@:q’ H+(,., ,.‘; iq) +eiq’R12 o’,P’[(~+~~:“,))(-~)P~S~TC)]
e-i’zq’ H-(r,
r’; $4)
+(1++2).
(2.25)
Having obtained the RGM kernel, the next step is the renormalization procedure. We can easily derive the effective electromagnetic current defined in eq. (2.16) from eqs. (2.17b, c) using the relation between the relative wave function and the renormalized wave function given in eq. (2.7): JjY(rr2,
&, RI*, 4) =
I
N-“‘(rr2,
r’)[.Izp’(r’, R12, q)S(r’-r”)
+fEX’( r’, r”, R 12, q)]N-“‘(
r”, &) dr’ dr” .
(2.26a)
Analogously to the effective NN potential, the effective electromagnetic current JF) consists of the conventional impulse and meson exchange currents and a non-local quark exchange current r;,,
JY)(rlz,
R,*, q) =
i (
JEmp)(ri, q)+JLMEC’(r
i=l
+J(QEC) p
where the quark
exchange
current
(r12, 42,
I3 r2,
4))
. @r12-
(2.26b)
R12, 41,
JIQEC) consists
d2)
of two terms
JLQEC)(r, r’, R, q) = JLQQlc)( r, r’, R, q) + JyN’(
r, r’, R, q) .
(2.27a)
512
Y. Yamauchi et al. / Quark exchange currents
The quark-interchange as
term JFIc)
JkQlc’(r,r’, R,
term JrEN)
are defined
N-1’2( r, rfr)JcEX)(r”, r”‘, R, q) N-“2( r”‘, r’) dr” dr”’ ,
q) =
JFEN’(r, r’, R, q) =
and the renormalization
I
N-“2( r, r”)JID)( r”, R, q)N-I’“(
-JID)(r,
R, q)S(r-
(2.27b)
r”, r’) dr”
r’) .
(2.27~)
These two terms are non-local operators that are effective only at short-distances between nucleons. We can easily understand from these equations that the quark exchange current JLQEC’ appears only when the exchange of quarks between the two clusters is taken into account, and that there is no corresponding current in conventional nuclear theory. Therefore it is this term which represents effect in nuclei. By definition, it is written in terms of nucleon dynamical so that it is possible to use it in a nuclear structure calculation.
the quark variables,
Note that in the two-nucleon system, the dependence of the effective current on RI2 = $( r, + r2) must be removed from these currents in order to eliminate the spurious c.m. motion. However, here we keep this dependence on the c.m. motion, because we want to apply these effective operators to nuclei with A> 2, where the c.m. coordinate of the total system is not equal to R,2 = R. The functions that are dependent only on R,2 do not become non-local because this coordinate is completely symmetric and remains unchanged under the application of the antisymmetrizer. Since the definition of the effective quark exchange current is made carefully in a way consistent with the definition of the effective NN interaction, and since the original quark hamiltonian and quark electromagnetic current satisfy current conservation, the effective NN interaction and current should satisfy the corresponding current conservation law on the nucleon level. Unfortunately, the expressions for the effective operators are quite lengthy so that we must defer an explicit proof to a separate paper. Nevertheless, it is evident that the effective quark exchange current appears as the counterpart of the short-range part of the effective NN interaction, which is governed by the quark exchange mechanism.
2.3. QUARK
DEGREES
OF
FREEDOM
IN THE
DESCRIPTION
OF NUCLEI
In sects. 2.1 and 2.2, we showed that the description of the two-nucleon system using the effective NN potential V, given in eq. (2.13a) and the effective electromagnetic current Jy’ given in eq. (2.26b) is equivalent to that of the six-quark system using the microscopic quark hamiltonian given in eqs. (2.1) and the quark electromagnetic current given in eqs. (2.15). We also showed that the quark effect in the electromagnetic interaction is represented by the quark exchange current JFEC). In this section we will show that these effective operators V, and J’,““‘, especially
Y Yamauchi et al. / Quark exchange currents
the quark
exchange
current
JLQEC),are applicable
we will show that the “disconnected” these effective
operators
diagrams
of neglecting
(1) The A-nucleon
three-body
Schrodinger
+(r,,...,rA)+
with A > 2. In addition,
of fig. 2 are already
and need not be explicitly
To see this, we will show that the following approximation
to nuclei
513
two statements
to A-body
included
in
are valid within
the
calculated. nucleon
operators:
equation
; i>j=l
X
?@rl,..., R,-++r:j,..., R,-;r:j ,...,
rA)dr;=&&(rl
,...,
rA), (2.28)
with the effective NN potential V, of eq. (2.13a) gives the same eigenvalues as the corresponding Schrodinger equation for the 3A-quark system with the quark hamiltonian given in eq. (2.1)*. (2) The description of the A-nucleon system with the effective electromagnetic current operator given as, Jy’
= i
Jo”“‘+
f
JIMEC’(ri, rj) +
i>j=1
i=l
i
JIQEC)(rb,
rii; R,) ,
(2.29)
i>j=l
is equivalent to the description of the 3A-quark system with the corresponding quark electromagnetic current given in eqs. (2.15)*. Here Jz”“’ and JIMEC’ are the conventional impulse and meson exchange currents given in eqs. (2.19) and (2.20), while JcQEC) is the quark exchange current which represents the quark effect in the nucleus and is given in eqs. (2.27). In other words the matrix element of the effective current operator of eq. (2.29) evaluated with the solution !? of eq. (2.28) gives the same result as the corresponding 3A-quark matrix element of the quark electromagnetic current operator given in eqs. (2.15) taken between 3A-quark wave functions. Let us start with a description of the 3A-quark system characterized by the quark hamiltonian given in eqs. (2.1). For the RGM wave function of the 3A-quark system we then have the following generalization of eq. (2.2) @3A-q)(,,r,
. . . , &A) = i?‘q’ c(fi
. .
Fig. 2. The “disconnected”
l
Of course
the summation
diagram should
(2.30)
~lu”‘(li.~,Si.,))~(llr.....IA)l.
/
L\i=l
_I
.
which corresponds to the term, If=, by the third term in eq. (2.37). then run over i = 1,.
,3A.
Jzmp)(k)
x CfJF’,
NiFx)
given
514
Y. Yamauchi et al. / Quark exchange currents
The 3A-quark coordinates p, , . . . , pjA are then rewritten in terms of the two internal Jacobi coordinates of the ith cluster 5i.l and Et.2and one c.m. coordinate of the ith cluster ri where (i= 1,. . . , A). The Pauli principle between quarks is included in eq. (2.30) by virtue of the antisymmetrizer of clusters (generalization of eq. (2.3a)) and 2(q) (generalization of eq. (2.3b)), given as GCq)= 1 - 9 i
Psisj + (permutations
operating on more than two clusters) . (2.31)
i>j=l
The antisymmet~zer of clusters is already incorporated in eq. (2.30) by choosing a wave function F which is totally antisymmetric for any permutation of the clusters as a whole. After eliminating quark degrees of freedom, this leads to the Pauli principle on the nucleon level. Once the total hamiltonian and the intrinsic wave function of the clusters are given, the equation of motion which determines the 3A-quark wave function W is given by the RGM equation \ (i,
~~‘(SII.Si.,))(H’q’-E)~‘3A~q’(~~,...,~,,)d~,.,
* * .~EA.z~O-
(2.32)
Integrating over the internal coordinates of each cluster (i = 1, , . . , A), then leads to an integro-differential equation of the form
,...,rA;r; ,...,ra)-E’N(r,,...,rA;r:,“ J[ff(r, (2.33)
xq(r;,...,rh)drl,***dra=O.
Next, we introduce the following renormalized
x!P(ri fi(rl,.
.
-,rA;
r;,...,
&)=
,....,
Np’i2(r,,. X
H((ry,.
x N-“2(r:‘,
xr;l--
wave function and hamiltonian
&)dr;-=*drj,, . ., r,; r!,.
(2.34a)
.., r>)
. , ,, r>; r:“,. . . ,rz) . . . , rz; r:, . . . , rh)
. &a; &.;.
. . drz,
(2.34b)
where N’/’ is the square root operator of the RGM norm kernel. It is obvious that G satisfies the orthogonality relation and the following Schradinger equation
I
Z?(r,,. . .) rA; r$, . . . , ra)?@(r$, . . . , ra> dri * * . drL= El?(r,,
. . . , rA) .
(2.34~)
In the case of the six-quark system, we identified the corresponding renormalized wave function given in eq. (2.7) as the appropriate NN wavefunction and eq. (2.8) as the effective SchrGdinger equation for the NN system, Therefore, we also identify
515
Y. Yamauchi et al. / Quark exchange currents
the renormalized wave function in eq. (2.34a) as the appropriate A-nucleon wave function, For A> 2, aside from two-body terms the effective hamiltonian also contains 3,4. . . A-body operators, This is a direct consequence of the antisymmet~zation principle on the quark level. It is now straightforward to define the effective electromagnetic current in the nucleon space via ($+~‘I$, = (~(‘A-q)IJ~)(y(3A-q)). (2.35) From the relation between the RGM wave function for the 3A-quark system of eq. (2.30) and the renormalized wave function of eq. (2.34a), the effective operator fieRf is given as ,
rA;
ri,. . . , rh) =
. . . , rA; rf,.
N-1’2(r,,
I X~WW P
(r;l,...,
r>;ry
. . , r>) ,....,
f-2)
x N-1’2(r~, . . . . , rx; r{, . . . , rf4)
xdrf...dridr;‘..+dr:, ,ra;r{,...,
4)
=
j
{
ii
@%ti.,
,
(2.36a)
&7_)fj(ri-
i-1
xJfs)JW
{ ii
j=l
@CYQ,,
d)}
, &*)8(q
-
$1)
xdtA.1* * * d!$* dr, * . *dr, .
(2.3617)
Again the spin-isospin color operators which are not shown here should be replaced by the co~esponding effective spin-isospin operators in the nucleon space. In this way, the effective nuclear force and electromagnetic current of the Anucleon system are obtained. These effective operators contain not only two-body but also three-body, . . . , A-body operators. Therefore, it remains to show that the one-body and two-body parts of the effective hamiltonian given in eq. (2.34b) and the effective electromagnetic current given in eq. (2.36a) are equivalent to the effective operators derived in the two-nucleon system. We will show this equivalence only for the effective current which comes from the quark impulse current. It can be shown for the other operators in the same way. First we perform the integration over the internal coordinates and evaluate the RGM kernel given in eq. (2.36b). Inserting eq. (2.15b, c) into eq. (2.36b), we obtain ~jPoMt = : J;mp)( i) + : I
i>j=l
+
(contributions
JFX~(~) +$ ,$mp)( ,k) Af’ NFX) i>j=l
from the permutations
more than two clusters) ,
involving (2.37)
where z(k) requires the omission of k in the sum over i and j. For simplicity, we do not show any functional dependence except for the index which denotes the particle number ij. The conventional impulse current J;,““(i) and the corresponding
Y. Yamauchi et al. / Quark exchange currents
516
exchange current, NY’
term of the RGM. kernel were already
is the exchange
case of the three-body “disconnected” of the current
J:“(Q),
which
comes
from the quark
impulse
given in eqs. (2.19a, b) and eqs. (2.23b) and (2.25), respectively. term of the norm kernel, current
which is given in eq. (2.6a). In the
in eq. (2.37), there are two distinct
contributions:
term like the third term in eq. (2.37), which is written and the exchange
a
as a product
part of the norm kernel in which no particle
appears
twice, and a “connected” term, which is not shown here. We will neglect the “connected” terms, because these can only be significantly different from zero if all three clusters come close together. On the other hand, the “disconnected” terms cannot be neglected because these represent the quite probable process in which the photon couples to a single cluster which does not exchange any quark with its neighboring clusters, while two other clusters not involved in the electromagnetic interaction exchange quarks, as shown in fig. 2. The remaining work involves the renormalization kernel for an A-nucleon system is given as
given in eq. (2.36a). The norm
q=‘+...,
(2.38)
N=l+
; i>j=l
where NFx) is the exchange term of the norm kernel, which is given in eq. (2.6). For simplicity, we treat NF”’ as a small quantity and neglect higher orders than N’. In this approximation N-l’* is given as N-l/*
= l-1
$
NljEx)+O(N2).
(2.39)
i>j=l
In the renormalization process, we must pay attention NpX) commute if they do not contain the same particle contain Ngx’.
the same particle because of the nonlocality For the first term of eq. (2.37), we then get
- : k=l z
to the fact that Jtrnp’ and but do not commute if they (momentum
dependence)
of
A;’ NFX’+O(N2)
Jf”P’(k)
i>j=l
i$,J:““)(i)
+
JOEY)
:
.’
(15)
i>j=*
-
;
k=l
J;“P’(k)
A;’ NFX’+O(N2)
.
(2.40a)
i>j=l
The first term of this equation is simply the conventional impulse current given in eq. (2.19a, b). The second term is the renormalization term given in eq. (2.27~). The third term has the same form as the “disconnected term” [the third term in eq.
Y. Yamauchi et al. / Quark exchange currents
(2.37)] except
for a minus N-‘/2
N--1/2
:
{
J;“P’(k)
sign. For the second
and third term of eq. (2.37), we get
i>fel J~x’(ii)JN-“2 =i>fz, J:Q’c’(ii)+O(N2), N-l/2
AF’ NrX’
= kg,
JimPI
Ag’
(2.40b)
NFX)
i>j=*
i>j=*
k=l
517
+O(N2).
(2.40~)
where we also count JF”‘( ij) as being of first order in N, because it is proportional to CP, the radial part of the norm kernel NEX. [See eqs. (2.6) and eq. (2.23) with eqs. (2.22b, c).] By summing up eqs. (2.40a, b, c), the total effective current becomes JF”) = i i=l
= ,g,
J:““)(i)
+
i
~jLQ’c)(q)
+
JE”“‘( i) + ;
f
~jL”ENl(ij)
i>j=1
i>j=l
JIPEC)( ij) .
(2.41)
t>j=l
As anticipated, the “disconnected” term has been eliminated by the renormalization process and the effective current is expressed by the currents which were already derived in two-nucleon system. [In ref. 31), Yazaki also showed the elimination of this “disconnected” term in a similar manner.] We can proceed in a similar way for the other operators and obtain eq. (2.28) and eqs. (2.29) in the approximation of neglecting more than three-body “connected” operators. In this way, we could prove that the effective NN potential and electromagnetic current which were obtained in two-nucleon system are applicable to nuclei with A > 2. Although we used the approximation of neglecting higher order terms in the expansion of N-“2 for simplicity, it is possible to show without this approximation that the “disconnected” terms will automatically drop out, and eq. (2.28) and eq. (2.29) can be derived by the renormalization procedure. Note that the expansion by orders of N should not be used in the actual calculation because NCEX) is no longer a small quantity at short-distances. It is reasonable for the “disconnected” terms to drop out automatically when we introduce the quark exchange current, because the wave function used to evaluate the currents has been derived from the Schrodinger equation including the effective NN potential, which itself results from quark exchange. Thus great care has to be taken to avoid double counting. Before ending this section, we want to comment on the multi-cluster problem in nuclear cluster theory. It is well known that for the 3a problem one does not obtain a satisfactory result if one uses effective (YLYpotentials with repulsive cores determined from CKXscattering. On the contrary, quite satisfactory results are obtained by models in which the forbidden states are taken into account. This result indicates that it is essential to take into account the forbidden states which give rise to the null states if complete antisymmetrization is performed. Since the permutations which operate on more than two clusters were neglected in our approximation, and
Y. Yamauchi
518
the same effective in the A-nucleon the quark nuclear
cluster
cluster
repulsion hand,
potential system, model
theory,
is almost
which
is obtained
our approach is quite
constant
for the two-nucleon
looks insufficient.
different
the oscillator
the same
the oscillator
et al. / Quark exchange currents
from that in nuclear
constant
which
as that of the nuclear which characterizes
system
However,
cluster
characterizes potential
the short-range
is used
the situation theory.
in In
the short-range
well. On the other repulsion
in NN
scattering is almost three times larger than that of the nuclear potential well, and the origin of the short-range repulsion is attributable not only to the Pauli principle, but also to the specific form of the quark dynamics (spin-dependent forces). Thus, the probability that two clusters will overlap is very small in quark cluster theory in contrast to the nuclear cluster theory. [The difference between quark cluster theory and nuclear cluster theory is investigated in ref. “).I Therefore, a good approximation will be obtained without taking into account any multi-quark component other than the 6-quark component, and by neglecting the permutations which operate on more than two clusters. In any case, we must wait for further studies to obtain a more quantitative conclusion about the role of 9-quark components, for example, or effective three-body forces and currents.
3. Quark exchange currents in nuclei 3.1. SHORT-RANGE
CORRELATIONS
WITH
THE
BETHE-GOLDSTONE
WAVE
FUNCTION
In this section, we discuss the calculation of the nuclear matrix elements of the quark exchange currents presented in the previous section. Since the many-body problem given in eq. (2.28) cannot be solved exactly, we use the nuclear shell model, which is well established for light nuclei. In this paper, we investigate the role of the quark exchange current in the magnetic form factors of certain light nuclei with a simple closed shell f 1 configuration (iSN, “0 and 39K). We take the nuclear wave function to be a single Slater determinant of single-particle states, and the singleparticle wave functions to be those of the harmonic oscillator model. While we expect the independent
single-particle
picture
to be a good (zeroth-order)
approxi-
mation, appropriate short-range correlations between the nucleons should be taken into account when we evaluate the matrix elements of the quark exchange currents, which contribute only in the short-range region. In our picture, the short-range repulsion itself comes from quark degrees of freedom and is described in terms of the effective NN interaction given in eqs. (2.13). Therefore it is reasonable to introduce short-range correlations using the effective NN interaction of eqs. (2.13). The consistency between the quark exchange currents and the nuclear wave function is shown to be important by examining the dependence of the gluon-quark exchange current contribution on the size (bparameter) of the nucleon in our previous study of deuteron I’). If we increase b, and do not take into account its influence on the NN interaction or the nuclear
Y. Yamauchi et al. / Quark exchange currents
wave function,
it is evident
of the quark exchange are proportional larger nucleon exchange
that a larger nucleon
currents.
to the function
overlap
understood
given in eqs. (2.27). The quark 9, the radial
the contribution
is that the quark exchange
of the cluster
size. This can be readily
current
size would increase
The main reason
to the probability
519
from the definition exchange
part of the norm kernel
one would expect a large enhancement of the quark we must now forget the effect that a larger nucleon
currents
which is increased current
for a
of the quark
is proportional
NEX given in eq. (2.6b). Thus, exchange current. However, size has on the short-range
repulsion between nucleons. The part of the effective NN interaction which leads to the short-range repulsion is also proportional to the cluster overlap. The quark exchange potential given in eqs. (2.13) is proportional to the function P’, the radial part of the exchange term of the norm kernel NEX given in eq. (2.6b). Therefore the range where the two nucleons begin to feel the short-range repulsion is extended to larger distances if the nucleon size b, is increased. This suppresses the nuclear wave function in the interior region, and thus prevents the quark exchange current contribution from becoming much larger. This was actually found and shown in fig. 11 in ref. I’). This analysis clearly shows that the consistency between the quark exchange currents and the corresponding short-range correlations is very important. Note that the two nucleons feel the short-range repulsion in exactly the same region where the quark exchange currents are effective. Therefore, in the present work we must take into account short-range correlations consistent with the quark exchange currents. We introduce short-range correlations by solving the Bethe-Goldstone equation given by p:+ -~_
r: + d
2mN 2m,bh
+
2mN
4
-+Qcq3(12)V,2 2m,b&
1
~~G~(12)=Eap?F’xIBpG)(12))
(3.1)
where the single-particle hamiltonian is taken to be the harmonic-oscillator hamiltonian. The oscillator length b, is chosen so that the oscillator wave functions reproduce
the experimental
r.m.s. radius
33,34): b, = 1.7 fm for “N; b, = 1.8 fm for
“0; b, = 1.9 fm for 39K. VI, is the effective NN potential given in eq. (2.13). In this work we consider only the central part and neglect the tensor and LS force of the effective NN potential V,,. We solve eq. (3.1) using an approximation given in ref. ‘“). In fig. 3, we plot the relative Bethe-Goldstone wave functions @zG)(r) given by @‘a”p”‘(12) = C (nl, NL; Aln,l,,
n,lz; A)[ Y,(C) x Y,(B)]~,~~G’(r)~~‘(R),
nlNL
(3.2) together
with the unperturbed one +27)(r) for s and p waves. Here, b, = 1.8 fm and n,l,; A) is the Talmi-Moshinsky bracket. The Bethe-Goldstone (4 NL; Alnil,, wave functions for higher partial waves are substantially the same as the unperturbed ones because of the centrifugal barrier. Since we define the c.m. and relative
520
Y. Yamauchi et al. / Quark exchange currents
The relative Bethe-Goldstone
I
’
I
1
w.f. ( n=O l=O )
I
I
H.O. JWJ JT=l
_ 9 0
(a)
0.0
I 0.5
I
I
I
1.0
1.5
2.0
r
hl
The relative Bethe-Goldstone
:: 6
0.0
_ I
_
I 2.5
3.0
w.f. ( n=O l=l )
I
I
I
I
I
0.5
1.0
1.5
2.0
2.5
3.0
r b-4 Fig. 3. (a) The relative Bethe-Goldstone wave function @sG’( r) with n = 0 and I = 0 according to eq. (3.2). Solid curve: for the JT= 0 channel; dashed curve: for the JT= 1 channel. Dotted curve: the unperturbed harmonic-oscillator wave function c$L$~’ (r). Note that the tensor and U-forces are not included when we solve the Bethe-Goldstone equation. In order to demonstrate that the Bethe-Goldstone wave function heals into the unperturbed one, these figures do not contain the normalization factor. [The Bethe-Goldstone wave function is normalized to one in the actual calculation.] (b) The relative Bethe-Goldstone wave function I?rzG) (r) with n = 0 and I= 1. Same notation as in fig. 3a.
coordinates as r = r, - r2 and R = ;(r, + Q), the unperturbed relative and c.m. wave functions, 4$‘(r) and 4%‘(R) are represented by the harmonic-oscillator wave functions with b parameters b, = Ab, and b, = fi b,. These results show that the healing distances are about 1 fm, which is reasonable and necessary for the independent-particle or pair picture to be valid. The validity
Y. Yamauchi et al. / Quark exchange
of the independent-pair
approximation
in which
currents
particle
521
correlations
that involve
more than two particles are neglected can be roughly estimated from the healing distance h. For example three-particle correlations can only be significantly different from zero if all three particles h3. Since distance
the healing the probability
distance
are found is short
simultaneously compared
for this to happen
within
a volume
with the average
of order
internucleon
is small and the independent-pair
picture
is valid. Our approximation of neglecting three-body to A-body quark exchange potentials and currents in sect. 2.3 can be justified using a similar argument. For example three-body quark exchange operators can only contribute if all three nucleons overlap simultaneously. Since the region where any two nucleons feel the short-range repulsion is the same as the region where these nucleons overlap, the probability for three nucleons to overlap is almost the same as the probability for all three nucleons to be found simultaneously within a volume of order h3. Then the probability for three-body quark exchange operators to contribute significantly is small since the healing distance h is short compared to the average internucleon distance. Since we started from a description of the nucleus in terms of quark degrees of freedom, it was not obvious that this microscopic picture of the nucleus could be reconciled with the independent particle shell-model. The magnitude of the healing distance is known to be very sensitive to the NN interaction, especially to the short-range part of the NN interaction which is described in terms of quark degrees of freedom in our model. However, our results indicate the validity of the method which we used to incorporate quark degrees of freedom into a nuclear structure calculation. This is important if one wishes to take into account quark degrees of freedom in nuclei. Next we want to comment on the approximation which we used when we solved the Bethe-Goldstone equation. We used the approximation given in ref. 26). This approximation
may be too simple to justify the comments
and Faessler solved the Bethe-Goldstone potential using a much better approximation are slightly
different
above. In ref. 35), Kurihara
equation with a similar effective NN given in refs. 25,27). (Their parameters
from the ones used here. They also considered
the admixture
of six-quark states with different b, parameters.) Their results also show that the healing distance is about 1 fm. We therefore expect that our results, given in fig. 3, will not change much if we use the better approximation for the Bethe-Goldstone equation, as in refs. 25,27,28). At this point, we think it is instructive to consider the dependence of the NN interaction and the quark exchange currents on the nucleon size or b, parameter. In ref. 36), Takeuchi et al. investigated the quark exchange effect on Gamow-Teller type P-decays. They took into account consistent short-range correlations from their corresponding quark hamiltonian with 6, = 0.6 fm. In addition, they evaluated the quark exchange effect for different nucleon sizes: b, = 0.6 fm and 0.8 fm without any short-range correlations in order to investigate the dependence of the quark exchange effect on the nucleon size. Comparing the results with and without the
522
Y. Yamauchi et al. / Quark exchange currents
short-range
correlations,
the estimation exchange
the short-range
of the quark
effect depends
exchange
strongly
on the nucleon
b, = 0.6 fm and b, = 0.8 fm. However, of the nucleon
size on the short-range
in ref. “), its influence
correlations
are found
effect. They
to be important
also reported
that
size by comparing
their results for
they did not take into account correlations
on the NN interaction
in this comparison. is quite important,
in
the quark
the influence As we showed
and no reliable
conclusion can be drawn without it. In addition it is not obvious that the independentparticle or -pair pictures are still valid for such a large nucleon size. As mentioned before, the nucleon size determines the range of the short-range repulsion which is extended to larger distances if the nucleon size is increased. Since the healing distance is always larger than the distance of the short-range repulsion, it is uncertain whether the independent-particle picture is still valid for large nucleon sizes. Therefore, we do not investigate the dependence nucleon size here but fix it at the relatively
3.2. NUCLEAR
MATRIX
ELEMENTS
OF THE
of the quark exchange current small value b, = 0.5 fm.
