Parity non-conservation in the pp system at low energy: Critical remarks about direct quark approaches

NUCLEAR PHYSICSA Nuclear Physics A586 (1995) 607-648

ELSEVIER

Parity non-conservation in the pp system at low energy: Critical remarks about direct quark approaches Bertrand Desplanques CNRS-IN2P3,

Uniuersitk Joseph Fourier, Institut des Sciences NucEaires, F-38026 Grenob1eGde.q France

53 Avenue des Martyrs,

Received 5 October 1994; revised 5 January 1995

Abstract Several papers dealing with predictions for the parity-non-conserving longitudinal asymmetry in pp scattering at low energy, using either a quark-model approach or the concept of parity admixture of vector mesons, are reviewed. About 10 papers which are concerned with the determination of the sign are discussed here. It is shown that they predict a sign for the asymmetry which is likely to be incorrect and, in any case, in contradiction with the standard theoretical understanding of the sign actually determined by experiment. The relationship of these approaches with that based on the old factorization approximation is emphasized. In this order, many details are given, and obviously special attention is given to the sign prediction (conventions, definitions, phases, etc.).

1. Introduction The signs of quantities

in physics

are crucial.

A particular

theoretical

model

that

reproduces the sign of an observed effect gets support from such an agreement; it may be an appropriate candidate for an explanation. If it does not, then it is sure that something is wrong or missing. Thus, the old factorization approach [ 1,2] which was used in earlier times to calculate the parity-non-conserving (PNC) coupling constants of vector mesons ( p essentially) to nucleons is known to be incorrect, or, at least, insufficient. It is known to give the wrong sign [3] for the longitudinal PNC asymmetry observed in pp scattering at the energies of about 15 and 45 MeV (compare theoretical results of Ref. [3] with experimental ones of Ref. [4]). A similar observation was made previously for PNC effects observed in radiative transitions in medium-heavy nuclei (4’K, “‘Lu and 181Ta), but at that time, it 0375.9474/95/$09.50 ‘.0-r

,,_l_c

0 1995 Elsevier Science B.V. All rights reserved

nr~r,nr\nnn~~

-

B. Desplanques / Nuclear Physics A586 (1995) 607-648

608

was thought that a sizeable

T-exchange

contribution

could provide the appropriate

sign

El. Since the DDH work [6], it is definitively vector-meson-nucleon literature

coupling

[7]). They

approximation

constants

have a sign opposite

and, therefore,

can account

known that there are other contributions (some

were

in fact discussed

to what is calculated for the observed

to

in earlier

in the factorization

sign in pp scattering

at 15

and 45 MeV. They also account for the observed signs in the radiative transitions mentioned above and in pa scattering at 45 MeV [8]. In all these cases, it is now believed that the r-exchange contribution can explain at most half of the observed effect and probably less [9,10]. Indeed, the absence of an effect in the radiative transition O-(1.08 MeV) + l+(g.s.) in 18F [ll], together with the present theoretical analysis, strongly limits the size of this contribution which has an isovector character. In the last decade, there were claims that significantly different approaches could also explain observations of PNC effects in pp scattering at low energy [12-151. They are essentially ignoring

dealing

with a direct description

of the effect in terms of quarks [12-141,

the exchange of vector mesons whose relevance

at short distances (I < 0.5-0.8

fm) might be questionable according to the authors. In an other case [15], the authors are dealing with the concept of parity admixture of vector and axial-vector mesons, which, at first sight, is quite legitimate.

The fact that these approaches

apparently

reproduce

the

sign of the parity-non-conserving asymmetry in pp scattering at low energy does not allow one to discard them. A careful examination, however, shows that they are reminiscent to some extent of the old factorization approximation. In this approach, the spatial structure of the PNC NN force resulted from introducing vertex form factors for vector and axial-vector nucleon currents [1,2], the nucleons interacting via the exchange of the weak-interaction vector bosons, W and Z. Removing from the form factors that part due to the coupling

to vector mesons (by taking the limit

rni and rnz+ a~>allows one to deal directly with quarks. Assuming instead the idea of vector-meson dominance of currents allows one to recover the standard meson-exchange picture

of the PNC NN interaction.

In the particular

case of vector-meson-dominated

form factors for both vector and axial-vector currents, one has to infer a PNC mixing of p and a,, or w and f,, for instance. With regard to this point, one is reminded that present estimates

of meson-nucleon

coupling

constants

contain

some contribution

due

to the parity admixture of the nucleon with baryon resonances. Furthermore, the concept of parity admixture is delicate to deal with. For example, it was shown that the nucleon-deuteron PNC scattering amplitude, calculated from the matrix element of the PNC NN force together with scattering states that are eigenstates of the strong interaction, did contain the contribution due to a parity-non-conserving component within the deuteron [16]. In other approaches, this contribution should be added explicitly [17]. Determining the signs of PNC NN forces and subsequently of PNC effects is a task which requires some caution [18]. Indeed, intermediate steps involve many signs that are convention dependent but, hopefully, cancel in making predictions for some observable. This sign uncertainty mainly concerns the PNC meson-nucleon coupling constants.

B. Desplanques / Nuclear Physics A586 (I 995) 607-648

609

These depend on the intrinsic phase used to describe mesons but, as the strong-interaction meson-nucleon couplings also depend on this phase and as it is the product of these two couplings which matters, the corresponding contribution to the NN force is independent of it. The sign of the couplings may also depend on whether one refers to lagrangian or hamiltonian densities, on the definition of the y,-matrices, on the definition of the y,-matrix, which usually carries the change of parity, on the order of y, and ys in the definition of axial currents and finally on the metric. The consistency of the inputs therefore has to be checked carefully. This includes the Dirac equation describing free spin- i particles (quarks or nucleon& In this paper, we reconsider results obtained in a few recent papers [12-151, and show that, according to the standard derivation of PNC forces, they predict an incorrect sign for the PNC asymmetry measured in pp scattering at low energy (1.5 and 45 MeV). We provide ample details (after all, something may have been overlooked in the standard derivation) and emphasize all the points that may lead to mistakes. We also complete some of these recent studies to make more transparent their relationship with the results of the factorization approximation. Section 2 is devoted to the weak interaction between quarks and to a precise definition of the ingredients it contains. Section 3 is concerned with the calculation of the PNC NN force, using a quark description of nucleons. A parenthesis will be made about a PNC quark-quark interaction mediated by vector mesons. In Section 4, we consider the contribution due to the parity mixture of vector mesons, p and a,, w and f,. 2. The weak interaction of u- and d-quarks As the models of nucleons or mesons in quark approaches of the PNC NN interaction under scrutiny here refer to constituent quarks, it is sufficient to limit oneself to the sector of the weak interaction involving u- and d-quarks. On the other hand, as we look at low-energy processes, it is legitimate to neglect any momentum transfer of the interaction, so that this one will effectively appear as a zero-range force in r-space. Quite generally this approximation, together with strong-interaction effects due to gluon exchange, masks the very origin of W- and Z-boson-exchange contributions. It is only in absence of these strong-interaction effects that some explicit track of these exchanges may be easily recovered. As this zeroth-order interaction provides the dominant contribution to some PNC meson-nucleon coupling constants (especially the isoscalar and isotensor couplings involving the pmeson) and as relying on W and Z exchanges allows one to distinguish different types of contributions (and diagrams), we give here the corresponding hamiltonian:

(1)

610

B. Desplanques/ Nuclear Physics A586 (1995) 607-648

where the metric tensor g ILyis given by go0 = + 1, g” = - 1, gpLy = 0 for p # V. With this convention,

the Fermi coupling

constant,

G,, is defined by

G, 1 t? _=_IE where M,

(2)

sh4; is the mass of the W-boson

to the electromagnetic

coupling

and g its coupling

to quarks. This one is related

by the relation

e2 = g2 sin%&. where 0, agreement

(3)

is the Weinberg angle. From its definition, G, is a positive number, in with the fact that the interaction between like particles arising from the

exchange of vector mesons is repulsive (remember the Coulomb interaction between electrons, or protons). Concerning the hamiltonian (l), let us add that 13, represents the Cabibbo angle and that the currents (IiG,d, . . . ) are supposed to contain a summation over colors. As the difference

between

hamiltonian-

and lagrangian-interaction

densities

is not

always made (this is unfortunate, but true), we remind that they usually differ by a sign (in the absence of derivative couplings): pint(x)

= -xint(

X).

(4)

This difference is essential here, as we are precisely looking at the sign of a PNC effect, which is directly proportional to the matrix element of 5?“‘(x) or Z’“‘(x), depending on the approach. For completeness, we also introduce the effective interaction between QCD strong-interaction effects are turned on. In the SU(4) description sector (u, d, s, cl, these effects are roughly measured

where

gf( p2) is the color-gluon

interaction,

coupling

by the factor K:

at the subtraction

limited to the PNC 2-quark sector (excluding

point

H&(K)

_~0.48

+

4 ~0.48

X

~-0.24

+

1 -

+

~~-0.24

3

2 sin28, 2

1 - 2 sin20, +

2

CL. The resulting

3- and more-quark

given by

X

quarks where of the quark

( uyfiy,AAu - +MAd)

sectors), is

B. Desplanques/

Nuclear Physics A586 (1995) 607-648

_.0.48

611

+K-0.24

X(uyvAAU-&hAd)

4

+6,,(~i~~y,~“u_dy,y,h~d)(iiy,A~u+L?y,hAd) where

the

AA represent

the W(3)

color

matrices

, I normalized

(ha) with

the condition

Tr(AAAB) = 26AB, while a summation over A is implicit. The expression of the ri, and sij coefficients as functions of K is related to that of the coefficients ylj and Sij given by DDH [6] by the relation

rij = $in(28,)yij

and sij = $sin(28C)6,j.