QUARK
EXCHANGE
on the
CURRENTS
Having defined consistent short-range correlations, it is now a straightforward task to evaluate the matrix elements for the quark exchange currents. In this work we investigate only the quark exchange current which results from the quark impulse current. We are still in a too primitive stage to be able to take into account the other quark exchange currents except in the deuteron. Therefore we will investigate only the simplest and most established one (i.e. the quark exchange current coming from the quark impulse current) and leave the other quark exchange currents for future study although they give a larger contribution in the deuteron 17-20). The magnetic form factors can be evaluated from the nuclear matrix elements of the magnetic multipole operator. The magnetic multipole operator Tyig is related to the space component of the electromagnetic current as follows:
7;9(q)=$$jdq^[Y,(q^)xJ];‘, First, we evaluate
the effect of the (short-range)
due to the Bethe-Goldstone corresponding nuclear matrix
wave function elements are
.
correlations given
in eqs.
(3.3) on the impulse (3.1)
and
current
(3.2).
The
Y. Yamauchi et al. / Quark exchange currents
where
K
stands
(-)i-mu,,K_m
for both
are single-particle
The abbreviation ation
.? = m
[ 01 denotes
definition
and
j=[~I-i
a coupling
is made
creation
and annihilation
of both angular
matrix
element
by multiplying
&,qfi(A
11YI
]]_I>W(JlJ1;
respectively.
The other abbrevi-
and isospin. factor
We use the
37). The shell model exp [q2bh/(4A)]
(in spin and isospin
A l)(-i)‘+‘-L-‘k’(A
in
space) reduced
(1Yr 1115)
G~(q2)~(1N’86~1)a(lN’IIJSiT,)(LfII YLIIG
X (&T~j(l(GL(q2)6~rO+
m I0
doubly
a”,,,=
I
N
X
operators,
by Edmonds
with an overall
the standard fashion 38). For the two-body matrix elements, we have
x&
momentum
given
a with
u’ and
is used here and in the following.
of the reduced
c.m. correction
l=j+f=sgn(K).
523
~~~~(R)jL(qR>~~p~i(R)R2
+ 6so 2
Ll
dR
X (411 V
IIlJFfpin(iq)
JL)i’+Ld-J(JIJYLdIll)
iW(lLJ1;
X (SfTfjIIGSE(q2)8dTO+GVE(q2)T\N)adTIIIISi T) a3 X
I0
~~~~(R)j,~(qR)((Lfll[
y&(R)
xv,lLIILi)~~~~,(R))R2 dR
X(If(IY,IIli)F~onv.~(~q)+SsoC2iW(L1~J1;Jl)i’d+L-J b X(Jll
x(LflI
YLII~~)(SFT~IIJGSE(~~)~~TO+
yLllLi)
Gr(q2)7~N’S,,,IIISiTi)
~~~,(R)iL(qR)~~p,‘,(R)R2
dR
X bus,,.,
.
(3.4b)
524
Y. Yamauchi et al. / Quark exchange currents
Here Ffpin(k), Fk.R(k) and &,.,(k)
are the integrals
over the relative
coordinates
given by
Im.h(kr)[@!$j) = lorn [@!z::) (rlj~~(kr>((~~IIIY,,(~)xV,lfll~i>~~~')
Fbpin(k) = F&mv.~ (k) =
(r)~~~)(r)-+$~(r)4~,)(r)]r*dr,
(3.5a)
0
&v.,(k)
-~~j(r)j,(kr)((l,ll[ The matrix
elements
of the impulse
Y,,(c)~~,l’II~J~~~(~))l~*dr.
current
are composed
of three
(3.5b) parts.
These
parts, which are denoted by the integrals Fgpi,(k), F&nv,R(k) and FE:,,,,(k) in eq. (3.4b) correspond to the matrix elements of the spin-magnetization current Jlspin], the convection current with its derivative operating on the c.m. motion J~Cconv:RI, and the convection current with its derivative operating on the relative motion ““:, ,, as defined in eqs. (2.23). J, CO Now let us turn to the two-body matrix elements of the corresponding quark exchange currents. Since the quark exchange current is written in a similar form as the impulse current (see eqs. (2.23)), the two-body matrix elements of the quark exchange current are given by essentially the same expression (3.4) with the exception of the functions Fipin( k), F~,,,,R (k), and F$,,.,(k). these take the form
For the quark exchange
current
= KTH!+l,“,/,(+;
k) + yPa~HL~,,ili(-;
k) + H’,~,n;r,(o; k) 7
(3.6a)
Ft-cm.&)
= ~~&,,n,r,(+;
k) + GH!t~~rn,,i(-;
k) + ff!+i@,
k) ,
(3.6b)
F:&,.,(k)
= S:J”n’;,,nir,(+;
k)+ G,P!$,,,r,(-;
Fg,i”(k)
k) + Plf;l,,,JO;
k) ,
(3.6~)
where
dt Y,,(~)H*(r, (0; k) Hl,,r,n~~
P$ln,rl(
f ; k)
= jrn j,(kr)[V~~‘(r)?P~~‘(r)
=
- ‘@~~‘!@~‘(r)]r’dr,
[ dr j dr’!PEy)(r)
(3.7a)
(3.7b)
Y&,(F)
mf,m,m, X-
P!$,,,JO;
(-)*‘i~‘(limilm(l,m,)
‘1
r’; k)!P$~‘(r’)Yl,,l(i’),
(-i)‘d
4?r
k) = loa [ p::rsi)
dh[Yld(k^)xP’(r,r’;
k)]l,~~~,~‘(r’)Y~i,i(;‘),
(3.7c)
(r)j,,(kr)((~,lI[~,(i)xV,l’lI~i>~~~’(r))
- S~~~‘(r)j,,(kr>((I,II[~d(i) ~~,l’II~i>~~~‘(~))l~*dr,
(3.7d)
Y. Yamauchi
et al. / Quark
exchange
currents
525
with
N;“2( On the right-hand interchange
r’)
r,
~~~G)( r) dr’
side of eqs. (3.6), the first and second
terms corresponding
.
(3.7e)
terms represent
to fig. lb and fig. lc, respectively,
the quark
while the third
term is the renormalization term. The matrix elements H( rt ) and P( f ) were already evaluated in previous studies of the deuteron ‘2,‘8,19)by expanding the relative wave function into locally peaked gaussians. The coefficients Si, and Y:r in eqs. (3.6) are defined ;s;,=,
as
= Fdi-=o = (s,~~lllor~ff’[(b>(-9)P36
(sTc’lI(Isi T)x (SfTfllElllsi T,)-’9
JF&=, = (S,T,tli20j~ff’[~7i’t)(-9)~~~ ‘sTc’]/j]Sj F:,>x {S,T,ill~l’l”‘iilSiT}-’
(3.8a)
*
(3.8b) (3.8~)
5-d?-=, = (s,~,~~~o’,~ff)[~T:4z)op36 (sTct]//jSi T) x (SfTfl]j$T~N’jljSiTi)-’ 7 Y;r,0
= (SrTf111201~ff”[~rr~q’(-9)P3,CsTc’]l]]SiTj) x (SfTfIll~a(lN’lllSi~)-l
9
(3.8d)
T,)-’ 9 Z&r=0 = (s,T,~l~o~“,“‘[~a~q’(-9)P36 ‘sTc’]JIISiTj) x {S~~~~~~~~~N’~l~Si X (S~~~))/~~(lN’U(IN’JJISi~)-l7
Y’ir=i = (S~T~~l120~ff’~SO’,“‘7:4,‘(-9~~~~Tc’ljll} ~~~~~ =(S~T~ll10~~ff”[~~~q’~~‘(-9)~~~Tc’]II~SiTi)
(3.8e)
X(S,~~JII~r~N’a~N’lllSi~)-’
.
(3.8f) (3.8g)
In eq. (3.4), the one-nucleon form factors Gk(q’), GG(q2), G”,(q2), and G”,(q’) are of gaussian type, as in eq. (2.19e) and (2.19f), which is unrealistic at high momentum transfers. Because we are interested in two-nucleon phenomena, rather than in the one-nucleon properties themselves, we simply replace these one-nucleon form factors by the corresponding empirical ones G”,( q2) = G’;(q2) = $( 1 + 9’10.71 GeV2)-‘,
(3.9a)
G”,( q’) = !+!%
(1+ q2/0.71 GeV2)-* ,
(3.9b)
(1 + q2/0.71 GeV2)-” ,
(3.9c)
(-$Jq2)
=
!!$b
in both the matrix elements quark exchange current.
of the impulse
3.3. THE QUARK
MAGNETIC
EXCHANGE
current
and the matrix
elements
of the
MOMENT
Now we investigate the q +O limit of the Ml operator, that is, the magnetic moment. This consideration is particularly useful in studying the structure of the
Y. Yamauchi et al. / Quark exchange currents
526
quark
exchange
current.
The quark
exchange
(QEc)(r, r’, R)=F[V, CL
It is expressed current
in units
of nuclear
JCQEC) is given
exchange current, two terms
The quark-interchange
P
(QW(r,
,J,
R)
magnetons
p-LN= e/(2m,).
together
exchange
r, r’, R) = P (QEC)(
moment
magnetic
with
~-l/2@,
The quark
exchange
Like the quark
operator
also consists
pcQ”)(r, r’, R) + pcREN)( r, r’, R) .
r”)(I.(EX)(rrr,
r”‘,
R)N-1/2(rrtr,
-FCD’(r,
of
(3.11a)
term pCREN) are given as r’)
N-1/2( r, r”)p(“‘( r”, R)N-“‘(
I* (REN)(r, r’, R) =
(3.10)
eqs. (2.23).
moment
term pCQlc) and the renormalization
=
is given as
xJCQEC)(r, r’, R, q)]‘,=o.
in eqs. (2.27)
the quark
magnetic
d,.”
d,.“‘,
(3.11b)
r”, r’) dr”
R)S(r-r’),
(3.11c)
where the direct term ~1~~) and the exchange term pCEX) can be evaluated from eq. (2.23) by taking the q +O limit as in eq. (3.10). As shown in the study of the deuteron I’), the contribution of this quark exchange current is zero in the deuteron magnetic moment and thermal np capture process. Let us investigate the quark exchange magnetic moment operator in the more general case. (i) 7’he isoscalar quark exchange magnetic moment. First, we consider the isoscalar part of the quark exchange magnetic moment. Taking the q + 0 limit in eqs. (2.23) and using
eq. (2.24b),
&&=+[L
the direct
r +LR+$Ty+cp
and exchange
)I >
terms are given as (EX) P-a~=0
(D) - PAT=O
NCEX’(r, r’),
(3.12)
where the index AT = 0 denotes the isoscalar part of the magnetic moment. By inserting these equations into eqs. (3.11), we can evaluate the (isoscalar) quark exchange given as
magnetic
Np”‘(r,
moment.
In eqs. (3.11), the renormalization
operator
N-“2
is
r’) = In ,C, T (1 -9(ST]O$r’[P$~TC’]]ST)($2”+‘}P1’z 3, 1, x4,, cb’Jz7?bq’(r)~~~‘Jbq)(r~)Y,m(r^)xs,YT,(r^’)x~T,
(3.13)
wave function with b = mb,. Here where 4,,,(b=JT75bq)(r) is the harmonic-oscillator we note that the functions +zCJTjSbq)( r) Y,,,,( r^) are eigen functions of P(r, r’), the radial part of the exchange term of the norm kernel NEX given in eq. (2.6b) with we can express N-‘12 by these functions as in eigenvalues ($)zn+f. Therefore
Y. Yamauchi et al. / Quark exchange currents
eq. (3.13). Since pLr&, is expressed the intrinsic
spin operators, pFE!ir,
as a sum of the orbital
it commutes
pLyTJ(r,
r’, R)=piffT’?i(r,
angular
momentum
and
with N-1’2. Thus, we get
r’, R) = -pFjT!!)(r, =p,Lyeo(r,
527
r’, R) R)
N-‘(r,
r”)NcEX)(r”, r’) dr”,
(3.14a)
r’, R)=O.
(3.14b)
r’, R)+pLFT)(r,
In this way, we have shown that the quark exchange magnetic moment does not contribute to the isoscalar part of the magnetic moment as long as we restrict ourselves
to quark
exchange
currents
connected
with the quark
impulse
current.
This is an expected result that is explained in greater detail in the study of the deuteron 12). As can be seen from eq. (3.12), the isoscalar magnetic moment operator is made up of the total orbital angular momentum and total spin of the system, which correspond to the q + 0 limit of the convection and spin-magnetization current in the impulse current. The total angular momentum or total spin of the system is fixed, whether it is described in terms of quark or nucleon degrees of freedom. Therefore, the quark exchange effect cannot be seen in the isoscalar part of the magnetic moments. Here, we wish to comment on the quark exchange currents connected with two-body pion and gluon quark-pair currents which we neglected here. In our study of the deuteron I*) we showed that there is a non-zero but small contribution from these quark exchange currents to the isoscalar deuteron magnetic moment. The above conclusion is therefore only valid as long as we restrict ourselves to quark exchange currents connected with the quark impulse current. (ii) 7’he isovector quark exchange magnetic moment.On the other hand, the quark exchange magnetic moment can contribute to the isovector magnetic moments. It can be expressed as a sum of the following terms: P
(QEC)=Il,jRQxE)+CL!~~~)+CL.~~EC)+~~EC).
(3.15)
We do not write the index AT = 1 here and in the following, because only the isovector part of the quark exchange magnetic moment remains. The meaning of the individual terms in eq. (3.15) is evident, given as follows:
if the corresponding
(D) P,XVR
lb
_‘I2 -
(N)
--‘2z
(N)
8i
- 4( Tya(lN)+
CD)_?
rx~
direct terms are
R,
T:y’a$N’) .
(3.16a)
(3.16b)
These four types of direct term can be derived from the following q + 0 limits of eqs. (2.23). The types &&RxV,and p L, are derived from the q + 0 limit of JICOnv:~17, where V, operates on eiq” or eiq’R, while F,,~~ is derived from the q-+0 limit of on eiq”. The q -+ 0 limit of JLspin] gives p:“‘. The Jt conv:R], where V, operates
528
Y. Yamauchi et al. f Quark exchange currents
corresponding the quark
exchange
exchange
magnetic
currents
moments
can be derived
from the q + 0 limits of
given in eqs. (2.23). In this case, V, does not operate
on eiq” but on H or P. One obtains
(3.17a)
xV,CP(r, r’)+(lo2), p:x)
= -pS(r,
(EX)_- -4S(r, &
(3.17b)
r’)L,,0je2ff)[(25-&‘-r$$)P$~TC)]+(lt,2),
(3.17c)
r’)O~~[(2alq~7~~~+a~‘~:4,~)P~~Tc~]+(1~2).
(3.17d)
In addition, both the direct and exchange derived from JlconvZR1, where V, operates p(LDR) = +( ~(1:) + @)LR, However,
the corresponding
quark
term contain another type I.L~,, which is on eiq’R. Using eq. (2.24c), one obtains .
p(LEb() = p(LD,)NcEX)( r, r’)
exchange
magnetic
moment
vanishes
(3.18) because
of the cancellation between the quark-interchange and the renormalization term. This can be readily shown using the same arguments as in the isoscalar case. Among the remaining four types of quark exchange magnetic moment, p.bQEC) has a particular property. It can be expressed like eq. (3.13) and is given as (b-~bq’(r)Y~m(~)Xs,7, pbQEC)=
c
c
c
s,r, S,T, nlm 1
X
“;‘,,,
_(_l)S,+T,+f
P 1
[s
T ff
~~=J~“q’(r’)YT,(3’)Xtsir
1(92n+r 3
JI-~F,[s~T,](~)~~+'
_(_l)S,+T+’ 2
2
x [-~(SiTiIO~‘[(20:4’719’
+o:“‘T:“Z’)P:S6T”]ls,Tf)(~)2n+f
+~(S~~~~~~~~~~N~~S~~~)~l-J1-~~~[~~~~~[~)2~+fJ1-~~~[S~~~~(~)2n+f~~ , (3.19a) with FP[ ST] = (ST\o~;@[ P$:Tc’]]ST) Let us investigate the In the case 1= even (Si , Ti) = (1,O) or vice in eq. (3.19a) reduces
.
(3.19b)
1. dependence of ~~~~~ on the relative angular momentum one finds that due to selection rules (S,, Tf) = (0,l) and versa, and FP[ Sf Tf] = FP[ Si Ti]. Thus the term in the brackets to
where the first and second term in the spin-isospin matrix element comes from the quark-interchange term, and the third or last term comes from the renormalization
Y. Yamauchi
term. The spin-isospin quark exchange
matrix
magnetic
et al. / Quark exchange currents
element
moment
of the cancellation
between
This is the reason
that the quark
in the thermal
np capture
spin structure.)
However,
529
in eq. (3.19c) is zero. Therefore
does not contribute
the quark-interchange exchange
in the case 1= even because
term and the renormalized
magnetic
moment
term.
does not contribute
12). (The other types do not contribute pbQEC) contributes
this type of
because
in the case I= odd. When
of their (S,, Tf) =
(q, T) = (1, l), the bracket in eq. (3.19a) can be reduced to a simple expression like eq. (3.19c), but the spin-isospin matrix element is not zero in this case. When (S,, Ti) = (1, l), and (Si, Ti) = (0,O) or vice versa, Fp[Sf, 7’f] # Fp[SjTi], and the parenthesis in eq. (3.19a) is not zero. Therefore this type of the quark exchange magnetic moment contributes in the case I= odd. This is also mentioned by Takeuchi et al. in their study of Gamow-Teller transitions 36). On the other hand, the other three types have only simple
selection
rules, which are the same as the ones of the
corresponding direct terms, because the permutation of quarks cannot change the rank of the operators, angular momentum, spin, or isospin, which determine the selection rules. It is well known that the two-body magnetic moment operator can be decomposed into the Sachs moment and the intrinsic part as 39*40)
pex= PSachs Using
current
+ hint
conservation,
kachs
=
k%achs =
3
the Sachs moment
im
d
x
[ VI,)
Dl
[R xje,(q = o)] .
y
pSilchs is shown
(3.20)
to be (3.21)
,
where VI2 is the NN interaction and D is the electric dipole operator. Eq. (3.21) shows that the Sachs moment is constructed only from the nuclear potential. In the case of the intrinsic part, current conservation does not determine this term, which needs a model to be specified. Following this decomposition, the quark exchange magnetic
moment p
can also be split into the above two pieces as
(QEC) = &2E$)
Sachs
This can be readily
verified
xr.)
(QEC) Pint
_ (QEC) -prxvR
+&yEc)+pbQEc).
from eqs. (2.23) and (2.27), together
(3.22)
with eq. (3.20).
Therefore ~~~~~ (QEC) is the Sachs moment, which is intimately related to the quark exchange potential given in eqs. (2.13). On the other hand, the other three types constitute the intrinsic part, which cannot be determined from current conservation alone. We summarize the above analysis. As long as we restrict ourselves to quark exchange currents connected with the quark impulse current there is no isoscalar quark exchange magnetic moment because of the cancellation between the quarkinterchange term and the renormalization term. The isovector quark exchange magnetic moment is made of four types: p$?kQ,“v”,‘, p (QEC) rxo, , piTEC), and ~2~~). Among
Y. Yamauchi et al. / Quark exchange currents
530
these types, pIpQx”v”,’ is the Sachs moment, exchange
potential
moment, rules
which
given
are not determined
for the corresponding
and ~$~~)
are non-zero
the case I,= Ii (ZO);
which can be constructed
in eqs. (2.13).
The other
by current
nuclear
matrix
three
conservation
elements
types
from the quark are the intrinsic
alone.
The selection
are as the following:
only in the case If= Ii f 1, while piTEC) is non-zero
in addition
pzEc)
is non-zero
I~$$kQx”v”.’ only in
only in the case Zf= Ii = odd.
Note that there is no quark exchange contribution in the case If= Ii = 0. Before ending this section, we want to comment on the work of Ito and Kisslinger 41). They evaluated the magnetic moments in certain nuclei with a closed shell* 1 configuration using a hybrid quark-baryon model. In their model, the two-nucleon system is represented by the six-quark state in the (OS&~ configuration for relative distances smaller than some boundary radius RB. For distances greater than or equal to R,, they used the two-nucleon description. The six-quark probability is determined from the probability current conservation and continuity across the boundary radius. Therefore their six-quark probability depends only on the twonucleon wave function and its derivative at r = RB. They investigated the influence of the above (OS&~ six-quark state on nuclear magnetic moments. However, they did not consider the intimate relation between the short-range part of the NN interaction and the quark degrees of freedom which we have strongly emphasized here. In our model, the six-quark system is described by a quark hamiltonian, which then leads to a dynamical description of the short-range part of the NN interaction. One should note that a six-quark probability defined as in the work of Ito and Kisslinger contains much of the conventional NN component, which is non-zero even at short distances. In contrast to their model, we could single out that part of the quark effect, which is already contained in the conventional model via the NN interaction and/or the nuclear wave function through the renormalization term given in eq. (3.11~). Actually, we showed that because of the cancellation between the quark-interchange term and the renormalization term there is no quark exchange effect in the case If= Ii = 0, which corresponds to their contribution from the (0~~1~)~ six-quark state. The quark exchange magnetic moment is non-zero not in the case lr= Ii = 0, but only in the other cases which they did not investigate. [The main contribution to the quark exchange magnetic moment comes from the off-diagonal elements
between
the relative
s- and p-waves
as we will show in the next section.]
4. Results and discussion 4.1. MAGNETIC
MOMENTS
First, we show the the effect of the quark into two parts, i.e., the and can be absorbed
FOR
NUCLEI
WITH
A = 15, 17 AND
39
numerical results for the magnetic moments. In our model, degrees of freedom on the magnetic moments is decomposed one which generates the short-range part of the NN interaction into the proper definition of the NN interaction and/or the
Y. Yamauchi et al. / Quark exchange currents
531
NN wave function, and the remaining part, which cannot be described in that way.
The former contribution is taken into account by considering short-range correlations in the evaluation of the impulse current. The latter is treated as a two-body current, that is the quark exchange current. Let us compare the contributions of the short-range correlations and the quark exchange current. As mentioned in the previous section, the contribution of the quark exchange current to the isoscalar magnetic moment vanishes. [Since we do not include the tensor and LS forces, short-range correlations do not contribute to the isoscalar magnetic moment either.] We therefore list only the isovector magnetic moment in table 1. If the contribution of the quark exchange current were negligible compared to that of the short-range correlations, the quark degrees of freedom could be included in a properly chosen NN interaction. In this case, there would be no need for us to consider quark degrees of freedom in the form of quark exchange currents. However, our results indicate that this is not the case. The quark exchange current contribution is much larger than the contribution of the short-range correlations. Therefore we have to consider quark exchange currents explicitly. It has long been recognized that two main mechanisms play important roles in nuclear magnetic moments: (1) the modification of the nuclear wave function by a residual interaction via (second-order) con~guration mixing 42*43)and (2) meson exchange currents44) (together with the effect of using modified nuclear wave function in their evaluation ““)). For reasonable estimates of these effects on nuclear magnetic moments, we refer to the results of Arima et al. 45) and list them in table 1. We emphasize that it is not our purpose to compare our results with experiment. If it were, we should have at least considered these effects in our model by choosing our effective NN interaction as the residual interaction. Although the (short-range) correlations taken into account via the Bethe-Goldstone wave function correspond TABLE 1 Corrections
to the isovector
magnetic
moments
expressed
as a percentage
of the Schmidt
values
WPA7”‘)/b4:~~d,) A
15 17 39
QEC
SC
-9.11 1.64 -20.91
-1.62 0.20 -3.32
CM 5.49 (20.0) -16.93 (-18.1) 42.22 (67.4)
MEC
CM + MEC
EXP
7.18 15.52 -48.32
12.67 -1.41 -6.10
11.13 1.34 -38.42
QEC: the quark exchange magnetic moment; SC: the effect of short-range correlations due to the Bethe-Goldstone wave function on the impulse current; CM: the effect of (second-order) configuration mixing; MEC: the meson exchange magnetic moment; CM + MEC; the sum of the configuration mixing and meson exchange magnetic moment; EXP: experimental data. For the CM, MEC and CM+MEC, we refer to the results of Arima ef nf. 45) given in their table 7.1. The numbers in parentheses refer to the Paris potential while the others are based on the Hamada-Johnston potential. The Schmidt values of the nuclei considered here are 0.4509 n.m. for A= 15,3.3530 n.m. for A = 17, and 0.5117 n.m. (Y t$7hcmfdt) for A = 39.
Y. Yamauchi et al. / Quark exchange currents
532
to configuration into account
of configuration detailed
mixing
within
the correlations mixing.
comparison
the two-particle-one-hole
induced
Thus,
the present
with experiment.
these effects in our model,
by the tensor analysis
Although
space,
we did not take
force, which is the main source is not sufficient
it is straightforward
we are still in a too primitive
to warrant
a
to evaluate
stage to be able to take
into account every possible effect using the microscopic quark model. We therefore ask whether there is any chance of seeing the quark exchange effect in nuclear magnetic moments, or in other words, whether the quark exchange effect is negligible or not. As seen from table 1, the quark exchange current is almost of the same size as the configuration mixing or meson exchange currents in nuclei with A = 15. In nuclei with A = 17 and A = 39, the quark exchange contribution is smaller than the contribution of meson exchange currents or second order configuration mixing. However, it is not negligible. Since a large amount of the configuration mixing effect is cancelled by the meson exchange current, the sum of these contributions is almost the same as the quark exchange current contribution. In any case, these two contributions are quite model dependent, which can be seen in table 1 by comparing the results for configuration mixing due to the Hamada-Johnston potential and the Paris potential, respectively. [See also table 5.2a in ref. “‘).I Therefore, in order to draw any definite conclusion we must wait for a more consistent calculation of these two contributions in our model. In table 2, we show more details of the quark exchange magnetic moment. As already investigated in the previous section, the quark exchange current can be decomposed into two parts, i.e., the Sachs moment and the intrinsic part. Using current conservation, the Sachs moment can be expressed via the commutator of the NN potential with the electric dipole operator as in eq. (3.21). The Sachs moment is completely determined by a given potential. The results in table 2 show that the major contribution to the quark exchange magnetic moment comes from the Sachs (QEC). The contribution of the intrinsic quark exchange moment, which moment pRXV, is the sum of the other types, is rather small. Depending on the nucleus, the effect of the intrinsic quark exchange magnetic moment is of the same size or smaller than the effect of the short-range correlations. In order to avoid any model dependence,
the evaluation
of the Sachs moment
has ususally been performed using the phenomenological approach 44). In this approach one evaluates the Sachs moment with the help of eq. (3.21) using a phenomenological potential. The meson exchange contribution in table 1, which is evaluated in ref. 44), contains the Sachs moment due to the Hamada-Johnston potential. Since the main contribution to the quark exchange moment ~2::’ is of Sachs moment type, it seems that this type of quark exchange effect might already be included in the Sachs moment due to a phenomenological potential. However, the Sachs moment is not determined by on-shell NN scattering. It is possible to perform a unitary transformation which alters the inner part of the wave function without changing its asymptotic form. Such a unitary transformation also changes
Y. Yamauchi et al. / Quark exchange currents TABLE
Contributions
of the quark
exchange magnetic as a percentage
533
2
moment to the isovector of the Schmidt values
magnetic
moments
(QEC)) (CL&
(BG)
(HO)
-8.042 -8.185 0.142 -17.503 -17.716 0.213
-9.106 sP pd -20.272 sP pd
(BG)
1.644
(HO)
sP pd df 3.489 sP pd df
@G)
-20.911 sP pd df
(HO)
fg -46.893 sP pd df fg
A = 15 -0.011 -0.014 0.03 -0.062 -0.067 0.005
1.729 1.806 -0.078 0.001 3.792 3.908 -0.117 0.001
A = 17 0.003 0.004 -8 x 1O-4 4 x 1om6 0.016 0.017 -0.001 4 x 1o-6
-18.967 -19.945 0.996 -0.018 1 x 1om4 -41.704 -43.175 1.486 -0.015 1 x 1o-4
A=39 -0.014 -0.020 -0.006 -2 x 10-s 3 x 1om6 -0.085 -0.095 0.009 -1 x lo+ 2 x 1o-6
PP dd
PP dd ff PP dd ff
PP dd ff ff PP dd ff ff
expressed
(QEC)) (P..
-0.442 -0.455 0.013 -0.982 -0.993 0.011
-0.610 -0.610
0.112 0.114 -0.022 3 x 1o-4 0.248 0.250 -0.002 3 x 1o-4
-0.200 -0.199
-1.724 -1.724
-2 x 10-4 -0.566 -0.566 -2 x 1o-4
-1.305 -1.304
-0.625 -0.645 0.022 -0.002 1 x10-s -1.397 -1.414 0.019 -0.002 1 x 1o-5
-9 x 1om4 -3.706 -3.705 -9 x 1om4
Separate contributions for the four types of operators given in eq. (3.15) are shown as a percentage of the Schmidt values. The quark exchange magnetic moment is evaluated with the relative BetheGoldstone wave function (BG) and with the harmonic oscillator wave functions (HO) for comparison. The contributions from individual partial waves are also shown.
the NN potential unitary
but does not affect the on-shell
transformation
also
changes
the
Sachs
NN scattering moment.
Thus,
phase there
shifts. This are many
phase-equivalent potentials which give the same on-shell NN scattering but give a different Sachs moment. Therefore it is worthwhile to investigate the Sachs moment due to the quark exchange mechanism. Since the short-range part of our NN interaction is highly nonlocal, it would give a rather different contribution to the Sachs moment as compared to the contribution of a phenomenological local potential. Let us study the dependence on the partial waves of the relative motion and the effect of the short-range correlations on the quark exchange magnetic moment. As
Y. Yamauchi et al. / Quark exchange currents
534
mentioned partial
in the previous
waves allowed
contributions
from higher partial
able since the quark nucleons centrifugal
section,
overlap, barrier
there is a selection
by selection exchange
rule for each type. Only the
rules are listed in table waves become
current
smaller
and smaller.
This is reason-
is only effective
in the region
where the two
and the two nucleons for higher partial
2. We can see that the
are prevented
waves. Comparing
Bethe-Goldstone wave function with the results function, we notice that the latter contribution
from coming
closer by the
the results using the relative
using a harmonic-oscillator is roughly twice as large
wave as the
former. This shows that short-range correlations are fairly important when evaluating the quark exchange magnetic moment. [We note that the effect of short-range correlations on the quark exchange current is small for higher partial waves. This is reasonable because the relative Bethe-Goldstone wave function is almost the same as the unperturbed one for partial waves higher than relative p-waves.]
4.2. MAGNETIC
FORM
FACTORS
OF 15N, “0
AND
=K
Now we turn to the quark exchange effect in the magnetic form factors of ‘*N, “0, and 39K. In the corresponding magnetic moments, we saw that the quark exchange effect is much larger than the effect of short-range correlations on the impulse current. We also saw that it is important to consider short-range correlations when evaluating the quark exchange magnetic moment. Let us see whether a similar behaviour can be seen in the magnetic form factors. For this purpose we examine the effect of short-range correlations due to the relative Bethe-Goldstone wave function on the impulse current and the quark exchange current. In figs. 4-6 we plot the nuclear matrix elements of the magnetic multipole operators, J T , and we show the effect of the short-range correlations on i(pJAJ Tf;“‘(q)ll W A ,) the impulse current and quark exchange current for each individual multipole form factor separately. The three contributions (quark exchange current with and without short-range correlations and the effect of short-range correlations on the impulse current) are plotted in fig. 4 for the Ml form factor of 15N, in figs. 5a, Sb, and 5c for the Ml, M3, and M5 form factors
of “0,
and in figs. 6a and 6b for the Ml and
M3 form factors of 39K. As a reference we also plot the single-particle value, that is, the impulse current evaluated with harmonic-oscillator wave functions. For the Ml form factors, the quark exchange contribution is larger than the effect of short-range correlations, although it becomes smaller after the diffraction minimum. [The quark exchange current has a minimum at q = 4.3 fm-’ in 15N.] The quark exchange contribution in the M3 and M5 form factors is much smaller than the quark exchange contribution in the Ml form factor. Later we will investigate the reason for this. Thus, the effect of the quark exchange current on the M3 and M5 form factors is comparable to or smaller than the effect of short-range correlations on the impulse current. It is also generally true that the quark exchange current shows a different dependence on the momentum transfer than the effect of the
Y. Yamauchi et al. / Quark exchange currents
535
Fig. 4. Ml form factor of “N. Dotted curve: the single-particle value that is the impulse current evaluated with harmonic-oscillator wave functions (IMP); dashed curve: the quark exchange current evaluated with Bethe-Goldstone wave functions (QEC): long-dashed curve: the quark exchange current evaluated with harmonic-oscillator wave functions (QEC(H0)); dash-dotted curve: the effect of short-range correlations due to the Bethe-Goldstone wave function on the impulse current (SC).
short-range correlations on the impulse current. Therefore the quark exchange effects cannot be simulated by taking the effect of short-range correlations on the impulse current into account. Let us examine the effect of short-range correlations on the quark exchange current. The contribution of the quark exchange current evaluated without shortrange correlations is two or three times larger than the contribution of the quark exchange current evaluated with short-range correlations. Thus, as a general tendency, we recognize that the effect of short-range correlations on the quark exchange current is as important in the magnetic form factors as it is in the magnetic moments. However, the dependence of the quark exchange current on the momentum transfer is exactly the same, whether it is evaluated with or without short-range correlations. The position of the diffraction minimum of the quark exchange current (if there is one) remains unchanged by the inclusion of short range correlations. This might indicate that the momentum transfer dependence of the quark exchange current is determined mainly by the c.m. motion, which remains the same whether short-range correlations are included or not.
Y. Yamauchi et al. / Quark exchange currents
536
10-l t
I
I
....
/_
..,
j IMP ...___.. QEC I _ -QEC(HO) --SC -
lo-”
10-s
1 , , ,t,li
1,
1o-5
0.0
2.0
1.0
9
3.0
0.0
Lb-‘I
1.0
,
2.0
b)
1 3.0
[fm-‘1
I
I
+
..
IMP ._....._ QEC _ _QEC(HO) _-SC
1o-2r
10-3 -
/
9
10-l -
2 -2
1
;
BR
Fig. 5. (a) Ml form factor
of “0.
(b) M3 form factor of “0. as in fig. 4.
(c) M5 form factor
of “0.
Same notation
Y. Yamauchi et al. / Quark exchange currents
L”
0.0
2.0
1.0 q
3.0
--
1.0
2.0
3.0
9 he’1
b-7
Fig. 6. (a) Ml form factor
0.0
of 39K. (b) M3 form factor
of 39K. Same notation
as in fig. 4.
Next, we would like to comment on the separate contributions from each type of quark exchange current to the Ml form factor. As shown in eqs. (2.27) with eq. (2.23), the quark exchange current can be decomposed into the three types: Jl$“’ from the quark spin-magnetic current, and Jlzn::k, and JE?,‘$:!l from the quark convection current. The major part of the quark exchange current comes from the isovector part of the type J~~O~?R1. Especially its off-diagonal matrix element between the relative s-wave and the relative p-wave is dominant. This fact is related to the dominance of the Sachs moment p$$z) in the quark exchange Note that JLzOt::kl corresponds to ppXEVt’in the q + 0 limit.
magnetic
moment.
Keeping these properties in mind, let us now investigate the dependence of the quark exchange current on the multipolarity of the magnetic form factor. The effect of the quark exchange current on each Ml form factor is small but nonnegligible. In the higher multipole form factors, that is in the M3 and M5 form factors, the quark exchange contribution is smaller than in the Ml form factors. In particular, in the M5 form factor, the quark exchange contribution can practically be neglected. This property is determined by the spin structure of the quark exchange current. As mentioned above, the type JCQEC) Iconv:R1provides the major contribution for the Ml form factors. The quark exchange current, which comes from the quark convection current, is a rank zero operator in spin space like the nuclear convection current although it has a spin dependence given in eq. (2.11) and (2.24a). Thus the orbital angular momentum operators acting on the relative and c.m. motion must be coupled
Y. Yamauchi et al. / Quark exchange currents
538
to rank 3 or 5 corresponding to the M3 or M5 form factors, respectively. trivial that the quark exchange contribution from the quark convection suppressed
when
the rank
of the relative
orbital
angular
large. By selection
rules, a larger rank for the relative
to matrix
elements
between
the initial
or final states. Thus, the corresponding
higher
partial
momentum
motion
operator
waves for the relative
It is rather current is operator
wave functions
quark exchange
is
corresponds
contributions
of are
suppressed by the centrifugal barrier. Nevertheless, even in these higher multipole form factors there is a quark exchange contribution for relative s- and p-waves because the remaining rank can be carried by the c.m. motion. However, this probability becomes smaller with increasing multipolarity because of the structure of the nuclear matrix element given in eq. (3.4b). In particular, the quark exchange contribution coming from the quark convection current cannot contribute to the M5 form factor of “0. The single-particle state Od,,, (I = 2) with core (I= 0) cannot couple to rank 5 without a spin operator. As a result, the quark exchange contribution becomes rather small for higher multipoles. This behavior might change significantly if the tensor correlations on the quark exchange current were included. In other words, the crossing term between the quark exchange current and the configuration mixing induced by tensor force might change this behavior significantly. In order to see whether the quark exchange current can be seen in these form factors, we show the square of the magnetic form factor IF,(q)\* in figs. 7-9. It is defined through the unpolarized electron-nucleus scattering cross section in the laboratory frame da Z* dw PC__ 77 d0 M [I&(q)12+G+tan2 d0 Ott and expressed
by the multipole
form factors
G@Wq)('l
,
(4.1)
as follows:
(4.2) where 2 is the atomic
number
of the nucleus,
and da/dRl,,,,
and n are the Mott
cross section and recoil factor. More details are given in ref. 38). In these figures we plot the contributions of the configuration mixing and the meson exchange current for comparison. We refer to the results of Arima et al. 47) for “0 and those of Blunden and Caste1 48) for “N and 39K. We also plot the experimental data 49-52). In fig. 7 we show the square of the magnetic form factor in “N. The difference between the dotted curve and the dashed curve demonstrates the well-known decrease of the cross section due to the configuration mixing. It reduces the cross section in the whole region and worsens the agreement with the experimental data. The meson exchange current gives a large enhancement and reproduces the experimental data well. The quark exchange current gives a small enhancement at low momentum transfers, while at high momentum transfers it has virtually no effect.
Y. Yamauchi et al. / Quark exchange currents
539
r
__- ._ IMP+CM _ _ IMP+CM+MEC _
IMP+CM+MEC+QEC
value, that is, the impulse Fig. 7. The magnetic form factor of “N. Dotted curve: the single-particle current evaluated with harmonic-oscillator wave functions (IMP); dashed curve: the impulse current evaluated with harmonic-oscillator wave functions + the configuration mixing = IMP+ CM; long-dashed curve: IMP+CM+ MEC (meson exchange current); full curve: IMP+ CM + MEC +QEC (quark exchange current). The experiment data are from Singhal et al. 49). For the estimates of CM and MEC, we refer to the results of Blunden 48). The quark exchange current is evaluated with the Bethe-Goldstone wave function.
This is related to the fact that the quark exchange current has a minimum at q = 4.3 fm-’ and that it is already small at around q = 3 fm-‘, as shown in fig. 4. Here we comment
on the contribution
of the impulse
current
and the configuration
mixing which are evaluated in ref. 48). In their calculation Blunden and Caste1 did not take into account the one-nucleon electromagnetic form factors GE and Gr,, given in eqs. (3.9) and their result could not reproduce the experimental data. However, this discrepancy is reduced by taking into account the one-nucleon form factor which is contained in our figures. For I70 the quark exchange current gives only a negligible contribution (see fig. 8). Although the quark exchange current gives a nonnegligible contribution in the Ml form factor as shown in fig. 5a, the Ml form factor itself is the minor contribution to the cross section. The contributions from the M3 and M5 form factors where the quark exchange current is suppressed hinder the contribution from the Ml form factor to the cross section. In the case of 39K, which is shown in fig. 9, the quark exchange current gives a small enhancement at low momentum transfers, while at high momentum transfers it has virtually no
540
Y. Yamauchi et al. / Quark exchange currents
__..._ IMP+CM _
_
5*I0-e
IMP+CM+MEC IMP+CM+MEC+QEC
:
I
0.0
1.0
:
-
I 2.0 9
3.0
Pm-‘1
Fig. 8. The magnetic form factor of “0. Same notation as in fig. 7. The experimental data (solid circle) are from Hynes et al. 50) and from Kalantar et al. ‘I). For the estimates of CM and MEC, we refer to the results of Arima et al. 47).
effect. This is also due to the hinderance by the M3 form factor where the quark
of the Ml form factor in the cross section exchange contribution is negligible.
Summarizing the results of this section, we have found that for magnetic moments the quark exchange current gives a non-negligible contribution. The major contribution of the quark exchange magnetic moment comes from the Sachs moment, which is intimately related to the quark exchange potential, which produces the short-range repulsion. For the magnetic form factor, the major contribution of the quark exchange contribution also comes from the quark convection current. However the quark exchange contribution is rather small from 0 to 3 fm-‘. Especially in the high momentum transfer region it has virtually no effect. This is mainly due to the spin structure of the quark exchange current connected with the quark convection current. This spin structure makes the quark exchange contribution very small in the M3 and M5 form factors, which are dominant at high momentum transfers. 5. Conclusion We have calculated the magnetic A = 15, 17, and 39 nuclei in a quark degrees of freedom into a conventional
moments and the magnetic form factors of cluster model. In order to incorporate quark nuclear structure calculation, we introduced
Y. Yamauchi et al. / Quark exchange currents
541
IMP
t
__-.__ IMP+CM
_ t
‘,
_ IMP+CM+MEC
_
IMP+CM+MEC+QEC
Fig. 9. The magnetic form factor of 39K. Same notation as in fig. 7. The experimental data are from LapikaS ‘*). For the estimates of CM and MEC, we refer to the results of Blunden @).
effective operators in the nucleon space. These effective operators, that is, the effective NN interaction and electromagnetic current, were derived in the following way. We started with a description of the six-quark system characterized by a quark hamiltonian using the quark cluster model. Interpreting the renormalized wave function and the equation of motion, which determines the renormalized function, as the NN wave function and the corresponding Schrodinger equation, we could derive the effective NN interaction. In the same manner we could derive the effective electromagnetic current, which was written in terms of nucleon dynamical variables, and which
yielded
exactly
the same results
as the microscopic
description
of the
electromagnetic interaction. We have shown that the internal structure of the nucleon had two major effects on the form of these effective operators. One major effect revealed itself in the impulse current, the one-pion exchange potential and the corresponding one-pion exchange current through microscopically calculated vertex factors for the yN, TN and yNn vertices. In conventional nuclear theory these are regarded as phenomenological input parameters from the beginning. Therefore this type of quark effect does not provide a clear signal for quark degrees of freedom in nuclear physics. However, the quark degrees of freedom also revealed themselves through the quark exchange potential and the corresponding quark exchange currents. The quark
542
Y. Yamauchi et al. / Quark exchange currents
exchange
potential
repulsion
in NN
was previously
necessary
counterparts
scattering.
The
of the quark
conservation.
These quark
tional
theory and represented
nuclear
ties of the nucleus.
exchange
shown quark
to be responsible exchange
exchange currents
currents
potential,
for the short-range were
introduced
as
in order to satisfy current
had not been treated
in the conven-
the quark effect in the electromagnetic
We have also shown that the description
of the A-nucleon
propersystem
with this effective NN interaction and electromagnetic current is equivalent to that of the 3A-quark system with the microscopic quark hamiltonian and electromagnetic current, as long as we restrict ourselves to effective two-body operators in the nucleon space. In this work, we studied to what extent the quark exchange current connected with the quark impulse current influences the magneic moments and magnetic form factors of nuclei with A = 15, 17 and 39. We found a non-negligible contribution from this quark exchange current in the magnetic moments. Although this quark exchange current contribution is within the theoretical ambiguity of configuration mixing and meson exchange current effects, it might be seen in the magnetic moments. The numerically most important part of the quark exchange magnetic moment was of Sachs moment type. The Sachs moment was expressed as a commutator between the electric dipole operator and the quark exchange potential, which produced the short-range repulsion. In the case of the magnetic form factors, the major contribution to the quark exchange current also came from the quark convection current, which is related to the Sachs moment in the q + 0 limit. Because of the spin structure of the quark exchange current, the quark exchange contribution turned out to be very small in the M3 and M5 form factors. As a result the quark exchange contribution to the square of the magnetic form factor, which is related to the unpolarized electron-nucleus cross section, was rather small for momentum transfers from 0 to 3 fm-‘. In particular, at high momentum transfers, the quark exchange current had virtually no effect. These results are quite different from the corresponding results in the deuteron. In the deuteron there was no quark exchange contribution from the quark exchange impulse current in the q -, 0 limit, and the quark exchange effect could only be seen at high momentum
transfers.
In addition,
the major contribution
to the quark exchange impulse current there came from the quark spin-magnetization current, while the quark exchange current connected with the convection current gave only a minor effect. Therefore, the behavior of quark exchange currents in heavier nuclei can be quite different from the corresponding results in the deuteron.
We wish to thank Profs. K. Yazaki, K. Shimizu, M. Ichimura and Drs. H. Sagawa and W. Bentz for help and numerous stimulating discussions. A.A. thanks the “Alexander von Humboldt Foundation” and the staff of the Institut fur Theoretische Physik in Tiibingen for their hospitality. A.B. gratefully acknowledges the receipt of a post-doctoral fellowship from the “Japan Society for the Promotion of Science”.
Y. Yamauchi
et al. / Quark exchange currents
543
Appendix In this appendix,
we show some details
direct terms can be easily calculated, terms. They are written
of the derivation
of eqs. (2.22). Since the
we show only the evaluation
in terms of the functions
H’, P* defined
of the exchange as
H+ e-&b;q2
Hp+
(r,
r’;
t4)
=
0 P-
First we rewrite
the 6 function
as follows
8(r-a)=(&)3(y)3’4[dpelp[-:b:p2]
x(&)“‘exp[--&(r-(n+iTp))2], and insert
it into eq. (A.l).
We then get 29)
(~~)(r,r1:~q,=(~)6(~~‘2 q where the GCM
e-ib;q2
kernels
(A.2)
Jdp ;;)“i’“i(
2% r+ iyp,
Jdp’exp[-j~~(p*+pf’)l % r’+ iyp
H*(GCM), P*(GCM) are defined
Csi,
$;
,* , I. 2q) ,
(A.3)
as
44)
(A.4)
Y. Yamauchi et al. / Quark exchange currents
544
It is well known that this GCM kernel given in (A.4) is equal to the matrix involving a two-centered shell model wave function given as 29)
element
x ,lfIl (IroT PI -&‘j> ,fi4 $“(PI+ fsj) dp, dp, - * . dp, . We can derive eq. (A.4) from eq. (AS) by transforming
the single particle
(A-5) coordinates
appearing in eq. (AS) to internal coordinates inside the clusters, the relative coordinate between the two clusters and the c.m. coordinate of the six-quark system. Because our operators do not contain any dependence on the c.m. coordinate, the integration over the c.m. coordinate gives unity and we can get eq. (A.4). Performing the integrals of eq. (AS), we get the GCM kernel as (GCM) -~(S~+S~-:s,.s,)-fb~k’ 9
I
1
X
( i $tsiFS))
exp [ i$k * (Si f Sj)] .
(A.6)
4
The functions omitted
P*(GCM) also contain
a term which is proportional
that term since it will not contribute
to the magnetic
to k. However
we
or electric form factor.
The RGM kernel is obtained by inserting this equation into eq. (A.3). For the functions PkcGCM), the derivation of eqs. (2.22d, e) is not straightforward. Inserting eq. (A.6) into eq. (A.3), we get (A-7) Eqs. (2.22d, e) can be derived from this equation using partial integration and omitting the term proportional to k which does not contribute to the magnetic form factor. We did not use eq. (A.7) but eqs. (2.22d, e) instead because in eqs. (2.22d, e) it is more transparent that H and P represent quark exchange corrections to the
Y. Yamauchi et al. / Quark exchange currents
functions
eik’r and elk.? V, respectively.
The exchange
545
term of the norm kernel given
in eq. (2.6) can be derived in the same manner. We can also derive eq. (2.6) using the q + 0 limit of the operator H’ since the right-hand side of eq. (A.l) contains then only the operator Before ending
PX6.
this appendix,
we want to add another
P given in eqs. (2.22d, e) and the corresponding Eq. (3.7~) can be rewritten P$+ri(
f; k) =
matrix elements
given in eqs. (3.7).
dr’ P$‘~‘(r)Y$,,,(i)SP(r,
r’)
r’) X (+,TV,,)]fn
X[ Yld(rf
-$k(tklr
+ r’l) exp (#k’)
-$j,($klr
- r’j) exp (ibtk2)
where V, and b,. operate one should pay attention
on the operators
as
C ( -)2’f~1(limiZ*]Z,mf) mf.m,mi
X
comment
>
p::‘(r’)
Yl,,i(i’)
,
(A.@
only on final and initial wave function, respectively. Here to the combination of Yr and V appearing in eq. (A.8).
In the renormalization term which is represented by the matrix elements, P$!l,,,lL(O; k) given in eq. (3.7d) a similar combination appears. Although P~l,,,,l(O; k) contains V, the d/dr does not contribute for l, = 1.This can be seen from the following relations
j,(kr)(~,ll[Y,(i)XV.l’II~i) =k[C+j,+I(kr)U~ll~+l(i)lJli>+c-j,-,(kr)(I,II Y,-,(;)II4)1, (A.94 (Z~+Z~+1+2)(1~~1*+1+1)(1*~1~+1+1)(1~+Z~~1) 41(1+ 1)(2l+
1)(21+3)
9
(l~+l~+Z+l)(l~~l~+l)(l~~l~+l)(l~+l~~1+1) 41(1-r 1)(21-t 1)(21Using
these
relations,
P~,~,/~i(O; k) can be written
1) as a sum of H$:,,,l(O;
(A.9b)
(A.9c) k) and
H$,i,i(O; k). However, there is no corresponding equation to eq. (A.9) in the case of the matrix elements P”d=’ ,+,,J f ; k). Plf;,;,‘+( f ; k) could only be expressed by a sum of H’-’ +,,+( f ; k) and H$~+( f ; k) if the combination of K and V appearing in eq. (~.8) were [ Yh(rf r’) x (V,*I,,)]~, [ Yd(C) XV,]; or [ Yb(i))xV,.]I, . Therefore, the difference
between
term is due to the different by r and r’ in eq. (A.8).
the quark combination
interchange term and the renormalization of Y[ and V or the nonlocality expressed
References 1) D.A. Liberman, Phys. Rev. D16 (1977) 1542 2) JET. Ribeiro, Z. Phys. CS (1980) 27 3) M. Oka and K. Yazaki, Phys. Lett. B90 (1980) 41; Prog. Theor. Phys. 66 (1981) 556; 572
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Y. Yamauchi et al. / Quark exchange currents
4) A. Faessler, F. Fernandez, G. Liibeck and K. Shimizu, Phys. Lett. B112 (1982) 201; Nucl. Phys. A402 (1983) 555 5) M. Oka and K. Yazaki, in Quarks and nuclei, ed. W. Weise (World Scientific, Singapore, 1984) p. 489 6) K. Shimizu, Rep. Prog. Phys. 52 (1989) 1 7) K. Shimizu, Phys. Lett. B148 (1984) 418 8) K. Brauer, A. Faessler, F. Fernandez and K. Shimizu, Z. Phys. A320 (1985) 609 9) F. Fernandez and E. Oset, Nucl. Phys. A455 (1986) 720 10) M. Oka and K. Yazaki, Nucl. Phys. A402 (1983) 477 11) S. Takeuchi and K. Yazaki, Nucl. Phys. A438 (1985) 605 12) Y. Yamauchi, R. Yamamoto and M. Wakamatsu, Phys. Lett. B146 (1984) 148, 153; Nucl. Phys. A443 (1985) 628 13) S. Takeuchi, K. Shimizu and K. Yazaki, Nucl. Phys. A449 (1986) 617 14) M. Chemtob and S. Furui, Nucl. Phys. A449 (1986) 683; A454 (1986) 548 15) P.J. Mulders and A.E.L. Dieperink, Nucl. Phys. A483 (1988) 461 16) H. Ito and A. Faessler, Nucl. Phys. A470 (1987) 626 17) A. Buchmann, Y. Yamauchi, H. Ito and A. Faessler, J. of Phys. G14 (1988) 1037 18) A. Buchmann, Y. Yamauchi and A. Faessler, Nucl. Phys. A496 (1989) 621 19) Y. Yamauchi, A. Buchmann and A. Faessler, Nucl. Phys. A494 (1989) 401 20) A. Buchmann, Y. Yamauchi and A. Faessler, Phys. Lett. B225 (1989) 301 21) Articles in Mesons in nuclei II, ed. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979) 22) K. Holinde, Nucl. Phys. A415 (1984) 477 23) L.C. Gomes, J.D. Walecka and V.F. Weisskop, Ann. of Phys. 3 (1958) 241 24) H.A. Bethe and J. Goldstone, Proc. Roy. Sot. A238 (1956) 551 25) R.J. Eden and V.J. Emery, Proc. Roy. Sot. A248 (1958) 266 26) S.T. Bulter, R.G.L. Hewitt, B.H.J. McKellar and J.S. Truelove, Phys. Rev. 186 (1969) 963 27) A. Kallio and B.D. Day, Nucl. Phys. Al24 (1969) 177 28) B.R. Barrett, R.G.L. Hewitt and R.J. McCarthy, Phys. Rev. C3 (1971) 1137 29) H. Horiuchi, Prog. Theor. Phys. Supp. 62 (1977) 90 30) K. Shimizu, private communication 31) K. Yazaki, Nucl. Phys. A497 (1989) 295~ 32) Y. Suzuki and K.Y. Hecht, Phys. Lett. B232 (1989) 159 33) R.P. Singhal, J.R. Moreira and H.S. Caplan, Phys. Rev. Lett. 24 (1970) 73 34) F. Ajzenberg-Selove, Nucl. Phys. A375 (1982) 1 35) Y. Kurihara and A. Faessler, Nucl. Phys. A467 (1987) 621 36) S. Takeuchi, K. Shimizu and K. Yazaki, Nucl. Phys. A481 (1988) 693 37) A.R. Edmonds, Angular momentum in quantum mechanics (Princeton Univ. Press, 1968) 38) T. de Forest and J.D. Walecka, Adv. in Phys. 15 (1966) 1 39) R.G. Sachs, Phys. Rev. 74 (1948) 433 40) R.H. Dalitz, Phys. Rev. 95 (1954) 799 41) H. Ito and L.S. Kisslinger, Ann. Phys. 174 (1987) 169 42) A. Arima and H. Horie, Prog. Theor. Phys. 11 (1954) 509; 12 (1954) 623 43) K. Shimizu, M. Ichimura and A. Arima, Nucl. Phys. A226 (1974) 282 44) H. Hyuga, A. Arima and K. Shimizu, Nucl. Phys. A336 (1980) 363 45) A. Arima, K. Shimizu, W. Bentz and H. Hyuga, Adv. in Nucl. Phys. vol. 18 (1987) 1 ed. J.W. Negele and E. Vogt (Plenum, New York, 1987) and references therein 46) I.S. Towner and F.C. Khanna, Nut. Phys. A399 (1983) 334; Phys. Reports 155 (1987) 264 47) A. Arima, Y. Horikawa, H. Hyuga and T. Suzuki, Phys. Rev. Lett. 40 (1978) 1001; 42 (1979) 1186 48) P.G. Blunden and B. Castel, Nucl. Phys. A445 (1985) 742 49) R.P. Singhal et al., Phys. Rev. C28 (1983) 513 50) M.V. Hynes et al., Phys. Rev. Lett. 42 (1979) 1444 51) N. Kalantar-Nayestanaki et al., Phys. Rev. Lett. 60 (1988) 1707 52) L. Lapikls, in Proc. Int. Conf. on nuclear physics with electromagnetic interactions, Mainz, 1979, ed. H. Arenhovel and D. Drechsel, Lecture notes in physics, vol. 108 (Springer, Berlin, 1979) p. 41
A526 (1991) 495-546
QUARK
EXCHANGE
CURRENTS
IN NUCLEI*
Effective operator description Yoshiaki
YAMAUCHI,
Alfons
BUCHMANN’
and Amand
lnnstitut.fiir Theoretische Physik, Universitiit Tiibingen, Aufder
FAESSLER
Morgenstelle 14, 7400 Tiibingen, Germany
Akito ARIMA Department of Physics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received
24 September
1990
Abstract: In this paper we introduce for the first time effective quark exchange current operators which can be used in nuclear structure calculations. To this end we first solve the equation of motion for a microscopic quark hamiltonian using the quark cluster model. We then construct the electromagnetic current operator on the quark level, i.e. the photon is coupled directly to the quarks. By eliminating the quark (internal) degrees of freedom, we derive an effective electromagnetic current operator on the nucleon level. A part of this effective current corresponds to the conventional impulse and meson exchange currents with vertex factors predicted by the quark model. In addition, this effective current contains new non-local and isospin-dependent terms which are generated by the Pauli principle on the quark level (quark exchange between nucleons). When we evaluate these quark exchange currents, we use harmonic-oscillator wave functions as nuclear wave functions including short-range correlations. We introduce these short-range correlations by solving the Bethe-Goldstone equation with our effective NN potential, which is derived from a microscopic quark hamiltonian. We investigate the role of these additional quark exchange currents in the magnetic moments and the elastic magnetic form factors of several closed-shell *l nuclei, such as 15N > “0 7and 39K.
1. Introduction In the last decade, several groups have searched for clear evidence of quark effects in nuclei in the framework of various quark models. Among these, the quark cluster model
seems to be the most appropriate,
because
it can successfully
describe
the
short-range part of the NN interaction, which is the fundamental input for treating any nuclear many-body problem. The main feature of the quark cluster model is that due to the Pauli principle on the quark level, various quark exchange processes between the three-quark clusters (nucleons) occur. For example, a gluon or a pion and a quark are simultaneously exchanged between two nucleons. It has been found that these gluon-quark lm6) and pion-quark 7-9) exchange processes produce a shortrange repulsion in NN scattering, thus providing a microscopic explanation of a long standing problem. l
Supported
by the DFG under contract number Department of Physics, Faculty
’ Present address:
Fa 67/10-5. of Science, University
of Tokyo,
113, Japan. 03759474/91/$03.50
@ 1991 - Elsevier
Science
Publishers
B.V. (North-Holland)
Bunkyo-ku,
Tokyo
496
Y. Yamauchi et al. / Quark exchange currents
Aside from NN scattering, electron-nucleus scattering seems to be an interesting subject for investigating quark effects ‘*-**), because electron scattering at high momentum
transfers
magnifies
the short-range
which in turn is related to the short-range
part of the nuclear
part of the NN interaction.
work 16-z*), we applied
the quark cluster model to electron-deuteron
to the Pauli
new electromagnetic
principle,
currents
wave function In our previous scattering.
Due
arise that are associated
with
the quark, gluon-quark, and pion-quark exchange processes, which generate the short-range repulsion between the nucleons. These currents are called quark exchange currents. They are not present in a conventional calculation of the electromagnetic properties of the deuteron. It has been shown that they appreciably influence the electromagnetic form factors of the deuteron ‘6-‘8*20) and the deuteron electrodisintegration cross section ‘9,20) at high momentum transfers (q 2 5 fm-‘). In the present work, we investigate the role of these quark exchange currents in the magnetic form factors of other light nuclei. Since a microscopic treatment in terms of quark degrees of freedom is not feasible for nuclei with A > 2, it is essential to devise a method which allows us to incorporate quark degrees of freedom into a conventional
nuclear
structure
calculation.
In nuclear
structure
calculations,
the
nucleus is usually regarded as an aggregate of nucleons interacting via a given NN interaction, and any physical quantity is expressed in terms of nucleon dynamical variables, i.e. the position, momentum, spin, and isospin of the nucleons, rather than in terms of the corresponding quark dynamical variables. A similar and more familiar method is usually adopted in the treatment of meson degrees of freedom in nuclei *‘). It is therefore instructive to recall this method first. It is well known that the long-range part of the NN interaction is transmitted by one-pion exchange, and the original lagrangian describing the nuclear system not only contains nucleon but also pion degrees of freedom. However, the usual nuclear physics prescription for including these pionic degrees of freedom has been to eliminate them explicitly and to express them implicitly in terms of an effective two-body potential (one-pion exchange potential) which depends only on nucleon dynamical variables. Similarly, if the nucleus is subjected to an external electromagnetic field the original
lagrangian
not only contains
the coupling
of the electromag-
netic field to the nucleon, but also to the pion field. The electromagnetic interaction of the external field with the pions is then ususally described in terms of an effective electromagnetic two-body operator (one-pion exchange current) acting in the nucleon space only. Due to the gauge invariance of the underlying field theory, the effective one-pion exchange current which implicitly describes pion degrees of freedom in the electromagnetic interaction, is linked to the effective one-pion exchange potential via the continuity equation for the electromagnetic current. The following is a graphic way of understanding the origin of the one-pion exchange current which bears some resemblance to the quark exchange currents to be discussed below. The one-pion exchange potential is an isospin-dependent charge exchange potential, i.e. by emitting a positive pion a proton can suddenly change
Y. Yamauchi et al. / Quark exchange currents
into a neutron.
Subsequently,
the positive
pion
point
by a neutron
which
changes
in space
equation must
for the electromagnetic
flow between
charge exchange between
can be reabsorbed
That is, in the presence
the total nuclear
into
at some other
a proton.
The continuity
then implies that an electromagnetic
the sites of the two nucleons
reaction.
the nucleons
then
current
497
current
which
current
have participated
of an isospin-dependent is not simply
in the
interaction
the sum of the single-
nucleon currents (impulse approximation) but must be accompanied by an additional two-body current in order to satisfy charge conservation. Thus, the one-pion exchange current describes the additional electromagnetic interaction of the external field with the underlying
pion
degrees
of freedom
and represents
the pionic e&ct
in the electromagnetic properties of the nucleus. In order to incorporate quark degrees of freedom into the nuclear many-body problem, we proceed in a similar fashion. We start with a description of the nucleus and its electromagnetic interaction with an external field in terms of quark degrees of freedom by using the quark cluster model. We then eliminate the quarks explicitly and introduce effective operators in the nucleon space. The equation of motion for the nuclear system is then described by a Schrodinger equation with an effective NN interaction. Likewise, the electromagnetic interaction of the nucleus with an external field is then described by an effective electromagnetic current operator in the nucleon space. Due to the Pauli principle on the quark level these effective operators acquire non-local and isospin-dependent parts even if the original operator between the quarks (e.g. the color magnetic interaction) was local and isospin for the case of the effective independent. This is shown, for example, in refs. 3,536722) NN interaction. Analogously, in the case of the effective electromagnetic current, additional non-local and isospin dependent two-body currents arise. Again, this is a consequence of the underlying quark degrees of freedom, and in particular, of the antisymmetrization principle on the quark level. We will refer to these additional electromagnetic currents as quark exchange currents. These quark exchange currents represent the quark efict in the electromagnetic properties of the nucleus. Finally, we stress that the continuity equation for the electromagnetic provides an intimate connection between the effective NN interaction effective electromagnetic current. As mentioned before, the short-range between
two nucleons
comes from the non-local
and isospin-dependent
current and the repulsion part of the
effective NN interaction. The physical origin of this non-locality and isospin dependence is the underlying quark structure of the nucleon and the Pauli principle on the quark level, which leads to various quark exchange processes between the nucleons. Then, because of the non-locality and isospin dependence of the effective NN interaction, corresponding effective two-body currents are required in order to satisfy current conservation. These effective two-body currents are the quark exchange currents. Another more graphic explanation of the origin of the quark exchange currents is as follows. When a proton comes close to a neutron, for example, they can exchange an up and a down quark between them, thereby
Y. Yamauchi et al. / Quark exchange currents
498
transforming
the original
then implies
that there must be a corresponding
the nucleons.
Thus,
proton
in order
to satisfy
we start with an electromagnetic the hamiltonian
into a neutron current
current
which
and vice versa. Charge quark
exchange
conservation satisfies
conservation
current
on the nucleon
current
conservation
on the quark level, and define the effective electromagnetic
in a way consistent
with the effective
between level, with current
NN interaction.
In this paper, we will investigate the influence of these quark exchange currents on the magnetic form factors of various light nuclei, where the shell-model picture is well established. Because the quark exchange currents contribute only in the short-range region, the evaluation of the quark exchange currents in the shell model must be done by taking appropriate short-range correlations between the nucleons into account, otherwise no reliable conclusion about the role of quark degrees of freedom in the nucleus can be drawn. In our picture, the short-range repulsion itself comes from quark degrees of freedom and is described in terms of the aforementioned effective NN interaction. Therefore, we will introduce the short-range correlation consistently with the quark exchange currents by solving the Bethe-Goldstone equation 23-28) with our effective NN interaction. The paper is organized as follows. In sect. 2, we first rewrite the quark exchange currents in terms of nucleon dynamical variables and show that these quark exchange currents represent a genuine quark effect in the electromagnetic nucleus. In sect. 3, these quark exchange currents are evaluated calculation employing consistent short-range correlations. In sect. results, and in sect. 5, we conclude with the main findings of the
properties of the in a shell model 4, we discuss our present paper.
2. Quark exchange currents In this section, we derive the quark exchange currents as effective two-body operators on the nucleon level using the quark cluster model based on the resonating group method (RGM). As mentioned in the introduction, these effective electromagnetic currents and the effective NN interaction are intimately related by the requirement of current conservation. Thus, the definition of the quark exchange currents as effective
operators
must
be performed
carefully
in a way consistent
with the
derivation of the NN interaction. Therefore, we first present the derivation of the effective NN interaction in the two-nucleon system in sect. 2.1. We start with a microscopic description of the NN system as a six-quark system in which the quarks interact via a given microscopic hamiltonian. We then obtain the Schrodinger equation with the effective NN potential by eliminating the quark degrees of freedom (or by integrating over the internal coordinates) and interpreting the renormalized wave function as the NN wave function. The effective NN potential is defined in such a way that it gives the same result as solving the equation of motion in the six-quark system directly. In deriving the effective NN interaction from the underlying quark dynamics, we pay special attention to the relation between the wave
499
Y. Yamauchi et al. / Quark exchange currents
function
in the two-nucleon
we turn
to the derivation
exchange current
current.
two-nucleon
and that in the six-quark
system.
In sect. 2.2,
of the effective
electromagnetic
current
or the quark
We start from the quark
electromagnetic
currents
which satisfy
conservation
electromagnetic
system
with the given quark hamiltonian.
current
by using
the relation
system and that in the six-quark
Then we derive the effective
between
the wave function
in the
system. The effective electromagnetic
current is defined in such a way that its matrix elements between NN wave functions gives the same result as the corresponding matrix elements of the microscopic quark currents taken between six-quark wave functions. Due to the Pauli principle on the quark level the effective electromagnetic current contains a non-local part, the quark exchange current. Finally, in sect. 2.3 we show that as long as we restrict ourselves to effective two-body operators, the description of the nucleus as an A-nucleon system using the effective NN interaction and effective electromagnetic current obtained in the two-nucleon system is equivalent to a microscopic description of the nucleus as a 3A-quark system using the microscopic quark hamiltonian and quark
electromagnetic
2.1. THE
EFFECTIVE
current.
NN
INTERACTION
Let us start with the total hamiltonian
for the six-quark
system given by (2.la)
Here mq = irn,is the quark mass. The center-of-mass (c.m.) motion of the six-quark system is eliminated via the second term where PC is the center-of-momentum interaction Vy) consists of the coordinate and MG = 6m,. The quark-quark confinement term one-pion exchange vy”f = -a,(X;
Vrnf, the one-gluon (OPE) term VyPEP:
exchange
(OGE)
term
V$‘GEP, and
the
(2.lb)
. Aj)& ,
(2.lc) -__
___A2 A”-m’,’
where fmq= gfrNNand fiNN/47r=0.08.
(2.ld)
The cut-off parameter A in the VoPEP is taken to be 4.2 fm-‘, which is estimated from the size of the pion ‘). The strong coupling constant (Y, is determined from the experimental N-A mass splitting and
500
Y. Yamauchi
the strength nucleon
of the confinement
mass against
parameter)
variations
b,. Here the nucleon
use the parameter of this hamiltonian The RGM
et al. / Quark exchange currents
potential
a, from
of the nucleon-size
the stability parameter
condition
is in a (0~)~ harmonic-oscillator
configuration.
set b, = 0.5 fm, LY,= 0.36, and a, = 67.27 MeV * fm-‘. and its parameters
wave function
of the
(harmonic-oscillator We
More details
are given in refs. 1s,‘9).
for the six-quark
system is then written
as
(2.2) where Qi$(&,
iji.J is the intrinsic
wave function
of the ith cluster. For these intrinsic
wave functions we choose only the nucleon (OS)’ harmonic-oscillator configuration. The single quark coordinates p,, . . . , p6 are then transformed into two internal coordinates for each cluster, &, and &2, and one c.m. coordinate for each cluster, ri (i = 1,2). Since the c.m. motion of the six-quark system as a whole is eliminated, the wave function ~(‘12’ does not depend on the c.m. coordinate R,2 = t( rl + rJ, but only on the intercluster (relative) coordinate r12 = r, - r2. Due to the Pauli principle between quarks, the six-quark wave function !P(rJ must be totally antisymmetric under the exchange of any two quarks. Since the intrinsic wave functions are antisymmetric, the antisymmetry of the six-quark wave function through the product of antisymmetrizers &2Q(c’“ster) and dCq)
can be achieved
%d(c’“ster) = v$( 1 - P,4P25P36) )
(2.3a) (2.3b)
acting on the wave function of eq. (2.2). Since the operator P,4P25P36 represents just the exchange of the two clusters as a whole, ~~~~~~~~~~ is the antisymmetrizer for the clusters. In eq. (2.2), this is already incorporated by choosing the relative wave function Ik ( r12) to be antisymmetric with respect to the exchange of the clusters as a whole. Thus the Pauli principle between nucleons comes from the antisymmetrizer &‘c’uster), when we introduce
the nucleon
space. Note that in addition
to the obvious
two-nucleon component where there is no overlap between the clusters, the RGM wave function also includes the six-quark component where the quarks are not confined to their respective clusters. To a large extent, this coexistence of NN and six-quark components in the RGM wave function follows from the antisymmetrization principle on the quark level. Once the total hamiltonian and the intrinsic wave function of the clusters are given, the equation of motion which determines the relative wave function P(r,,) is given by the RGM equation of motion:
1(ii1@$(Si.l,$i2))(H’q’--E)ly(6q)(pl,. .. ~6) dgr,r.. . d&.2=0. 3
(2.4)
501
Y. Yamauchi et al. / Quark exchange currents
Integration
over the internal
coordinates
leads to an integro-differential
equation
of the form
J
H(r,z,
The hamiltonian
kernel
r{JP(rL)
drL= E
J
N(r12, 42) p(&)
H (r, r’) and norm kernel
d42 .
N( r, r’) are defined
(2Sa) as
We are concerned only with the orbital part of the wave function and have suppressed the spin, isospin, and color dependence here. By dividing the antisymmetrizer (1 -9P3J into the identity operator part (1) and the permutation part (-9P3,J, these kernels are decomposed into two terms, a direct term and an exchange term. For example, the norm kernel is decomposed as follows: N(r,r’)=
N’D’(r)S(r-r’)+N~EX~(r,r’)=S(r-r’)-90~~fff)[P~~rc~]~(r,r’),
(2.6a) (2.6b)
where Ol;*‘[ P:FC) ] 1s . the effective spin-isospin operator which gives the same result as the quark exchange operator P$s6Tc’ in the spin-isospin and color space of the quarks. We will give an explicit expression for this effective spin-isospin operator later. Eqs. (2.6) are derived using familiar techniques from nuclear cluster theory *‘). Details of this derivation are given in the appendix. These RGM expressions clearly show that the operators resulting after integration over the internal coordinates become non-local due to the quark exchange process (Pauli principle). While the direct term contributes everywhere, the exchange term is only effective from eqs. (2.6). the norm kernel the cluster radius
in the overlap region of the clusters which can be understood easily (The function C?(r, r’), i.e. the radial part of the exchange term of NCEX’, decreases rapidly as the intercluster distance r or r’ exceeds and is practically zero for twice the cluster radius.) The hamiltonian
kernel can be further decomposed into different types. For the one-body operator (kinetic energy) there is one direct term (fig. la) and two different exchange terms (fig. lb, c). For the two-body operator (potential) one must consider two different direct terms (fig. Id, e) and five different exchange terms (fig. If-j). By solving eq. (2Sa) or eq. (2.4), we get the six-quark RGM wave function as defined by eq. (2.2). As mentioned before, it is fairly cumbersome to extend this microscopic treatment on the quark level to nuclei with A > 2. Therefore, we will introduce the nucleon space and use a Schrodinger equation with an effective NN
502
Y. Yamauchi
123
et al. / Quark exchange
654
123
654
123
(a)
1 123
Cd)
I
654
(b)
654 (e>
currents
123
N
123
654
Cc)
654 (f)
1
123
654 (9)
1 123
654
Fig. 1. The different types of RGM kernels in the quark cluster model. For a one-body quark operator: (a) direct term; (b), (c) exchange terms. For a two-body quark operator: (d) direct (intracluster) term, (e) direct (intercluster) term, (f-j) exchange terms. The solid lines represent quarks and the boxes quark operators.
potential written in terms of nucleon dynamical variables. This effective description must then yield the same result as solving the six-quark wave function from eq. (2.4) directly. One obvious candidate for an appropriate NN wave function and an effective Schrodinger equation is the relative wave function ?P(r,,) and eq. (2.5a). There is, however, an important difference between ly(r12) in eq. (2Sa) and the conventional NN wave function. The solution of eq. (2.5a) !P(r,J does not satisfy the usual orthonormality condition and Schrodinger equation because of the appearance of the norm kernel in eq. (2.5a). Therefore the solution of eq. (2.5a) cannot be interpreted as a NN wave function. In order to define a proper NN wave function,
Y. Yamauchi et al. / Quark exchange currents
the renormalized
wave function G(r)
is introduced
503
where
N”’
J
N-“‘(r,
with I!? = E - 2Eint, where
J
N”2(r,
IS . the square
consequence, the renormalized for a non-local potential
V,,(r,r’)=
=
r’)V(r’)
root operator
wave function
r”)H(r”, E,,, defines
dr’,
(2.7) of the norm
kernel.
P satisfies the Schrodinger
rrn)N-“2(r”‘,
r’)dr”di~~-~-2~i,,,,
the total internal
energy
As a
equation
(2.9)
of one cluster.
Next, we turn to the discussion of the spin-isospin and color degrees of freedom, which we have omitted for simplicity up to now. In order to rewrite the spin-isospin and color part of the six-quark matrix elements, we introduce effective spin-isospin operators in the NN space. These effective operators OCeff) must then satisfy the condition that for any state they yield the same matrix elements as the corresponding spin-isospin and color operators OCq) when evaluated on the quark level. Due to the quark exchange property of the permutation operator the resulting effective NN operators will in general be two-body operators acting on the spin-isospin coordinates of nucleons 1 and 2 (xsf~~O~;*“~x%& = (*~~~‘c’(o(q)lx~~~,c,)
(2.10)
for (ST,), (S,T’J=(OO), (Ol), (lo), (11). As an example, consider the effective of quarks 3 and 6 operator corresponding to P$zTc), which acts on the coordinates belonging to the first and second cluster Oie:)[ P ~SQTC)]=~[(1+~1IN).1/N))+~(1+~7(1N).71N))aiN).a~N)], which can be easily checked
using eq. (2.10). This equation
(2.11)
is also given in ref. 22).
Eq. (2.11) clearly exhibits the charge exchange character (i.e. the isospin dependence (N) * 7iN)) of the effective operators even when the initial quark operators are 71 isospin-independent. In this way, we can systematically rewrite all quark operators in terms of effective NN operators, and from this derivation it is evident that the effective Schrodinger equation in the two-nucleon space given by eq. (2.8) gives the same result as solving the equation of motion for the six-quark system given by eq. (2.4). Although our effective NN potential V,, is written in terms of nuclear dynamical variables, it represents the effect of the quark degrees of freedom in the same way as a microscopic calculation would. We have chosen the effective description mainly because it is more convenient for practical applications to nuclei with A > 2. After the relation between the six-quark wave function and the effective NN wave function has been
Y. Yamauchi et al. / Quark exchange currents
504
clarified,
the derivation
in a way consistent
of the effective
with the definition
quark
exchange
of the effective
currents
can be performed
NN interaction.
This will be
done in the next subsection. However, effective
before
ending
NN interaction
hamiltonian
kernel
this subsection,
let us investigate
given by eq. (2.9) in greater
into a direct H(r,
and an exchange
evaluated
the
r’).
(2.12a)
and is given by Pf*
PD’(
of the
term
r’)=HcD)(r)G(r-r’)+HcEX)(r,
The direct term can be readily
the properties
detail. First we decompose
Vp2p”‘( r,J ,
r12)= 2Ei”t + ~+
(2.12b)
where V I”
=
$
$“)
. $‘),,\N)
. v,~$N”’
.
v,
j-
n
dk e-tk-exp
(-{btk’)
2~~ k’+mf,
A. (2.12c)
Note that Eint comes from the intracluster
terms represented
by fig. la for the kinetic
energy and fig. Id for the potential. The direct potential VypP results from the intercluster term represented by fig. le. The integration over the internal coordinates results in the factor exp (-ibtk2). It is easily understood that the direct potential is the same as the conventional vertex factor F,,(k) =exp confinement and one-gluon
one-pion exchange (-~6’,k’)(A’/(k’+A’))‘/ exchange potentials,
potential, which contains the TN twice. In the case of the there is no corresponding inter-
cluster term (fig. le) because of the color operator hi * A,, which represents color confinement. Using eqs. (2.12), we can rewrite the non-local potential of eq. (2.9) as follows V,,( r, r’) = Vp2pEp(r) . 6( r - r’) + V?Fp( r, r’) . Here, the first term is the conventional (2.12c), and the effective quark-exchange
(2.13a)
one-pion exchange potential given in eq. potential VpFp consists of two terms
V?zp( r, r’) = V?i’( r, r’) + VP,““( r, r’) . The quark-interchange as V?i’:‘( r, r’) =
J
potential
V $” and the renormalization
N-‘/2( r, r”)H(EX)( #‘, r’rr)N-“2(
V FTN(r, r’) = J ~-‘/2(~, ry~(D)(r9~-W(
(2.13b) term Vy2ENare defined
rnr, r’) dr” dr”’
,
r”, r’) dr”-H(D)(r)G(r-r’).
The quark-exchange potential is only effective in the region overlap or at short relative distances between the two nucleons.
(2.13~)
(2.13d)
where the clusters Note that N-‘j2 is
Y. Yamauchi et al. / Quark exchange currents
different (2.6)),
from the 6 function
only in the short-range
so that the renormalization
repulsion
this short-range
in NN scattering.
repulsion
or overlap
term gives an appreciable
this region. Previous work has shown that the quark-exchange a short-range
505
is attributable
region
(see eqs.
contribution
only in
Vp,“‘( r, r’) produces
potential
It is well understood to the Pauli
principle,
that the origin
of
as well as to the
spin-spin term of the OGEP and OPEP between quarks, which also create the N-A mass splitting in the three-quark system. Since our effective potential so far lacks enough intermediate-range attraction, we simply add a phenomenological c exchange
V” to our effective
potential V”(r,J
= -$
V,, in eq. (2.8).
NN potential
l _ e-m,,‘-l2- e -2u sinh (m,r,,)
--&
,2 1 [cash (2~) - l] eC”-‘lr
r,,G2R0
r12) 2&,
(2.14)
with cy = m,R,,, m, = 520.0 MeV and R,, = 1.34 b, [ref. ‘)I. The aN coupling constant g, is determined in order to reproduce the binding energy of the deuteron: gt/4m = w exchange between 2.55 [refs. 18,‘9)]. Even if we had started with a microscopic quarks, we would still end up with an effective NN potential with essentially the same characteristics as the one in eq. (2.14). This is due to the fact that the effects resulting from quark antisymmetrization are negligible for a spin-isospin independent central potential like the (T potential 30). In our formalism this means that the corresponding quark exchange potential V,,QEP is negligible because of the cancellation between the quark-interchange V$” term and the renormalization term Vr:N; thus, only the direct term of the sigma potential survives. Finally, we give a summary of the procedure needed to derive the effective operators in the nucleon space from the corresponding operators in the quark space. There are two steps. The first step is to integrate over the internal coordinates and to create the RGM kernel as shown in eq. (2.5b). A part of this RGM kernel corresponds to the conventional one-pion exchange potential with vertex factors predicted by the quark model. In addition, due to the Pauli principle on the quark level, this RGM
kernel
acquires
a non-local
part,
which then leads to the quark-
term as given in eq. (2.13~). The second step consists of the renormalizagiven in eq. (2.9). This procedure then leads to the renormalization term given in eq. (2.13d). interchange
tion procedure
2.2. DEFINITION
OF THE
QUARK
EXCHANGE
CURRENTS
After the six-quark wave function is obtained by solving eq. (2.4) or (2.5a), the electromagnetic interaction of the six-quark system with an external field is described in terms of the six-quark elements of the quark charge and current operators ( p(6q)lJ$)( p(6q)), where JF’ IS . the quark electromagnetic current given as JzLq)= ; JIpi’“P’( pi, 4) + I=1
; i
{Jz:+“)(
pi, pi, 4) +$=‘“o”)(
pi, pi, 4)) _
(2.15a)
506
Y. Yamauchi et al. / Quark exchange currents
Here, brevity
the three-momentum
of the photon
we write the quark
order to satisfy current
charge
conservation
we introduce
not only the quark
also the pion
JzLpion) and gluon
is denoted
and current
by q. For convenience
as a four-vector
with the quark hamiltonian impulse
current
quark
current)
currents
(two-body
Jt’imp),
q)
=Z& tq
(2.15b) ajq)
x
but
currents):
&q:imp)( pi, q) = ejq) eir.p, , fq:imp)(pi,
In
given in eq. (2.la)
(one-body
JjLq’g’“on)exchange
and
current.
eiw,
+2
eiw,
v,)
,
(2.1%)
4
Jpio”ypi,
pj, q) = 0 )
(2.15d)
J(q:pion)(pi, pj, q) = J(q:pion-pair)(pi, pj, q) +J(q:pionic)(pi,
.P:pio-ypi,
pj, q) =
pj, q) ,
(2.15e) -k,
eff$! ($)
XT?)),
eiq.P, ,p),y)
.
vp,,( !?$z_T$)
2 xj5&T+(‘++d
(2.15f)
> r
fq:pionid(Pi,
pi,
q)
=
-
je&
(@)x
p)pp
. vp,ajq) * v,
T
ei(rl-k).
k2+m; A2 kz+nz
Jpq’uo”ypi, pj, q) = 0 )
A2 (q-k)2+A2
P,
(q-k)‘+mZ, l/2 ’
(2.1W (2.15h)
where
the isospin structure of the quark charge operator eiq) is given by ei”‘= ei(l + 37g)). In ref. I’), we also considered other two-body currents, (e.g. the isoscular pion-pair current) which are of higher order in (l/ mq). These higher order terms in the charge and current operators are the counterparts of the corresponding higher order terms in the kinetic energy, OPEP and OGEP as required by current conservation. Since we do not consider these higher order terms in the hamiltonian, we also do not consider the corresponding higher order terms in the electromagnetic current in order to satisfy current conservation. For the same reason, we do not choose the gluon-pair current, which we used in refs. 18*19),but the gauge invariant gluon current which satisfies current conservation with (the LS force of) the OGEP. The gauge invariant gluon current differs from the gluon-pair current by a factor t. More details of the one-gluon exchange current are given in ref. 20).
Y. Yamauchi et al. / Quark exchange currents
Having
defined
wave function
the description
of eq. (2.7), which
(2.2), we now introduce terms of nucleon description six-quark
of the two-nucleon is related
by the renormalized wave function
currents,
of eq.
which are written
in
and which yield exactly the same result as a microscopic
of the electromagnetic wave function.
system
to the six-quark
the effective electromagnetic
variables
507
interaction
Thus the effective
in terms
of quark
electromagnetic
currents
current
Jt”’
and the is defined
by ly@l)) .
($jJ1”“‘1 I@ = (l@Q)j#$j We follow
the same procedure
as in the definition
(2.16)
of the effective
NN potential.
First, we integrate over the internal coordinates and rewrite the six-quark matrix element in eq. (2.16) as a matrix element which contains only the relative wave function: (Pq)(J;)J?Pq))
)
= ( P(J;b)+JFX’J?P)
(2.17a)
with
I?QZ(Ei.l,li.,)}J:)
J:“‘( rn, Rn, 4) = J { X
i=l
1I?@C(Ej.l,5j.2)I
dtl.1 . * . 42.2
(2.17b)
3
j=l
JFx)(r,
r’,
R,2, q) =
I? J-t
@;Vnt(gi.l?
gi.2)
i=l
6(r12-
r)Jjp)(-9p36)
I
* * . G2
~(r12-r")&l
dr12,
(2.17~)
j=l
where we decomposed the RGM kernel into the direct term JLD’ and the exchange term JF"' . The direct term can be readily evaluated and will be shown to be the same as the conventional electromagnetic current, that is, the impulse current JE”‘“’ and meson exchange current JIMEC) (pion-pair and pionic current) with the one-nucleon vertex factors calculated in the quark model:
JlD)(r12,RI*, q)= $ JE”“(ri, q)+JIMEC)(rl,
r,, q).
(2.18)
i=*
The conventional
impulse
current
is given as
Jgmp)(ri, q) = e(GL(q’)+
TizG”,(q2)) ei4’c ,
(2.19a)
JCimp)( ri, q) = &{(G^,(q2)+~i,GK(q2))qxajN’ei*.~ N
+(G”,(q2)+
7,,GL(q2))2
ei9“, V,} ,
(2.19b)
508
Y. Yamauchi et al. / Quark exchange currents
with Gsh,(q2)4$mP)(q2)+
,3$&'on)(q2),
Q’(+
G$duon)(q2)+
G$"P'($)+
(2.19~) G$Pion)(q2),
Gg( q’) = G’,( q2) = Gsimp)( q2) = i exp (-ab’,q”) G zmp)( q’) = 1 exp (-ibis’)
(2.19d) ,
(2.19e)
.
(2.19f)
The conventional impulse current comes from the direct term of the quark impulse current represented by fig. la and from the intracluster terms of the quark-pion and quark-gluon currents represented by fig. Id. The contribution from the quark-pion and quark-gluon intracluster terms shown in fig. Id is included in eqs. (2.19a, b) through the corresponding corrections to the one-nucleon form factors, GCg’“““)(q2) and GCpio”)(q2) g’iven in eqs. (2.19c, d). Since the explicit expressions for Gcg’“‘“)(q2) and GCpion)(q2) were already given in refs. ‘83’9), we will not repeat them here. The conventional meson exchange current is given as JLMEC)(r,, rj, q) = 0 ,
(2.20a)
J’MEC’( ri, PI, q) = J(pio”-payJ.i, rj, q) +J(pio”ic)(& #pion-pair)(ri, ‘;, q) =!&!$E
J(pionyri,
I - ------
X
,-‘k.‘a,
27r2 k2+mt
Pi, q) = -i,*
I
(2.20b)
(TjN) x TIN))Z ei4’rLajN)aJN) . v,, dk
X
rj, q) )
F,N(k2)F,N,(k
($J) x $Q$J) 71
q)+(i++j)
. Vp/N’
(2.2Oc)
. v,,
$(2k-q)&$
x Fmv(k2)F,,((q
7T (2.20d)
- k)2) ,
with ,
(2.20e)
(2.20f) The conventional meson exchange current comes from the intercluster term represented by fig. le. The integration over the internal coordinates results in the vertex are factors F-N and F,,,,N g iven in eqs. (2.20e, f). More details of this derivation given in refs. ‘*,19). In the case of the gluon current, there is no corresponding intercluster term fig. le because of the color operator Ai. Ai, which represents color confinement.
Y. Yamauchi et al. / Quark exchange currents
Now, let us turn to the exchange
terms. Here we evaluate
which comes from the quark impulse for comparison. evaluated
The exchange
in the same
coordinate
of the
unchanged
under
only the exchange
and also the corresponding
six-quark
term
direct term
terms which come from other quark currents
manner.
can be
We start
with the spatial component. The c.m. symmetric and remains R,2 is completely
system
the application
on R,2 can be factored
current
509
of the antisymmetrizer.
out from the integral
Therefore
the dependence
in eq. (2.17~). Thus,
we first rewrite
the quark impulse current by transforming the single quark coordinates p, , . . . , p6 to a c.m. coordinate, R,, and the coordinates relative to the c.m. frame of the six quarks
(pi - R,J as follows
&+“P)(p
) q)
= s{3q
x ,id
eiq.(pc-R)
eiwR+2
. 3 eiq.(p,-R)(vpc
-iv,)
eir.R
N +eiq-(p,--R)
e iq.R
V,} .
(2.21)
We then insert eq. (2.21) into eq. (2.17~) and evaluate the integrals over the internal and the relative coordinate r,2. We need to evaluate the
coordinates &,.r . . . & following integrals:
ir &X&1.1, S,.J S(h2 - r) i=, I eiq.(~3-R,2) 3e’q”(P’-RlZ)(Vp, -iVR,2) X eiq.(PZ-RI2) Px6 3eiq”p~-R1z’(Vp,-~VR,~)P36 i
e4+--R
121 P36
3eiq’(p3-R12)(Vpj - iVR,,) Ps6 1
x
S(r12- r’) d&
. * . dh2 dr12
eiiq.rv,6cr
The operators
H’, P* can be derived
H+(
r, r’;
k)
=
LP( r, r’)
ry
P+(r,
r’;
$4)
P-( r,
r’;
jq)
using familiar
theory 29). Since we show the details only the final result here
_
techniques
of this derivation
.
(2.22a)
1 from nuclear
in the appendix,
exp [ isk( r + r’)] exp (&bik’)
,
cluster we give
(2.22b)
H-(r,
r’;
k) = Y(r, r’) exp [ i$k(r - r’)] exp (fbzk*) ,
P’(r,
r’;
k) = -i{e,H+(r,
r’) - H+(r, r’)v,.,} ,
(2.22d)
P-(r,
r’;
k) = -~{~,H-(r,
r’)+H-(r,
(2.22e)
r’)?,,},
(2.22c)
where LP(r, r’) is the radial part of the exchange term of the norm kernel given in eq. (2.6b). The derivative operators, 9, and d,, operate only on the final and initial
510
wave
Y. Yamauchi et al. / Quark exchange currents
functions,
evaluate
respectively.
Using
the direct and exchange
the operators
defined
in eq. (2.22),
terms which come from the quark impulse
we can current.
They are given as J(D.imP)(r, RI23 4) = JgAl, + Jg!w:R, + J~2”:,, #EX.imp)(,., ,.f, R
12, (I) = Jt:&!, + JE%z,
(2.23b)
+ JtcEoXn?.,~ ,
Jt$inI = -?2m,i
e @R,, q X o(lN)(;+$&))
J[Fpy& = -_?2m,i
e iq.R,z q x 20~*)[~:“)(t+~~:))(-9)P~~~~)]
+ _eiq~R,, e 2mNi
(2.23a)
,
e-ib$7’ eifr.r+
,
(l-2)
q x o~)[~~)(~+~~~))(_~)~~~TC)]
(2.23~)
epibiq2 ~+(r,
e-t@* H-(,.,
e J(D) =-eiq.R,, [conv:r]
(;+$&J))
e-~b~q2eilr.r~,+(lt,2),
(2.23e)
mNi
=_?_
[conv:r]
,J; iq) (2.23d)
+(1*2),
J(=)
,J; tq)
eiq’% 20~*)[(~+~~~))(_9)p~~~~)]
epk’$q* pi-(,,
,J; &)
mNi +e eWR12 o’,e;“‘[(++$7~‘)(-9)P$~Tc)] mNi
e-fbiq’ P-(*, r’;
$4)
(2.23f)
+(I-2)) J(D)
= -
J(=)
= -
[conv:Rl
e
2mNi
[conv:Rl
e
2m,i
e’q’R,zVR,,(~+~~(l~))
e-6‘bZq* q eifq”+(lH2),
eiq.% v,,,~o’,“,“)[(~+~~~)“!)(-~)P~~~~)]
(2.238) e-fbiq2 ~+(r,
e + -eiq'R,,VR,,0(leff)[(~+~~~))(-9)p~Tc)]e-~b~q2 2mNi
lt(l++2)
H-(r,r';iq)
(2.23h)
.
The effective spin-isospin cated. For example
operators
appearing
in eqs. (2.23) are rather
-~(l+~a(lN).aiN’)i(lrN’X?:N’)Z. there
are some
operators,
such as
useful
relations
20~~fi)[~~)p~sd_c) ] + o$$T
compli-
* cJy))(Ty)+TyyZ
O(le2R’[7~~)P~TC)]=~(l+$~iN)
However,
r’; $4)
between
$l)p$;w]=
2O&ff)[ rE’p$sQTC)] + O(lezff)[ @pit”‘]
these
(2.24a) effective
ay’O~;fi)[P:~=C’] = 7~~‘o$f’[p$~‘“‘]
spin-isospin
)
(2.24b)
.
(2.24~)
Y. Yamauchi et al. / Quark exchange currents
Now we consider that the direct
the properties
term given
as the conventional
of these direct and exchange
in eq. (2.23a)
impulse
511
current
together
terms. We can verify
with eq. (2.23c, e, g) is the same
given in eq. (2.19b) by transforming
nates r12 and RI2 to rl and r,. As can be seen from eq. (2.23a), the spatial of the impulse
current
J[sDdinI, the convection
can be decomposed current
the coordicomponent
into three types; the spin-magnetic
with the derivative
operating
current
on the c.m. motion
J(D) Fconv:Rl,
and the convection current with the derivative operating on the relative motion Jf,Dd,,:,, . This form is useful when we introduce the short-range correlations between nucleons in sect. 3. The exchange term can be decomposed in the same
(EX) and J[,,,,:,I (EX) correspond Jt:$&, J[conv:R~,
to corrections to Jfs”p!lnl,J~~\v:R1, and These exchange terms contain the operators H’ and P’ which correspond to corrections to eifq” and eifq” V,, respectively, as is easily seen from eqs. (2.22). way.
J(D) [conv:r]~
The contributions from fig. lb correspond to the parts which are expressed by the operators Ht and Pt and those from fig. lc are the parts which are expressed by the operators H- and P-. These exchange terms, given in eqs. (2.23), contain the radial part of the exchange term of the norm kernel 9 given in eq. (2.6b) through the operators H’ and P’. Therefore these exchange terms are effective only at short distances. The time component of the quark exchange current can also be expressed by the operators Jrx.imp)(r,
H” and is given as ,.I, R,,, 4) = eiq.R,, 20’,~~)ff’[(~+t~~=))(-9)p~S,~~)] e-@:q’ H+(,., ,.‘; iq) +eiq’R12 o’,P’[(~+~~:“,))(-~)P~S~TC)]
e-i’zq’ H-(r,
r’; $4)
+(1++2).
(2.25)
Having obtained the RGM kernel, the next step is the renormalization procedure. We can easily derive the effective electromagnetic current defined in eq. (2.16) from eqs. (2.17b, c) using the relation between the relative wave function and the renormalized wave function given in eq. (2.7): JjY(rr2,
&, RI*, 4) =
I
N-“‘(rr2,
r’)[.Izp’(r’, R12, q)S(r’-r”)
+fEX’( r’, r”, R 12, q)]N-“‘(
r”, &) dr’ dr” .
(2.26a)
Analogously to the effective NN potential, the effective electromagnetic current JF) consists of the conventional impulse and meson exchange currents and a non-local quark exchange current r;,,
JY)(rlz,
R,*, q) =
i (
JEmp)(ri, q)+JLMEC’(r
i=l
+J(QEC) p
where the quark
exchange
current
(r12, 42,
I3 r2,
4))
. @r12-
(2.26b)
R12, 41,
JIQEC) consists
d2)
of two terms
JLQEC)(r, r’, R, q) = JLQQlc)( r, r’, R, q) + JyN’(
r, r’, R, q) .
(2.27a)
512
Y. Yamauchi et al. / Quark exchange currents
The quark-interchange as
term JFIc)
JkQlc’(r,r’, R,
term JrEN)
are defined
N-1’2( r, rfr)JcEX)(r”, r”‘, R, q) N-“2( r”‘, r’) dr” dr”’ ,
q) =
JFEN’(r, r’, R, q) =
and the renormalization
I
N-“2( r, r”)JID)( r”, R, q)N-I’“(
-JID)(r,
R, q)S(r-
(2.27b)
r”, r’) dr”
r’) .
(2.27~)
These two terms are non-local operators that are effective only at short-distances between nucleons. We can easily understand from these equations that the quark exchange current JLQEC’ appears only when the exchange of quarks between the two clusters is taken into account, and that there is no corresponding current in conventional nuclear theory. Therefore it is this term which represents effect in nuclei. By definition, it is written in terms of nucleon dynamical so that it is possible to use it in a nuclear structure calculation.
the quark variables,
Note that in the two-nucleon system, the dependence of the effective current on RI2 = $( r, + r2) must be removed from these currents in order to eliminate the spurious c.m. motion. However, here we keep this dependence on the c.m. motion, because we want to apply these effective operators to nuclei with A> 2, where the c.m. coordinate of the total system is not equal to R,2 = R. The functions that are dependent only on R,2 do not become non-local because this coordinate is completely symmetric and remains unchanged under the application of the antisymmetrizer. Since the definition of the effective quark exchange current is made carefully in a way consistent with the definition of the effective NN interaction, and since the original quark hamiltonian and quark electromagnetic current satisfy current conservation, the effective NN interaction and current should satisfy the corresponding current conservation law on the nucleon level. Unfortunately, the expressions for the effective operators are quite lengthy so that we must defer an explicit proof to a separate paper. Nevertheless, it is evident that the effective quark exchange current appears as the counterpart of the short-range part of the effective NN interaction, which is governed by the quark exchange mechanism.
2.3. QUARK
DEGREES
OF
FREEDOM
IN THE
DESCRIPTION
OF NUCLEI
In sects. 2.1 and 2.2, we showed that the description of the two-nucleon system using the effective NN potential V, given in eq. (2.13a) and the effective electromagnetic current Jy’ given in eq. (2.26b) is equivalent to that of the six-quark system using the microscopic quark hamiltonian given in eqs. (2.1) and the quark electromagnetic current given in eqs. (2.15). We also showed that the quark effect in the electromagnetic interaction is represented by the quark exchange current JFEC). In this section we will show that these effective operators V, and J’,““‘, especially
Y Yamauchi et al. / Quark exchange currents
the quark
exchange
current
JLQEC),are applicable
we will show that the “disconnected” these effective
operators
diagrams
of neglecting
(1) The A-nucleon
three-body
Schrodinger
+(r,,...,rA)+
with A > 2. In addition,
of fig. 2 are already
and need not be explicitly
To see this, we will show that the following approximation
to nuclei
513
two statements
to A-body
included
in
are valid within
the
calculated. nucleon
operators:
equation
; i>j=l
X
?@rl,..., R,-++r:j,..., R,-;r:j ,...,
rA)dr;=&&(rl
,...,
rA), (2.28)
with the effective NN potential V, of eq. (2.13a) gives the same eigenvalues as the corresponding Schrodinger equation for the 3A-quark system with the quark hamiltonian given in eq. (2.1)*. (2) The description of the A-nucleon system with the effective electromagnetic current operator given as, Jy’
= i
Jo”“‘+
f
JIMEC’(ri, rj) +
i>j=1
i=l
i
JIQEC)(rb,
rii; R,) ,
(2.29)
i>j=l
is equivalent to the description of the 3A-quark system with the corresponding quark electromagnetic current given in eqs. (2.15)*. Here Jz”“’ and JIMEC’ are the conventional impulse and meson exchange currents given in eqs. (2.19) and (2.20), while JcQEC) is the quark exchange current which represents the quark effect in the nucleus and is given in eqs. (2.27). In other words the matrix element of the effective current operator of eq. (2.29) evaluated with the solution !? of eq. (2.28) gives the same result as the corresponding 3A-quark matrix element of the quark electromagnetic current operator given in eqs. (2.15) taken between 3A-quark wave functions. Let us start with a description of the 3A-quark system characterized by the quark hamiltonian given in eqs. (2.1). For the RGM wave function of the 3A-quark system we then have the following generalization of eq. (2.2) @3A-q)(,,r,
. . . , &A) = i?‘q’ c(fi
. .
Fig. 2. The “disconnected”
l
Of course
the summation
diagram should
(2.30)
~lu”‘(li.~,Si.,))~(llr.....IA)l.
/
L\i=l
_I
.
which corresponds to the term, If=, by the third term in eq. (2.37). then run over i = 1,.
,3A.
Jzmp)(k)
x CfJF’,
NiFx)
given
514
Y. Yamauchi et al. / Quark exchange currents
The 3A-quark coordinates p, , . . . , pjA are then rewritten in terms of the two internal Jacobi coordinates of the ith cluster 5i.l and Et.2and one c.m. coordinate of the ith cluster ri where (i= 1,. . . , A). The Pauli principle between quarks is included in eq. (2.30) by virtue of the antisymmetrizer of clusters (generalization of eq. (2.3a)) and 2(q) (generalization of eq. (2.3b)), given as GCq)= 1 - 9 i
Psisj + (permutations
operating on more than two clusters) . (2.31)
i>j=l
The antisymmet~zer of clusters is already incorporated in eq. (2.30) by choosing a wave function F which is totally antisymmetric for any permutation of the clusters as a whole. After eliminating quark degrees of freedom, this leads to the Pauli principle on the nucleon level. Once the total hamiltonian and the intrinsic wave function of the clusters are given, the equation of motion which determines the 3A-quark wave function W is given by the RGM equation \ (i,
~~‘(SII.Si.,))(H’q’-E)~‘3A~q’(~~,...,~,,)d~,.,
* * .~EA.z~O-
(2.32)
Integrating over the internal coordinates of each cluster (i = 1, , . . , A), then leads to an integro-differential equation of the form
,...,rA;r; ,...,ra)-E’N(r,,...,rA;r:,“ J[ff(r, (2.33)
xq(r;,...,rh)drl,***dra=O.
Next, we introduce the following renormalized
x!P(ri fi(rl,.
.
-,rA;
r;,...,
&)=
,....,
Np’i2(r,,. X
H((ry,.
x N-“2(r:‘,
xr;l--
wave function and hamiltonian
&)dr;-=*drj,, . ., r,; r!,.
(2.34a)
.., r>)
. , ,, r>; r:“,. . . ,rz) . . . , rz; r:, . . . , rh)
. &a; &.;.
. . drz,
(2.34b)
where N’/’ is the square root operator of the RGM norm kernel. It is obvious that G satisfies the orthogonality relation and the following Schradinger equation
I
Z?(r,,. . .) rA; r$, . . . , ra)?@(r$, . . . , ra> dri * * . drL= El?(r,,
. . . , rA) .
(2.34~)
In the case of the six-quark system, we identified the corresponding renormalized wave function given in eq. (2.7) as the appropriate NN wavefunction and eq. (2.8) as the effective SchrGdinger equation for the NN system, Therefore, we also identify
515
Y. Yamauchi et al. / Quark exchange currents
the renormalized wave function in eq. (2.34a) as the appropriate A-nucleon wave function, For A> 2, aside from two-body terms the effective hamiltonian also contains 3,4. . . A-body operators, This is a direct consequence of the antisymmet~zation principle on the quark level. It is now straightforward to define the effective electromagnetic current in the nucleon space via ($+~‘I$, = (~(‘A-q)IJ~)(y(3A-q)). (2.35) From the relation between the RGM wave function for the 3A-quark system of eq. (2.30) and the renormalized wave function of eq. (2.34a), the effective operator fieRf is given as ,
rA;
ri,. . . , rh) =
. . . , rA; rf,.
N-1’2(r,,
I X~WW P
(r;l,...,
r>;ry
. . , r>) ,....,
f-2)
x N-1’2(r~, . . . . , rx; r{, . . . , rf4)
xdrf...dridr;‘..+dr:, ,ra;r{,...,
4)
=
j
{
ii
@%ti.,
,
(2.36a)
&7_)fj(ri-
i-1
xJfs)JW
{ ii
j=l
@CYQ,,
d)}
, &*)8(q
-
$1)
xdtA.1* * * d!$* dr, * . *dr, .
(2.3617)
Again the spin-isospin color operators which are not shown here should be replaced by the co~esponding effective spin-isospin operators in the nucleon space. In this way, the effective nuclear force and electromagnetic current of the Anucleon system are obtained. These effective operators contain not only two-body but also three-body, . . . , A-body operators. Therefore, it remains to show that the one-body and two-body parts of the effective hamiltonian given in eq. (2.34b) and the effective electromagnetic current given in eq. (2.36a) are equivalent to the effective operators derived in the two-nucleon system. We will show this equivalence only for the effective current which comes from the quark impulse current. It can be shown for the other operators in the same way. First we perform the integration over the internal coordinates and evaluate the RGM kernel given in eq. (2.36b). Inserting eq. (2.15b, c) into eq. (2.36b), we obtain ~jPoMt = : J;mp)( i) + : I
i>j=l
+
(contributions
JFX~(~) +$ ,$mp)( ,k) Af’ NFX) i>j=l
from the permutations
more than two clusters) ,
involving (2.37)
where z(k) requires the omission of k in the sum over i and j. For simplicity, we do not show any functional dependence except for the index which denotes the particle number ij. The conventional impulse current J;,““(i) and the corresponding
Y. Yamauchi et al. / Quark exchange currents
516
exchange current, NY’
term of the RGM. kernel were already
is the exchange
case of the three-body “disconnected” of the current
J:“(Q),
which
comes
from the quark
impulse
given in eqs. (2.19a, b) and eqs. (2.23b) and (2.25), respectively. term of the norm kernel, current
which is given in eq. (2.6a). In the
in eq. (2.37), there are two distinct
contributions:
term like the third term in eq. (2.37), which is written and the exchange
a
as a product
part of the norm kernel in which no particle
appears
twice, and a “connected” term, which is not shown here. We will neglect the “connected” terms, because these can only be significantly different from zero if all three clusters come close together. On the other hand, the “disconnected” terms cannot be neglected because these represent the quite probable process in which the photon couples to a single cluster which does not exchange any quark with its neighboring clusters, while two other clusters not involved in the electromagnetic interaction exchange quarks, as shown in fig. 2. The remaining work involves the renormalization kernel for an A-nucleon system is given as
given in eq. (2.36a). The norm
q=‘+...,
(2.38)
N=l+
; i>j=l
where NFx) is the exchange term of the norm kernel, which is given in eq. (2.6). For simplicity, we treat NF”’ as a small quantity and neglect higher orders than N’. In this approximation N-l’* is given as N-l/*
= l-1
$
NljEx)+O(N2).
(2.39)
i>j=l
In the renormalization process, we must pay attention NpX) commute if they do not contain the same particle contain Ngx’.
the same particle because of the nonlocality For the first term of eq. (2.37), we then get
- : k=l z
to the fact that Jtrnp’ and but do not commute if they (momentum
dependence)
of
A;’ NFX’+O(N2)
Jf”P’(k)
i>j=l
i$,J:““)(i)
+
JOEY)
:
.’
(15)
i>j=*
-
;
k=l
J;“P’(k)
A;’ NFX’+O(N2)
.
(2.40a)
i>j=l
The first term of this equation is simply the conventional impulse current given in eq. (2.19a, b). The second term is the renormalization term given in eq. (2.27~). The third term has the same form as the “disconnected term” [the third term in eq.
Y. Yamauchi et al. / Quark exchange currents
(2.37)] except
for a minus N-‘/2
N--1/2
:
{
J;“P’(k)
sign. For the second
and third term of eq. (2.37), we get
i>fel J~x’(ii)JN-“2 =i>fz, J:Q’c’(ii)+O(N2), N-l/2
AF’ NrX’
= kg,
JimPI
Ag’
(2.40b)
NFX)
i>j=*
i>j=*
k=l
517
+O(N2).
(2.40~)
where we also count JF”‘( ij) as being of first order in N, because it is proportional to CP, the radial part of the norm kernel NEX. [See eqs. (2.6) and eq. (2.23) with eqs. (2.22b, c).] By summing up eqs. (2.40a, b, c), the total effective current becomes JF”) = i i=l
= ,g,
J:““)(i)
+
i
~jLQ’c)(q)
+
JE”“‘( i) + ;
f
~jL”ENl(ij)
i>j=1
i>j=l
JIPEC)( ij) .
(2.41)
t>j=l
As anticipated, the “disconnected” term has been eliminated by the renormalization process and the effective current is expressed by the currents which were already derived in two-nucleon system. [In ref. 31), Yazaki also showed the elimination of this “disconnected” term in a similar manner.] We can proceed in a similar way for the other operators and obtain eq. (2.28) and eqs. (2.29) in the approximation of neglecting more than three-body “connected” operators. In this way, we could prove that the effective NN potential and electromagnetic current which were obtained in two-nucleon system are applicable to nuclei with A > 2. Although we used the approximation of neglecting higher order terms in the expansion of N-“2 for simplicity, it is possible to show without this approximation that the “disconnected” terms will automatically drop out, and eq. (2.28) and eq. (2.29) can be derived by the renormalization procedure. Note that the expansion by orders of N should not be used in the actual calculation because NCEX) is no longer a small quantity at short-distances. It is reasonable for the “disconnected” terms to drop out automatically when we introduce the quark exchange current, because the wave function used to evaluate the currents has been derived from the Schrodinger equation including the effective NN potential, which itself results from quark exchange. Thus great care has to be taken to avoid double counting. Before ending this section, we want to comment on the multi-cluster problem in nuclear cluster theory. It is well known that for the 3a problem one does not obtain a satisfactory result if one uses effective (YLYpotentials with repulsive cores determined from CKXscattering. On the contrary, quite satisfactory results are obtained by models in which the forbidden states are taken into account. This result indicates that it is essential to take into account the forbidden states which give rise to the null states if complete antisymmetrization is performed. Since the permutations which operate on more than two clusters were neglected in our approximation, and
Y. Yamauchi
518
the same effective in the A-nucleon the quark nuclear
cluster
cluster
repulsion hand,
potential system, model
theory,
is almost
which
is obtained
our approach is quite
constant
for the two-nucleon
looks insufficient.
different
the oscillator
the same
the oscillator
et al. / Quark exchange currents
from that in nuclear
constant
which
as that of the nuclear which characterizes
system
However,
cluster
characterizes potential
the short-range
is used
the situation theory.
in In
the short-range
well. On the other repulsion
in NN
scattering is almost three times larger than that of the nuclear potential well, and the origin of the short-range repulsion is attributable not only to the Pauli principle, but also to the specific form of the quark dynamics (spin-dependent forces). Thus, the probability that two clusters will overlap is very small in quark cluster theory in contrast to the nuclear cluster theory. [The difference between quark cluster theory and nuclear cluster theory is investigated in ref. “).I Therefore, a good approximation will be obtained without taking into account any multi-quark component other than the 6-quark component, and by neglecting the permutations which operate on more than two clusters. In any case, we must wait for further studies to obtain a more quantitative conclusion about the role of 9-quark components, for example, or effective three-body forces and currents.
3. Quark exchange currents in nuclei 3.1. SHORT-RANGE
CORRELATIONS
WITH
THE
BETHE-GOLDSTONE
WAVE
FUNCTION
In this section, we discuss the calculation of the nuclear matrix elements of the quark exchange currents presented in the previous section. Since the many-body problem given in eq. (2.28) cannot be solved exactly, we use the nuclear shell model, which is well established for light nuclei. In this paper, we investigate the role of the quark exchange current in the magnetic form factors of certain light nuclei with a simple closed shell f 1 configuration (iSN, “0 and 39K). We take the nuclear wave function to be a single Slater determinant of single-particle states, and the singleparticle wave functions to be those of the harmonic oscillator model. While we expect the independent
single-particle
picture
to be a good (zeroth-order)
approxi-
mation, appropriate short-range correlations between the nucleons should be taken into account when we evaluate the matrix elements of the quark exchange currents, which contribute only in the short-range region. In our picture, the short-range repulsion itself comes from quark degrees of freedom and is described in terms of the effective NN interaction given in eqs. (2.13). Therefore it is reasonable to introduce short-range correlations using the effective NN interaction of eqs. (2.13). The consistency between the quark exchange currents and the nuclear wave function is shown to be important by examining the dependence of the gluon-quark exchange current contribution on the size (bparameter) of the nucleon in our previous study of deuteron I’). If we increase b, and do not take into account its influence on the NN interaction or the nuclear
Y. Yamauchi et al. / Quark exchange currents
wave function,
it is evident
of the quark exchange are proportional larger nucleon exchange
that a larger nucleon
currents.
to the function
overlap
understood
given in eqs. (2.27). The quark 9, the radial
the contribution
is that the quark exchange
of the cluster
size. This can be readily
current
size would increase
The main reason
to the probability
519
from the definition exchange
part of the norm kernel
one would expect a large enhancement of the quark we must now forget the effect that a larger nucleon
currents
which is increased current
for a
of the quark
is proportional
NEX given in eq. (2.6b). Thus, exchange current. However, size has on the short-range
repulsion between nucleons. The part of the effective NN interaction which leads to the short-range repulsion is also proportional to the cluster overlap. The quark exchange potential given in eqs. (2.13) is proportional to the function P’, the radial part of the exchange term of the norm kernel NEX given in eq. (2.6b). Therefore the range where the two nucleons begin to feel the short-range repulsion is extended to larger distances if the nucleon size b, is increased. This suppresses the nuclear wave function in the interior region, and thus prevents the quark exchange current contribution from becoming much larger. This was actually found and shown in fig. 11 in ref. I’). This analysis clearly shows that the consistency between the quark exchange currents and the corresponding short-range correlations is very important. Note that the two nucleons feel the short-range repulsion in exactly the same region where the quark exchange currents are effective. Therefore, in the present work we must take into account short-range correlations consistent with the quark exchange currents. We introduce short-range correlations by solving the Bethe-Goldstone equation given by p:+ -~_
r: + d
2mN 2m,bh
+
2mN
4
-+Qcq3(12)V,2 2m,b&
1
~~G~(12)=Eap?F’xIBpG)(12))
(3.1)
where the single-particle hamiltonian is taken to be the harmonic-oscillator hamiltonian. The oscillator length b, is chosen so that the oscillator wave functions reproduce
the experimental
r.m.s. radius
33,34): b, = 1.7 fm for “N; b, = 1.8 fm for
“0; b, = 1.9 fm for 39K. VI, is the effective NN potential given in eq. (2.13). In this work we consider only the central part and neglect the tensor and LS force of the effective NN potential V,,. We solve eq. (3.1) using an approximation given in ref. ‘“). In fig. 3, we plot the relative Bethe-Goldstone wave functions @zG)(r) given by @‘a”p”‘(12) = C (nl, NL; Aln,l,,
n,lz; A)[ Y,(C) x Y,(B)]~,~~G’(r)~~‘(R),
nlNL
(3.2) together
with the unperturbed one +27)(r) for s and p waves. Here, b, = 1.8 fm and n,l,; A) is the Talmi-Moshinsky bracket. The Bethe-Goldstone (4 NL; Alnil,, wave functions for higher partial waves are substantially the same as the unperturbed ones because of the centrifugal barrier. Since we define the c.m. and relative
520
Y. Yamauchi et al. / Quark exchange currents
The relative Bethe-Goldstone
I
’
I
1
w.f. ( n=O l=O )
I
I
H.O. JWJ JT=l
_ 9 0
(a)
0.0
I 0.5
I
I
I
1.0
1.5
2.0
r
hl
The relative Bethe-Goldstone
:: 6
0.0
_ I
_
I 2.5
3.0
w.f. ( n=O l=l )
I
I
I
I
I
0.5
1.0
1.5
2.0
2.5
3.0
r b-4 Fig. 3. (a) The relative Bethe-Goldstone wave function @sG’( r) with n = 0 and I = 0 according to eq. (3.2). Solid curve: for the JT= 0 channel; dashed curve: for the JT= 1 channel. Dotted curve: the unperturbed harmonic-oscillator wave function c$L$~’ (r). Note that the tensor and U-forces are not included when we solve the Bethe-Goldstone equation. In order to demonstrate that the Bethe-Goldstone wave function heals into the unperturbed one, these figures do not contain the normalization factor. [The Bethe-Goldstone wave function is normalized to one in the actual calculation.] (b) The relative Bethe-Goldstone wave function I?rzG) (r) with n = 0 and I= 1. Same notation as in fig. 3a.
coordinates as r = r, - r2 and R = ;(r, + Q), the unperturbed relative and c.m. wave functions, 4$‘(r) and 4%‘(R) are represented by the harmonic-oscillator wave functions with b parameters b, = Ab, and b, = fi b,. These results show that the healing distances are about 1 fm, which is reasonable and necessary for the independent-particle or pair picture to be valid. The validity
Y. Yamauchi et al. / Quark exchange
of the independent-pair
approximation
in which
currents
particle
521
correlations
that involve
more than two particles are neglected can be roughly estimated from the healing distance h. For example three-particle correlations can only be significantly different from zero if all three particles h3. Since distance
the healing the probability
distance
are found is short
simultaneously compared
for this to happen
within
a volume
with the average
of order
internucleon
is small and the independent-pair
picture
is valid. Our approximation of neglecting three-body to A-body quark exchange potentials and currents in sect. 2.3 can be justified using a similar argument. For example three-body quark exchange operators can only contribute if all three nucleons overlap simultaneously. Since the region where any two nucleons feel the short-range repulsion is the same as the region where these nucleons overlap, the probability for three nucleons to overlap is almost the same as the probability for all three nucleons to be found simultaneously within a volume of order h3. Then the probability for three-body quark exchange operators to contribute significantly is small since the healing distance h is short compared to the average internucleon distance. Since we started from a description of the nucleus in terms of quark degrees of freedom, it was not obvious that this microscopic picture of the nucleus could be reconciled with the independent particle shell-model. The magnitude of the healing distance is known to be very sensitive to the NN interaction, especially to the short-range part of the NN interaction which is described in terms of quark degrees of freedom in our model. However, our results indicate the validity of the method which we used to incorporate quark degrees of freedom into a nuclear structure calculation. This is important if one wishes to take into account quark degrees of freedom in nuclei. Next we want to comment on the approximation which we used when we solved the Bethe-Goldstone equation. We used the approximation given in ref. 26). This approximation
may be too simple to justify the comments
and Faessler solved the Bethe-Goldstone potential using a much better approximation are slightly
different
above. In ref. 35), Kurihara
equation with a similar effective NN given in refs. 25,27). (Their parameters
from the ones used here. They also considered
the admixture
of six-quark states with different b, parameters.) Their results also show that the healing distance is about 1 fm. We therefore expect that our results, given in fig. 3, will not change much if we use the better approximation for the Bethe-Goldstone equation, as in refs. 25,27,28). At this point, we think it is instructive to consider the dependence of the NN interaction and the quark exchange currents on the nucleon size or b, parameter. In ref. 36), Takeuchi et al. investigated the quark exchange effect on Gamow-Teller type P-decays. They took into account consistent short-range correlations from their corresponding quark hamiltonian with 6, = 0.6 fm. In addition, they evaluated the quark exchange effect for different nucleon sizes: b, = 0.6 fm and 0.8 fm without any short-range correlations in order to investigate the dependence of the quark exchange effect on the nucleon size. Comparing the results with and without the
522
Y. Yamauchi et al. / Quark exchange currents
short-range
correlations,
the estimation exchange
the short-range
of the quark
effect depends
exchange
strongly
on the nucleon
b, = 0.6 fm and b, = 0.8 fm. However, of the nucleon
size on the short-range
in ref. “), its influence
correlations
are found
effect. They
to be important
also reported
that
size by comparing
their results for
they did not take into account correlations
on the NN interaction
in this comparison. is quite important,
in
the quark
the influence As we showed
and no reliable
conclusion can be drawn without it. In addition it is not obvious that the independentparticle or -pair pictures are still valid for such a large nucleon size. As mentioned before, the nucleon size determines the range of the short-range repulsion which is extended to larger distances if the nucleon size is increased. Since the healing distance is always larger than the distance of the short-range repulsion, it is uncertain whether the independent-particle picture is still valid for large nucleon sizes. Therefore, we do not investigate the dependence nucleon size here but fix it at the relatively
3.2. NUCLEAR
MATRIX
ELEMENTS
OF THE
of the quark exchange current small value b, = 0.5 fm.
QUARK
EXCHANGE
on the
CURRENTS
Having defined consistent short-range correlations, it is now a straightforward task to evaluate the matrix elements for the quark exchange currents. In this work we investigate only the quark exchange current which results from the quark impulse current. We are still in a too primitive stage to be able to take into account the other quark exchange currents except in the deuteron. Therefore we will investigate only the simplest and most established one (i.e. the quark exchange current coming from the quark impulse current) and leave the other quark exchange currents for future study although they give a larger contribution in the deuteron 17-20). The magnetic form factors can be evaluated from the nuclear matrix elements of the magnetic multipole operator. The magnetic multipole operator Tyig is related to the space component of the electromagnetic current as follows:
7;9(q)=$$jdq^[Y,(q^)xJ];‘, First, we evaluate
the effect of the (short-range)
due to the Bethe-Goldstone corresponding nuclear matrix
wave function elements are
.
correlations given
in eqs.
(3.3) on the impulse (3.1)
and
current
(3.2).
The
Y. Yamauchi et al. / Quark exchange currents
where
K
stands
(-)i-mu,,K_m
for both
are single-particle
The abbreviation ation
.? = m
[ 01 denotes
definition
and
j=[~I-i
a coupling
is made
creation
and annihilation
of both angular
matrix
element
by multiplying
&,qfi(A
11YI
]]_I>W(JlJ1;
respectively.
The other abbrevi-
and isospin. factor
We use the
37). The shell model exp [q2bh/(4A)]
(in spin and isospin
A l)(-i)‘+‘-L-‘k’(A
in
space) reduced
(1Yr 1115)
G~(q2)~(1N’86~1)a(lN’IIJSiT,)(LfII YLIIG
X (&T~j(l(GL(q2)6~rO+
m I0
doubly
a”,,,=
I
N
X
operators,
by Edmonds
with an overall
the standard fashion 38). For the two-body matrix elements, we have
x&
momentum
given
a with
u’ and
is used here and in the following.
of the reduced
c.m. correction
l=j+f=sgn(K).
523
~~~~(R)jL(qR>~~p~i(R)R2
+ 6so 2
Ll
dR
X (411 V
IIlJFfpin(iq)
JL)i’+Ld-J(JIJYLdIll)
iW(lLJ1;
X (SfTfjIIGSE(q2)8dTO+GVE(q2)T\N)adTIIIISi T) a3 X
I0
~~~~(R)j,~(qR)((Lfll[
y&(R)
xv,lLIILi)~~~~,(R))R2 dR
X(If(IY,IIli)F~onv.~(~q)+SsoC2iW(L1~J1;Jl)i’d+L-J b X(Jll
x(LflI
YLII~~)(SFT~IIJGSE(~~)~~TO+
yLllLi)
Gr(q2)7~N’S,,,IIISiTi)
~~~,(R)iL(qR)~~p,‘,(R)R2
dR
X bus,,.,
.
(3.4b)
524
Y. Yamauchi et al. / Quark exchange currents
Here Ffpin(k), Fk.R(k) and &,.,(k)
are the integrals
over the relative
coordinates
given by
Im.h(kr)[@!$j) = lorn [@!z::) (rlj~~(kr>((~~IIIY,,(~)xV,lfll~i>~~~'
Fbpin(k) = F&mv.~ (k) =
(r)~~~)(r)-+$~(r)4~,)(r)]r*dr,
(3.5a)
0
&v.,(k)
-~~j(r)j,(kr)((l,ll[ The matrix
elements
of the impulse
Y,,(c)~~,l’II~J~~~(~))l~*dr.
current
are composed
of three
(3.5b) parts.
These
parts, which are denoted by the integrals Fgpi,(k), F&nv,R(k) and FE:,,,,(k) in eq. (3.4b) correspond to the matrix elements of the spin-magnetization current Jlspin], the convection current with its derivative operating on the c.m. motion J~Cconv:RI, and the convection current with its derivative operating on the relative motion ““:, ,, as defined in eqs. (2.23). J, CO Now let us turn to the two-body matrix elements of the corresponding quark exchange currents. Since the quark exchange current is written in a similar form as the impulse current (see eqs. (2.23)), the two-body matrix elements of the quark exchange current are given by essentially the same expression (3.4) with the exception of the functions Fipin( k), F~,,,,R (k), and F$,,.,(k). these take the form
For the quark exchange
current
= KTH!+l,“,/,(+;
k) + yPa~HL~,,ili(-;
k) + H’,~,n;r,(o; k) 7
(3.6a)
Ft-cm.&)
= ~~&,,n,r,(+;
k) + GH!t~~rn,,i(-;
k) + ff!+i@,
k) ,
(3.6b)
F:&,.,(k)
= S:J”n’;,,nir,(+;
k)+ G,P!$,,,r,(-;
Fg,i”(k)
k) + Plf;l,,,JO;
k) ,
(3.6~)
where
dt Y,,(~)H*(r, (0; k) Hl,,r,n~~
P$ln,rl(
f ; k)
= jrn j,(kr)[V~~‘(r)?P~~‘(r)
=
- ‘@~~‘!@~‘(r)]r’dr,
[ dr j dr’!PEy)(r)
(3.7a)
(3.7b)
Y&,(F)
mf,m,m, X-
P!$,,,JO;
(-)*‘i~‘(limilm(l,m,)
‘1
r’; k)!P$~‘(r’)Yl,,l(i’),
(-i)‘d
4?r
k) = loa [ p::rsi)
dh[Yld(k^)xP’(r,r’;
k)]l,~~~,~‘(r’)Y~i,i(;‘),
(3.7c)
(r)j,,(kr)((~,lI[~,(i)xV,l’lI~i>~~~’(r))
- S~~~‘(r)j,,(kr>((I,II[~d(i) ~~,l’II~i>~~~‘(~))l~*dr,
(3.7d)
Y. Yamauchi
et al. / Quark
exchange
currents
525
with
N;“2( On the right-hand interchange
r’)
r,
~~~G)( r) dr’
side of eqs. (3.6), the first and second
terms corresponding
.
(3.7e)
terms represent
to fig. lb and fig. lc, respectively,
the quark
while the third
term is the renormalization term. The matrix elements H( rt ) and P( f ) were already evaluated in previous studies of the deuteron ‘2,‘8,19)by expanding the relative wave function into locally peaked gaussians. The coefficients Si, and Y:r in eqs. (3.6) are defined ;s;,=,
as
= Fdi-=o = (s,~~lllor~ff’[(b>(-9)P36
(sTc’lI(Isi T)x (SfTfllElllsi T,)-’9
JF&=, = (S,T,tli20j~ff’[~7i’t)(-9)~~~ ‘sTc’]/j]Sj F:,>x {S,T,ill~l’l”‘iilSiT}-’
(3.8a)
*
(3.8b) (3.8~)
5-d?-=, = (s,~,~~~o’,~ff)[~T:4z)op36 (sTct]//jSi T) x (SfTfl]j$T~N’jljSiTi)-’ 7 Y;r,0
= (SrTf111201~ff”[~rr~q’(-9)P3,CsTc’]l]]SiTj) x (SfTfIll~a(lN’lllSi~)-l
9
(3.8d)
T,)-’ 9 Z&r=0 = (s,T,~l~o~“,“‘[~a~q’(-9)P36 ‘sTc’]JIISiTj) x {S~~~~~~~~~N’~l~Si X (S~~~))/~~(lN’U(IN’JJISi~)-l7
Y’ir=i = (S~T~~l120~ff’~SO’,“‘7:4,‘(-9~~~~Tc’ljll} ~~~~~ =(S~T~ll10~~ff”[~~~q’~~‘(-9)~~~Tc’]II~SiTi)
(3.8e)
X(S,~~JII~r~N’a~N’lllSi~)-’
.
(3.8f) (3.8g)
In eq. (3.4), the one-nucleon form factors Gk(q’), GG(q2), G”,(q2), and G”,(q’) are of gaussian type, as in eq. (2.19e) and (2.19f), which is unrealistic at high momentum transfers. Because we are interested in two-nucleon phenomena, rather than in the one-nucleon properties themselves, we simply replace these one-nucleon form factors by the corresponding empirical ones G”,( q2) = G’;(q2) = $( 1 + 9’10.71 GeV2)-‘,
(3.9a)
G”,( q’) = !+!%
(1+ q2/0.71 GeV2)-* ,
(3.9b)
(1 + q2/0.71 GeV2)-” ,
(3.9c)
(-$Jq2)
=
!!$b
in both the matrix elements quark exchange current.
of the impulse
3.3. THE QUARK
MAGNETIC
EXCHANGE
current
and the matrix
elements
of the
MOMENT
Now we investigate the q +O limit of the Ml operator, that is, the magnetic moment. This consideration is particularly useful in studying the structure of the
Y. Yamauchi et al. / Quark exchange currents
526
quark
exchange
current.
The quark
exchange
(QEc)(r, r’, R)=F[V, CL
It is expressed current
in units
of nuclear
JCQEC) is given
exchange current, two terms
The quark-interchange
P
(QW(r,
,J,
R)
magnetons
p-LN= e/(2m,).
together
exchange
r, r’, R) = P (QEC)(
moment
magnetic
with
~-l/2@,
The quark
exchange
Like the quark
operator
also consists
pcQ”)(r, r’, R) + pcREN)( r, r’, R) .
r”)(I.(EX)(rrr,
r”‘,
R)N-1/2(rrtr,
-FCD’(r,
of
(3.11a)
term pCREN) are given as r’)
N-1/2( r, r”)p(“‘( r”, R)N-“‘(
I* (REN)(r, r’, R) =
(3.10)
eqs. (2.23).
moment
term pCQlc) and the renormalization
=
is given as
xJCQEC)(r, r’, R, q)]‘,=o.
in eqs. (2.27)
the quark
magnetic
d,.”
d,.“‘,
(3.11b)
r”, r’) dr”
R)S(r-r’),
(3.11c)
where the direct term ~1~~) and the exchange term pCEX) can be evaluated from eq. (2.23) by taking the q +O limit as in eq. (3.10). As shown in the study of the deuteron I’), the contribution of this quark exchange current is zero in the deuteron magnetic moment and thermal np capture process. Let us investigate the quark exchange magnetic moment operator in the more general case. (i) 7’he isoscalar quark exchange magnetic moment. First, we consider the isoscalar part of the quark exchange magnetic moment. Taking the q + 0 limit in eqs. (2.23) and using
eq. (2.24b),
&&=+[L
the direct
r +LR+$Ty+cp
and exchange
)I >
terms are given as (EX) P-a~=0
(D) - PAT=O
NCEX’(r, r’),
(3.12)
where the index AT = 0 denotes the isoscalar part of the magnetic moment. By inserting these equations into eqs. (3.11), we can evaluate the (isoscalar) quark exchange given as
magnetic
Np”‘(r,
moment.
In eqs. (3.11), the renormalization
operator
N-“2
is
r’) = In ,C, T (1 -9(ST]O$r’[P$~TC’]]ST)($2”+‘}P1’z 3, 1, x4,, cb’Jz7?bq’(r)~~~‘Jbq)(r~)Y,m(r^)xs,YT,(r^’)x~T,
(3.13)
wave function with b = mb,. Here where 4,,,(b=JT75bq)(r) is the harmonic-oscillator we note that the functions +zCJTjSbq)( r) Y,,,,( r^) are eigen functions of P(r, r’), the radial part of the exchange term of the norm kernel NEX given in eq. (2.6b) with we can express N-‘12 by these functions as in eigenvalues ($)zn+f. Therefore
Y. Yamauchi et al. / Quark exchange currents
eq. (3.13). Since pLr&, is expressed the intrinsic
spin operators, pFE!ir,
as a sum of the orbital
it commutes
pLyTJ(r,
r’, R)=piffT’?i(r,
angular
momentum
and
with N-1’2. Thus, we get
r’, R) = -pFjT!!)(r, =p,Lyeo(r,
527
r’, R) R)
N-‘(r,
r”)NcEX)(r”, r’) dr”,
(3.14a)
r’, R)=O.
(3.14b)
r’, R)+pLFT)(r,
In this way, we have shown that the quark exchange magnetic moment does not contribute to the isoscalar part of the magnetic moment as long as we restrict ourselves
to quark
exchange
currents
connected
with the quark
impulse
current.
This is an expected result that is explained in greater detail in the study of the deuteron 12). As can be seen from eq. (3.12), the isoscalar magnetic moment operator is made up of the total orbital angular momentum and total spin of the system, which correspond to the q + 0 limit of the convection and spin-magnetization current in the impulse current. The total angular momentum or total spin of the system is fixed, whether it is described in terms of quark or nucleon degrees of freedom. Therefore, the quark exchange effect cannot be seen in the isoscalar part of the magnetic moments. Here, we wish to comment on the quark exchange currents connected with two-body pion and gluon quark-pair currents which we neglected here. In our study of the deuteron I*) we showed that there is a non-zero but small contribution from these quark exchange currents to the isoscalar deuteron magnetic moment. The above conclusion is therefore only valid as long as we restrict ourselves to quark exchange currents connected with the quark impulse current. (ii) 7’he isovector quark exchange magnetic moment.On the other hand, the quark exchange magnetic moment can contribute to the isovector magnetic moments. It can be expressed as a sum of the following terms: P
(QEC)=Il,jRQxE)+CL!~~~)+CL.~~EC)+~~EC).
(3.15)
We do not write the index AT = 1 here and in the following, because only the isovector part of the quark exchange magnetic moment remains. The meaning of the individual terms in eq. (3.15) is evident, given as follows:
if the corresponding
(D) P,XVR
lb
_‘I2 -
(N)
--‘2z
(N)
8i
- 4( Tya(lN)+
CD)_?
rx~
direct terms are
R,
T:y’a$N’) .
(3.16a)
(3.16b)
These four types of direct term can be derived from the following q + 0 limits of eqs. (2.23). The types &&RxV,and p L, are derived from the q + 0 limit of JICOnv:~17, where V, operates on eiq” or eiq’R, while F,,~~ is derived from the q-+0 limit of on eiq”. The q -+ 0 limit of JLspin] gives p:“‘. The Jt conv:R], where V, operates
528
Y. Yamauchi et al. f Quark exchange currents
corresponding the quark
exchange
exchange
magnetic
currents
moments
can be derived
from the q + 0 limits of
given in eqs. (2.23). In this case, V, does not operate
on eiq” but on H or P. One obtains
(3.17a)
xV,CP(r, r’)+(lo2), p:x)
= -pS(r,
(EX)_- -4S(r, &
(3.17b)
r’)L,,0je2ff)[(25-&‘-r$$)P$~TC)]+(lt,2),
(3.17c)
r’)O~~[(2alq~7~~~+a~‘~:4,~)P~~Tc~]+(1~2).
(3.17d)
In addition, both the direct and exchange derived from JlconvZR1, where V, operates p(LDR) = +( ~(1:) + @)LR, However,
the corresponding
quark
term contain another type I.L~,, which is on eiq’R. Using eq. (2.24c), one obtains .
p(LEb() = p(LD,)NcEX)( r, r’)
exchange
magnetic
moment
vanishes
(3.18) because
of the cancellation between the quark-interchange and the renormalization term. This can be readily shown using the same arguments as in the isoscalar case. Among the remaining four types of quark exchange magnetic moment, p.bQEC) has a particular property. It can be expressed like eq. (3.13) and is given as (b-~bq’(r)Y~m(~)Xs,7, pbQEC)=
c
c
c
s,r, S,T, nlm 1
X
“;‘,,,
_(_l)S,+T,+f
P 1
[s
T ff
~~=J~“q’(r’)YT,(3’)Xtsir
1(92n+r 3
JI-~F,[s~T,](~)~~+'
_(_l)S,+T+’ 2
2
x [-~(SiTiIO~‘[(20:4’719’
+o:“‘T:“Z’)P:S6T”]ls,Tf)(~)2n+f
+~(S~~~~~~~~~~N~~S~~~)~l-J1-~~~[~~~~~[~)2~+fJ1-~~~[S~~~~(~)2n+f~~ , (3.19a) with FP[ ST] = (ST\o~;@[ P$:Tc’]]ST) Let us investigate the In the case 1= even (Si , Ti) = (1,O) or vice in eq. (3.19a) reduces
.
(3.19b)
1. dependence of ~~~~~ on the relative angular momentum one finds that due to selection rules (S,, Tf) = (0,l) and versa, and FP[ Sf Tf] = FP[ Si Ti]. Thus the term in the brackets to
where the first and second term in the spin-isospin matrix element comes from the quark-interchange term, and the third or last term comes from the renormalization
Y. Yamauchi
term. The spin-isospin quark exchange
matrix
magnetic
et al. / Quark exchange currents
element
moment
of the cancellation
between
This is the reason
that the quark
in the thermal
np capture
spin structure.)
However,
529
in eq. (3.19c) is zero. Therefore
does not contribute
the quark-interchange exchange
in the case 1= even because
term and the renormalized
magnetic
moment
term.
does not contribute
12). (The other types do not contribute pbQEC) contributes
this type of
because
in the case I= odd. When
of their (S,, Tf) =
(q, T) = (1, l), the bracket in eq. (3.19a) can be reduced to a simple expression like eq. (3.19c), but the spin-isospin matrix element is not zero in this case. When (S,, Ti) = (1, l), and (Si, Ti) = (0,O) or vice versa, Fp[Sf, 7’f] # Fp[SjTi], and the parenthesis in eq. (3.19a) is not zero. Therefore this type of the quark exchange magnetic moment contributes in the case I= odd. This is also mentioned by Takeuchi et al. in their study of Gamow-Teller transitions 36). On the other hand, the other three types have only simple
selection
rules, which are the same as the ones of the
corresponding direct terms, because the permutation of quarks cannot change the rank of the operators, angular momentum, spin, or isospin, which determine the selection rules. It is well known that the two-body magnetic moment operator can be decomposed into the Sachs moment and the intrinsic part as 39*40)
pex= PSachs Using
current
+ hint
conservation,
kachs
=
k%achs =
3
the Sachs moment
im
d
x
[ VI,)
Dl
[R xje,(q = o)] .
y
pSilchs is shown
(3.20)
to be (3.21)
,
where VI2 is the NN interaction and D is the electric dipole operator. Eq. (3.21) shows that the Sachs moment is constructed only from the nuclear potential. In the case of the intrinsic part, current conservation does not determine this term, which needs a model to be specified. Following this decomposition, the quark exchange magnetic
moment p
can also be split into the above two pieces as
(QEC) = &2E$)
Sachs
This can be readily
verified
xr.)
(QEC) Pint
_ (QEC) -prxvR
+&yEc)+pbQEc).
from eqs. (2.23) and (2.27), together
(3.22)
with eq. (3.20).
Therefore ~~~~~ (QEC) is the Sachs moment, which is intimately related to the quark exchange potential given in eqs. (2.13). On the other hand, the other three types constitute the intrinsic part, which cannot be determined from current conservation alone. We summarize the above analysis. As long as we restrict ourselves to quark exchange currents connected with the quark impulse current there is no isoscalar quark exchange magnetic moment because of the cancellation between the quarkinterchange term and the renormalization term. The isovector quark exchange magnetic moment is made of four types: p$?kQ,“v”,‘, p (QEC) rxo, , piTEC), and ~2~~). Among
Y. Yamauchi et al. / Quark exchange currents
530
these types, pIpQx”v”,’ is the Sachs moment, exchange
potential
moment, rules
which
given
are not determined
for the corresponding
and ~$~~)
are non-zero
the case I,= Ii (ZO);
which can be constructed
in eqs. (2.13).
The other
by current
nuclear
matrix
three
conservation
elements
types
from the quark are the intrinsic
alone.
The selection
are as the following:
only in the case If= Ii f 1, while piTEC) is non-zero
in addition
pzEc)
is non-zero
I~$$kQx”v”.’ only in
only in the case Zf= Ii = odd.
Note that there is no quark exchange contribution in the case If= Ii = 0. Before ending this section, we want to comment on the work of Ito and Kisslinger 41). They evaluated the magnetic moments in certain nuclei with a closed shell* 1 configuration using a hybrid quark-baryon model. In their model, the two-nucleon system is represented by the six-quark state in the (OS&~ configuration for relative distances smaller than some boundary radius RB. For distances greater than or equal to R,, they used the two-nucleon description. The six-quark probability is determined from the probability current conservation and continuity across the boundary radius. Therefore their six-quark probability depends only on the twonucleon wave function and its derivative at r = RB. They investigated the influence of the above (OS&~ six-quark state on nuclear magnetic moments. However, they did not consider the intimate relation between the short-range part of the NN interaction and the quark degrees of freedom which we have strongly emphasized here. In our model, the six-quark system is described by a quark hamiltonian, which then leads to a dynamical description of the short-range part of the NN interaction. One should note that a six-quark probability defined as in the work of Ito and Kisslinger contains much of the conventional NN component, which is non-zero even at short distances. In contrast to their model, we could single out that part of the quark effect, which is already contained in the conventional model via the NN interaction and/or the nuclear wave function through the renormalization term given in eq. (3.11~). Actually, we showed that because of the cancellation between the quark-interchange term and the renormalization term there is no quark exchange effect in the case If= Ii = 0, which corresponds to their contribution from the (0~~1~)~ six-quark state. The quark exchange magnetic moment is non-zero not in the case lr= Ii = 0, but only in the other cases which they did not investigate. [The main contribution to the quark exchange magnetic moment comes from the off-diagonal elements
between
the relative
s- and p-waves
as we will show in the next section.]
4. Results and discussion 4.1. MAGNETIC
MOMENTS
First, we show the the effect of the quark into two parts, i.e., the and can be absorbed
FOR
NUCLEI
WITH
A = 15, 17 AND
39
numerical results for the magnetic moments. In our model, degrees of freedom on the magnetic moments is decomposed one which generates the short-range part of the NN interaction into the proper definition of the NN interaction and/or the
Y. Yamauchi et al. / Quark exchange currents
531
NN wave function, and the remaining part, which cannot be described in that way.
The former contribution is taken into account by considering short-range correlations in the evaluation of the impulse current. The latter is treated as a two-body current, that is the quark exchange current. Let us compare the contributions of the short-range correlations and the quark exchange current. As mentioned in the previous section, the contribution of the quark exchange current to the isoscalar magnetic moment vanishes. [Since we do not include the tensor and LS forces, short-range correlations do not contribute to the isoscalar magnetic moment either.] We therefore list only the isovector magnetic moment in table 1. If the contribution of the quark exchange current were negligible compared to that of the short-range correlations, the quark degrees of freedom could be included in a properly chosen NN interaction. In this case, there would be no need for us to consider quark degrees of freedom in the form of quark exchange currents. However, our results indicate that this is not the case. The quark exchange current contribution is much larger than the contribution of the short-range correlations. Therefore we have to consider quark exchange currents explicitly. It has long been recognized that two main mechanisms play important roles in nuclear magnetic moments: (1) the modification of the nuclear wave function by a residual interaction via (second-order) con~guration mixing 42*43)and (2) meson exchange currents44) (together with the effect of using modified nuclear wave function in their evaluation ““)). For reasonable estimates of these effects on nuclear magnetic moments, we refer to the results of Arima et al. 45) and list them in table 1. We emphasize that it is not our purpose to compare our results with experiment. If it were, we should have at least considered these effects in our model by choosing our effective NN interaction as the residual interaction. Although the (short-range) correlations taken into account via the Bethe-Goldstone wave function correspond TABLE 1 Corrections
to the isovector
magnetic
moments
expressed
as a percentage
of the Schmidt
values
WPA7”‘)/b4:~~d,) A
15 17 39
QEC
SC
-9.11 1.64 -20.91
-1.62 0.20 -3.32
CM 5.49 (20.0) -16.93 (-18.1) 42.22 (67.4)
MEC
CM + MEC
EXP
7.18 15.52 -48.32
12.67 -1.41 -6.10
11.13 1.34 -38.42
QEC: the quark exchange magnetic moment; SC: the effect of short-range correlations due to the Bethe-Goldstone wave function on the impulse current; CM: the effect of (second-order) configuration mixing; MEC: the meson exchange magnetic moment; CM + MEC; the sum of the configuration mixing and meson exchange magnetic moment; EXP: experimental data. For the CM, MEC and CM+MEC, we refer to the results of Arima ef nf. 45) given in their table 7.1. The numbers in parentheses refer to the Paris potential while the others are based on the Hamada-Johnston potential. The Schmidt values of the nuclei considered here are 0.4509 n.m. for A= 15,3.3530 n.m. for A = 17, and 0.5117 n.m. (Y t$7hcmfdt) for A = 39.
Y. Yamauchi et al. / Quark exchange currents
532
to configuration into account
of configuration detailed
mixing
within
the correlations mixing.
comparison
the two-particle-one-hole
induced
Thus,
the present
with experiment.
these effects in our model,
by the tensor analysis
Although
space,
we did not take
force, which is the main source is not sufficient
it is straightforward
we are still in a too primitive
to warrant
a
to evaluate
stage to be able to take
into account every possible effect using the microscopic quark model. We therefore ask whether there is any chance of seeing the quark exchange effect in nuclear magnetic moments, or in other words, whether the quark exchange effect is negligible or not. As seen from table 1, the quark exchange current is almost of the same size as the configuration mixing or meson exchange currents in nuclei with A = 15. In nuclei with A = 17 and A = 39, the quark exchange contribution is smaller than the contribution of meson exchange currents or second order configuration mixing. However, it is not negligible. Since a large amount of the configuration mixing effect is cancelled by the meson exchange current, the sum of these contributions is almost the same as the quark exchange current contribution. In any case, these two contributions are quite model dependent, which can be seen in table 1 by comparing the results for configuration mixing due to the Hamada-Johnston potential and the Paris potential, respectively. [See also table 5.2a in ref. “‘).I Therefore, in order to draw any definite conclusion we must wait for a more consistent calculation of these two contributions in our model. In table 2, we show more details of the quark exchange magnetic moment. As already investigated in the previous section, the quark exchange current can be decomposed into two parts, i.e., the Sachs moment and the intrinsic part. Using current conservation, the Sachs moment can be expressed via the commutator of the NN potential with the electric dipole operator as in eq. (3.21). The Sachs moment is completely determined by a given potential. The results in table 2 show that the major contribution to the quark exchange magnetic moment comes from the Sachs (QEC). The contribution of the intrinsic quark exchange moment, which moment pRXV, is the sum of the other types, is rather small. Depending on the nucleus, the effect of the intrinsic quark exchange magnetic moment is of the same size or smaller than the effect of the short-range correlations. In order to avoid any model dependence,
the evaluation
of the Sachs moment
has ususally been performed using the phenomenological approach 44). In this approach one evaluates the Sachs moment with the help of eq. (3.21) using a phenomenological potential. The meson exchange contribution in table 1, which is evaluated in ref. 44), contains the Sachs moment due to the Hamada-Johnston potential. Since the main contribution to the quark exchange moment ~2::’ is of Sachs moment type, it seems that this type of quark exchange effect might already be included in the Sachs moment due to a phenomenological potential. However, the Sachs moment is not determined by on-shell NN scattering. It is possible to perform a unitary transformation which alters the inner part of the wave function without changing its asymptotic form. Such a unitary transformation also changes
Y. Yamauchi et al. / Quark exchange currents TABLE
Contributions
of the quark
exchange magnetic as a percentage
533
2
moment to the isovector of the Schmidt values
magnetic
moments
(QEC)) (CL&
(BG)
(HO)
-8.042 -8.185 0.142 -17.503 -17.716 0.213
-9.106 sP pd -20.272 sP pd
(BG)
1.644
(HO)
sP pd df 3.489 sP pd df
@G)
-20.911 sP pd df
(HO)
fg -46.893 sP pd df fg
A = 15 -0.011 -0.014 0.03 -0.062 -0.067 0.005
1.729 1.806 -0.078 0.001 3.792 3.908 -0.117 0.001
A = 17 0.003 0.004 -8 x 1O-4 4 x 1om6 0.016 0.017 -0.001 4 x 1o-6
-18.967 -19.945 0.996 -0.018 1 x 1om4 -41.704 -43.175 1.486 -0.015 1 x 1o-4
A=39 -0.014 -0.020 -0.006 -2 x 10-s 3 x 1om6 -0.085 -0.095 0.009 -1 x lo+ 2 x 1o-6
PP dd
PP dd ff PP dd ff
PP dd ff ff PP dd ff ff
expressed
(QEC)) (P..
-0.442 -0.455 0.013 -0.982 -0.993 0.011
-0.610 -0.610
0.112 0.114 -0.022 3 x 1o-4 0.248 0.250 -0.002 3 x 1o-4
-0.200 -0.199
-1.724 -1.724
-2 x 10-4 -0.566 -0.566 -2 x 1o-4
-1.305 -1.304
-0.625 -0.645 0.022 -0.002 1 x10-s -1.397 -1.414 0.019 -0.002 1 x 1o-5
-9 x 1om4 -3.706 -3.705 -9 x 1om4
Separate contributions for the four types of operators given in eq. (3.15) are shown as a percentage of the Schmidt values. The quark exchange magnetic moment is evaluated with the relative BetheGoldstone wave function (BG) and with the harmonic oscillator wave functions (HO) for comparison. The contributions from individual partial waves are also shown.
the NN potential unitary
but does not affect the on-shell
transformation
also
changes
the
Sachs
NN scattering moment.
Thus,
phase there
shifts. This are many
phase-equivalent potentials which give the same on-shell NN scattering but give a different Sachs moment. Therefore it is worthwhile to investigate the Sachs moment due to the quark exchange mechanism. Since the short-range part of our NN interaction is highly nonlocal, it would give a rather different contribution to the Sachs moment as compared to the contribution of a phenomenological local potential. Let us study the dependence on the partial waves of the relative motion and the effect of the short-range correlations on the quark exchange magnetic moment. As
Y. Yamauchi et al. / Quark exchange currents
534
mentioned partial
in the previous
waves allowed
contributions
from higher partial
able since the quark nucleons centrifugal
section,
overlap, barrier
there is a selection
by selection exchange
rule for each type. Only the
rules are listed in table waves become
current
smaller
and smaller.
This is reason-
is only effective
in the region
where the two
and the two nucleons for higher partial
2. We can see that the
are prevented
waves. Comparing
Bethe-Goldstone wave function with the results function, we notice that the latter contribution
from coming
closer by the
the results using the relative
using a harmonic-oscillator is roughly twice as large
wave as the
former. This shows that short-range correlations are fairly important when evaluating the quark exchange magnetic moment. [We note that the effect of short-range correlations on the quark exchange current is small for higher partial waves. This is reasonable because the relative Bethe-Goldstone wave function is almost the same as the unperturbed one for partial waves higher than relative p-waves.]
4.2. MAGNETIC
FORM
FACTORS
OF 15N, “0
AND
=K
Now we turn to the quark exchange effect in the magnetic form factors of ‘*N, “0, and 39K. In the corresponding magnetic moments, we saw that the quark exchange effect is much larger than the effect of short-range correlations on the impulse current. We also saw that it is important to consider short-range correlations when evaluating the quark exchange magnetic moment. Let us see whether a similar behaviour can be seen in the magnetic form factors. For this purpose we examine the effect of short-range correlations due to the relative Bethe-Goldstone wave function on the impulse current and the quark exchange current. In figs. 4-6 we plot the nuclear matrix elements of the magnetic multipole operators, J T , and we show the effect of the short-range correlations on i(pJAJ Tf;“‘(q)ll W A ,) the impulse current and quark exchange current for each individual multipole form factor separately. The three contributions (quark exchange current with and without short-range correlations and the effect of short-range correlations on the impulse current) are plotted in fig. 4 for the Ml form factor of 15N, in figs. 5a, Sb, and 5c for the Ml, M3, and M5 form factors
of “0,
and in figs. 6a and 6b for the Ml and
M3 form factors of 39K. As a reference we also plot the single-particle value, that is, the impulse current evaluated with harmonic-oscillator wave functions. For the Ml form factors, the quark exchange contribution is larger than the effect of short-range correlations, although it becomes smaller after the diffraction minimum. [The quark exchange current has a minimum at q = 4.3 fm-’ in 15N.] The quark exchange contribution in the M3 and M5 form factors is much smaller than the quark exchange contribution in the Ml form factor. Later we will investigate the reason for this. Thus, the effect of the quark exchange current on the M3 and M5 form factors is comparable to or smaller than the effect of short-range correlations on the impulse current. It is also generally true that the quark exchange current shows a different dependence on the momentum transfer than the effect of the
Y. Yamauchi et al. / Quark exchange currents
535
Fig. 4. Ml form factor of “N. Dotted curve: the single-particle value that is the impulse current evaluated with harmonic-oscillator wave functions (IMP); dashed curve: the quark exchange current evaluated with Bethe-Goldstone wave functions (QEC): long-dashed curve: the quark exchange current evaluated with harmonic-oscillator wave functions (QEC(H0)); dash-dotted curve: the effect of short-range correlations due to the Bethe-Goldstone wave function on the impulse current (SC).
short-range correlations on the impulse current. Therefore the quark exchange effects cannot be simulated by taking the effect of short-range correlations on the impulse current into account. Let us examine the effect of short-range correlations on the quark exchange current. The contribution of the quark exchange current evaluated without shortrange correlations is two or three times larger than the contribution of the quark exchange current evaluated with short-range correlations. Thus, as a general tendency, we recognize that the effect of short-range correlations on the quark exchange current is as important in the magnetic form factors as it is in the magnetic moments. However, the dependence of the quark exchange current on the momentum transfer is exactly the same, whether it is evaluated with or without short-range correlations. The position of the diffraction minimum of the quark exchange current (if there is one) remains unchanged by the inclusion of short range correlations. This might indicate that the momentum transfer dependence of the quark exchange current is determined mainly by the c.m. motion, which remains the same whether short-range correlations are included or not.
Y. Yamauchi et al. / Quark exchange currents
536
10-l t
I
I
....
/_
..,
j IMP ...___.. QEC I _ -QEC(HO) --SC -
lo-”
10-s
1 , , ,t,li
1,
1o-5
0.0
2.0
1.0
9
3.0
0.0
Lb-‘I
1.0
,
2.0
b)
1 3.0
[fm-‘1
I
I
+
..
IMP ._....._ QEC _ _QEC(HO) _-SC
1o-2r
10-3 -
/
9
10-l -
2 -2
1
;
BR
Fig. 5. (a) Ml form factor
of “0.
(b) M3 form factor of “0. as in fig. 4.
(c) M5 form factor
of “0.
Same notation
Y. Yamauchi et al. / Quark exchange currents
L”
0.0
2.0
1.0 q
3.0
--
1.0
2.0
3.0
9 he’1
b-7
Fig. 6. (a) Ml form factor
0.0
of 39K. (b) M3 form factor
of 39K. Same notation
as in fig. 4.
Next, we would like to comment on the separate contributions from each type of quark exchange current to the Ml form factor. As shown in eqs. (2.27) with eq. (2.23), the quark exchange current can be decomposed into the three types: Jl$“’ from the quark spin-magnetic current, and Jlzn::k, and JE?,‘$:!l from the quark convection current. The major part of the quark exchange current comes from the isovector part of the type J~~O~?R1. Especially its off-diagonal matrix element between the relative s-wave and the relative p-wave is dominant. This fact is related to the dominance of the Sachs moment p$$z) in the quark exchange Note that JLzOt::kl corresponds to ppXEVt’in the q + 0 limit.
magnetic
moment.
Keeping these properties in mind, let us now investigate the dependence of the quark exchange current on the multipolarity of the magnetic form factor. The effect of the quark exchange current on each Ml form factor is small but nonnegligible. In the higher multipole form factors, that is in the M3 and M5 form factors, the quark exchange contribution is smaller than in the Ml form factors. In particular, in the M5 form factor, the quark exchange contribution can practically be neglected. This property is determined by the spin structure of the quark exchange current. As mentioned above, the type JCQEC) Iconv:R1provides the major contribution for the Ml form factors. The quark exchange current, which comes from the quark convection current, is a rank zero operator in spin space like the nuclear convection current although it has a spin dependence given in eq. (2.11) and (2.24a). Thus the orbital angular momentum operators acting on the relative and c.m. motion must be coupled
Y. Yamauchi et al. / Quark exchange currents
538
to rank 3 or 5 corresponding to the M3 or M5 form factors, respectively. trivial that the quark exchange contribution from the quark convection suppressed
when
the rank
of the relative
orbital
angular
large. By selection
rules, a larger rank for the relative
to matrix
elements
between
the initial
or final states. Thus, the corresponding
higher
partial
momentum
motion
operator
waves for the relative
It is rather current is operator
wave functions
quark exchange
is
corresponds
contributions
of are
suppressed by the centrifugal barrier. Nevertheless, even in these higher multipole form factors there is a quark exchange contribution for relative s- and p-waves because the remaining rank can be carried by the c.m. motion. However, this probability becomes smaller with increasing multipolarity because of the structure of the nuclear matrix element given in eq. (3.4b). In particular, the quark exchange contribution coming from the quark convection current cannot contribute to the M5 form factor of “0. The single-particle state Od,,, (I = 2) with core (I= 0) cannot couple to rank 5 without a spin operator. As a result, the quark exchange contribution becomes rather small for higher multipoles. This behavior might change significantly if the tensor correlations on the quark exchange current were included. In other words, the crossing term between the quark exchange current and the configuration mixing induced by tensor force might change this behavior significantly. In order to see whether the quark exchange current can be seen in these form factors, we show the square of the magnetic form factor IF,(q)\* in figs. 7-9. It is defined through the unpolarized electron-nucleus scattering cross section in the laboratory frame da Z* dw PC__ 77 d0 M [I&(q)12+G+tan2 d0 Ott and expressed
by the multipole
form factors
G@Wq)('l
,
(4.1)
as follows:
(4.2) where 2 is the atomic
number
of the nucleus,
and da/dRl,,,,
and n are the Mott
cross section and recoil factor. More details are given in ref. 38). In these figures we plot the contributions of the configuration mixing and the meson exchange current for comparison. We refer to the results of Arima et al. 47) for “0 and those of Blunden and Caste1 48) for “N and 39K. We also plot the experimental data 49-52). In fig. 7 we show the square of the magnetic form factor in “N. The difference between the dotted curve and the dashed curve demonstrates the well-known decrease of the cross section due to the configuration mixing. It reduces the cross section in the whole region and worsens the agreement with the experimental data. The meson exchange current gives a large enhancement and reproduces the experimental data well. The quark exchange current gives a small enhancement at low momentum transfers, while at high momentum transfers it has virtually no effect.
Y. Yamauchi et al. / Quark exchange currents
539
r
__- ._ IMP+CM _ _ IMP+CM+MEC _
IMP+CM+MEC+QEC
value, that is, the impulse Fig. 7. The magnetic form factor of “N. Dotted curve: the single-particle current evaluated with harmonic-oscillator wave functions (IMP); dashed curve: the impulse current evaluated with harmonic-oscillator wave functions + the configuration mixing = IMP+ CM; long-dashed curve: IMP+CM+ MEC (meson exchange current); full curve: IMP+ CM + MEC +QEC (quark exchange current). The experiment data are from Singhal et al. 49). For the estimates of CM and MEC, we refer to the results of Blunden 48). The quark exchange current is evaluated with the Bethe-Goldstone wave function.
This is related to the fact that the quark exchange current has a minimum at q = 4.3 fm-’ and that it is already small at around q = 3 fm-‘, as shown in fig. 4. Here we comment
on the contribution
of the impulse
current
and the configuration
mixing which are evaluated in ref. 48). In their calculation Blunden and Caste1 did not take into account the one-nucleon electromagnetic form factors GE and Gr,, given in eqs. (3.9) and their result could not reproduce the experimental data. However, this discrepancy is reduced by taking into account the one-nucleon form factor which is contained in our figures. For I70 the quark exchange current gives only a negligible contribution (see fig. 8). Although the quark exchange current gives a nonnegligible contribution in the Ml form factor as shown in fig. 5a, the Ml form factor itself is the minor contribution to the cross section. The contributions from the M3 and M5 form factors where the quark exchange current is suppressed hinder the contribution from the Ml form factor to the cross section. In the case of 39K, which is shown in fig. 9, the quark exchange current gives a small enhancement at low momentum transfers, while at high momentum transfers it has virtually no
540
Y. Yamauchi et al. / Quark exchange currents
__..._ IMP+CM _
_
5*I0-e
IMP+CM+MEC IMP+CM+MEC+QEC
:
I
0.0
1.0
:
-
I 2.0 9
3.0
Pm-‘1
Fig. 8. The magnetic form factor of “0. Same notation as in fig. 7. The experimental data (solid circle) are from Hynes et al. 50) and from Kalantar et al. ‘I). For the estimates of CM and MEC, we refer to the results of Arima et al. 47).
effect. This is also due to the hinderance by the M3 form factor where the quark
of the Ml form factor in the cross section exchange contribution is negligible.
Summarizing the results of this section, we have found that for magnetic moments the quark exchange current gives a non-negligible contribution. The major contribution of the quark exchange magnetic moment comes from the Sachs moment, which is intimately related to the quark exchange potential, which produces the short-range repulsion. For the magnetic form factor, the major contribution of the quark exchange contribution also comes from the quark convection current. However the quark exchange contribution is rather small from 0 to 3 fm-‘. Especially in the high momentum transfer region it has virtually no effect. This is mainly due to the spin structure of the quark exchange current connected with the quark convection current. This spin structure makes the quark exchange contribution very small in the M3 and M5 form factors, which are dominant at high momentum transfers. 5. Conclusion We have calculated the magnetic A = 15, 17, and 39 nuclei in a quark degrees of freedom into a conventional
moments and the magnetic form factors of cluster model. In order to incorporate quark nuclear structure calculation, we introduced
Y. Yamauchi et al. / Quark exchange currents
541
IMP
t
__-.__ IMP+CM
_ t
‘,
_ IMP+CM+MEC
_
IMP+CM+MEC+QEC
Fig. 9. The magnetic form factor of 39K. Same notation as in fig. 7. The experimental data are from LapikaS ‘*). For the estimates of CM and MEC, we refer to the results of Blunden @).
effective operators in the nucleon space. These effective operators, that is, the effective NN interaction and electromagnetic current, were derived in the following way. We started with a description of the six-quark system characterized by a quark hamiltonian using the quark cluster model. Interpreting the renormalized wave function and the equation of motion, which determines the renormalized function, as the NN wave function and the corresponding Schrodinger equation, we could derive the effective NN interaction. In the same manner we could derive the effective electromagnetic current, which was written in terms of nucleon dynamical variables, and which
yielded
exactly
the same results
as the microscopic
description
of the
electromagnetic interaction. We have shown that the internal structure of the nucleon had two major effects on the form of these effective operators. One major effect revealed itself in the impulse current, the one-pion exchange potential and the corresponding one-pion exchange current through microscopically calculated vertex factors for the yN, TN and yNn vertices. In conventional nuclear theory these are regarded as phenomenological input parameters from the beginning. Therefore this type of quark effect does not provide a clear signal for quark degrees of freedom in nuclear physics. However, the quark degrees of freedom also revealed themselves through the quark exchange potential and the corresponding quark exchange currents. The quark
542
Y. Yamauchi et al. / Quark exchange currents
exchange
potential
repulsion
in NN
was previously
necessary
counterparts
scattering.
The
of the quark
conservation.
These quark
tional
theory and represented
nuclear
ties of the nucleus.
exchange
shown quark
to be responsible exchange
exchange currents
currents
potential,
for the short-range were
introduced
as
in order to satisfy current
had not been treated
in the conven-
the quark effect in the electromagnetic
We have also shown that the description
of the A-nucleon
propersystem
with this effective NN interaction and electromagnetic current is equivalent to that of the 3A-quark system with the microscopic quark hamiltonian and electromagnetic current, as long as we restrict ourselves to effective two-body operators in the nucleon space. In this work, we studied to what extent the quark exchange current connected with the quark impulse current influences the magneic moments and magnetic form factors of nuclei with A = 15, 17 and 39. We found a non-negligible contribution from this quark exchange current in the magnetic moments. Although this quark exchange current contribution is within the theoretical ambiguity of configuration mixing and meson exchange current effects, it might be seen in the magnetic moments. The numerically most important part of the quark exchange magnetic moment was of Sachs moment type. The Sachs moment was expressed as a commutator between the electric dipole operator and the quark exchange potential, which produced the short-range repulsion. In the case of the magnetic form factors, the major contribution to the quark exchange current also came from the quark convection current, which is related to the Sachs moment in the q + 0 limit. Because of the spin structure of the quark exchange current, the quark exchange contribution turned out to be very small in the M3 and M5 form factors. As a result the quark exchange contribution to the square of the magnetic form factor, which is related to the unpolarized electron-nucleus cross section, was rather small for momentum transfers from 0 to 3 fm-‘. In particular, at high momentum transfers, the quark exchange current had virtually no effect. These results are quite different from the corresponding results in the deuteron. In the deuteron there was no quark exchange contribution from the quark exchange impulse current in the q -, 0 limit, and the quark exchange effect could only be seen at high momentum
transfers.
In addition,
the major contribution
to the quark exchange impulse current there came from the quark spin-magnetization current, while the quark exchange current connected with the convection current gave only a minor effect. Therefore, the behavior of quark exchange currents in heavier nuclei can be quite different from the corresponding results in the deuteron.
We wish to thank Profs. K. Yazaki, K. Shimizu, M. Ichimura and Drs. H. Sagawa and W. Bentz for help and numerous stimulating discussions. A.A. thanks the “Alexander von Humboldt Foundation” and the staff of the Institut fur Theoretische Physik in Tiibingen for their hospitality. A.B. gratefully acknowledges the receipt of a post-doctoral fellowship from the “Japan Society for the Promotion of Science”.
Y. Yamauchi
et al. / Quark exchange currents
543
Appendix In this appendix,
we show some details
direct terms can be easily calculated, terms. They are written
of the derivation
of eqs. (2.22). Since the
we show only the evaluation
in terms of the functions
H’, P* defined
of the exchange as
H+ e-&b;q2
Hp+
(r,
r’;
t4)
=
0 P-
First we rewrite
the 6 function
as follows
8(r-a)=(&)3(y)3’4[dpelp[-:b:p2]
x(&)“‘exp[--&(r-(n+iTp))2], and insert
it into eq. (A.l).
We then get 29)
(~~)(r,r1:~q,=(~)6(~~‘2 q where the GCM
e-ib;q2
kernels
(A.2)
Jdp ;;)“i’“i(
2% r+ iyp,
Jdp’exp[-j~~(p*+pf’)l % r’+ iyp
H*(GCM), P*(GCM) are defined
Csi,
$;
,* , I. 2q) ,
(A.3)
as
44)
(A.4)
Y. Yamauchi et al. / Quark exchange currents
544
It is well known that this GCM kernel given in (A.4) is equal to the matrix involving a two-centered shell model wave function given as 29)
element
x ,lfIl (IroT PI -&‘j> ,fi4 $“(PI+ fsj) dp, dp, - * . dp, . We can derive eq. (A.4) from eq. (AS) by transforming
the single particle
(A-5) coordinates
appearing in eq. (AS) to internal coordinates inside the clusters, the relative coordinate between the two clusters and the c.m. coordinate of the six-quark system. Because our operators do not contain any dependence on the c.m. coordinate, the integration over the c.m. coordinate gives unity and we can get eq. (A.4). Performing the integrals of eq. (AS), we get the GCM kernel as (GCM) -~(S~+S~-:s,.s,)-fb~k’ 9
I
1
X
( i $tsiFS))
exp [ i$k * (Si f Sj)] .
(A.6)
4
The functions omitted
P*(GCM) also contain
a term which is proportional
that term since it will not contribute
to the magnetic
to k. However
we
or electric form factor.
The RGM kernel is obtained by inserting this equation into eq. (A.3). For the functions PkcGCM), the derivation of eqs. (2.22d, e) is not straightforward. Inserting eq. (A.6) into eq. (A.3), we get (A-7) Eqs. (2.22d, e) can be derived from this equation using partial integration and omitting the term proportional to k which does not contribute to the magnetic form factor. We did not use eq. (A.7) but eqs. (2.22d, e) instead because in eqs. (2.22d, e) it is more transparent that H and P represent quark exchange corrections to the
Y. Yamauchi et al. / Quark exchange currents
functions
eik’r and elk.? V, respectively.
The exchange
545
term of the norm kernel given
in eq. (2.6) can be derived in the same manner. We can also derive eq. (2.6) using the q + 0 limit of the operator H’ since the right-hand side of eq. (A.l) contains then only the operator Before ending
PX6.
this appendix,
we want to add another
P given in eqs. (2.22d, e) and the corresponding Eq. (3.7~) can be rewritten P$+ri(
f; k) =
matrix elements
given in eqs. (3.7).
dr’ P$‘~‘(r)Y$,,,(i)SP(r,
r’)
r’) X (+,TV,,)]fn
X[ Yld(rf
-$k(tklr
+ r’l) exp (#k’)
-$j,($klr
- r’j) exp (ibtk2)
where V, and b,. operate one should pay attention
on the operators
as
C ( -)2’f~1(limiZ*]Z,mf) mf.m,mi
X
comment
>
p::‘(r’)
Yl,,i(i’)
,
(A.@
only on final and initial wave function, respectively. Here to the combination of Yr and V appearing in eq. (A.8).
In the renormalization term which is represented by the matrix elements, P$!l,,,lL(O; k) given in eq. (3.7d) a similar combination appears. Although P~l,,,,l(O; k) contains V, the d/dr does not contribute for l, = 1.This can be seen from the following relations
j,(kr)(~,ll[Y,(i)XV.l’II~i) =k[C+j,+I(kr)U~ll~+l(i)lJli>+c-j,-,(kr)(I,II Y,-,(;)II4)1, (A.94 (Z~+Z~+1+2)(1~~1*+1+1)(1*~1~+1+1)(1~+Z~~1) 41(1+ 1)(2l+
1)(21+3)
9
(l~+l~+Z+l)(l~~l~+l)(l~~l~+l)(l~+l~~1+1) 41(1-r 1)(21-t 1)(21Using
these
relations,
P~,~,/~i(O; k) can be written
1) as a sum of H$:,,,l(O;
(A.9b)
(A.9c) k) and
H$,i,i(O; k). However, there is no corresponding equation to eq. (A.9) in the case of the matrix elements P”d=’ ,+,,J f ; k). Plf;,;,‘+( f ; k) could only be expressed by a sum of H’-’ +,,+( f ; k) and H$~+( f ; k) if the combination of K and V appearing in eq. (~.8) were [ Yh(rf r’) x (V,*I,,)]~, [ Yd(C) XV,]; or [ Yb(i))xV,.]I, . Therefore, the difference
between
term is due to the different by r and r’ in eq. (A.8).
the quark combination
interchange term and the renormalization of Y[ and V or the nonlocality expressed
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