It is given here

for completeness: rll

= (3 - 2 sin20,)(0.056

K”.48 - 0.051 Ko.35 - 0.067 K-o.24

+ 0.062 K-o.4o), 712 =

-

fsin’&,(

-0.049

ZC”.” + 0.190 K”.43 - 0.426 K-o.‘3 + 0.274 K-o.35),

7% = - $sin2B,(0.086

K”.85 + 0.146 K”.43 + 0.623 K-0.‘3 + 0.151 K-035),

s,, = (3 - 2 sin2$,)(

-0.042

K”.48 + 0.028 K”.35 - 0.025 K-0.24

+ 0.039 K0.40)) S,, = - +sin”e,(

-0.113

K”.85 - 0.099 K”.43 + 0.129 K-‘.13 + 0.079 K-0.35), K”.85 - 0.126 K”.43 - 0.084 K-‘.13 + 0.148 K-0.35).

s,, = -fsin2&,(0.063

(w Except perhaps for global phases, the currents entering in (1) and (6) are independent of the choice of the y,- and y,-matrices, which is achieved by using the corresponding solutions

of the Dirac equation.

Although

here in order to remove any ambiguity Yo=(;

;),

The positive-energy

Y=(-“a

there are a few standard choices, we give ours which might result in inconsistencies:

;).

spinors (p. = \lmz+),

tYoPo-YP-+(P)

(7) solutions

of the free Dirac equation:

=O,

(8)

are given by

’

1

(9)

0.P

where

1 K~)

E, + m \ is a 2-component spinor describing

the nucleon

spin.

612

B. Desplanques / Nuclear Physics A586 (1995) 607-648

The definition

of the y,-matrix

requires more caution,

as the sign of PNC effects is

related to its sign. First, it is important to notice its place in writing the current, -ypys, which is by convention. As y,- and y,-matrices anticommute, using the other convention (employed

occasionally,

see Ref. [20]) would suppose a redefinition

of the y,-matrix

or

the relative sign of vector and axial-vector currents. For our convention, which corresponds to writing the left-handed components of the +-field as I,$ = i(l + ys)$, the -y,-matrix reads

%=(_F1,‘). The phase is determined by the fact that weak currents, as they are defined, must reproduce various experimental data, what has been formalized in assigning the quarks u and d and leptons

ve and e entering

charged

weak currents

to left-handed

SU(2),

doublets. In a practical case, the above assignment of signs can be deduced from the leptonic decay of polarized neutrons, ii + pev,, by looking at the direction emission of e and V, with respect to the neutron polarization (see the review in Ref. [21] where there is, however, a sign inconsistency in writing the relative ratios of vector and axial-vector leptonic and hadronic currents). With our definition of the y,-matrix, the matrix element for the above process is proportional A =M?&&(l+

Y&oV)

to i-&)V(l

+gLY+“(P”),

where gi = + 1.26. The upper index, 1, has been introduced relative

to the isovector

axial current from the isoscalar

later on. Present conventions

for the y,- and y,-matrices

to distinguish

(11) the coupling

one, which will be referred to (Eqs. (7) and (10)) are identical

to those used by Donoghue, Holstein and Golowich [29]. They differ from those given by BjGrken and Drell for the sign of the y,-matrix [30]. A few typical matrix elements u( P, s’)Y@( P, S) = (K,t i(p,

+‘u(p,

S)

=

may be given as references: 1 KS),

;hsc

1 KS),

P

E(ZJ,

+‘oT’+(P,

S)

=

-($‘I

+s), P

E(

p,

S’)yY+(

p,

S)

=

-

( K,J

1 u

I KS>.

(12)

Describing nucleons by the positive-energy solutions of the Dirac equation is also a convention. Using negative-energy solutions instead would have supposed a redefinition of the sign of the y,-matrix, or of the relative sign of the vector and axial-vector currents, in order to be always consistent with experiments. We can now make a non-relativistic expansion of the PNC part of the interaction given by (6). We limit ourselves here to the part which is symmetric under the exchange of vector and axial currents. On the one hand, this is sufficient for our purpose as this part contains the dominant contribution to pp scattering. On the other hand, this allows us to skip a contribution involving the pion exchange which does not contribute to pp

613

B. Desplanques / Nuclear Physics A586 (I 995) 607448

scattering at the lowest order, and furthermore will not make the present paper longer without any real necessity. Thus, the PNC potential interaction between quarks, which is relevant for the NN system, reads in momentum space for K = 1 and for the part of H&j, symmetrical in vector and axial-vector currents: V‘91(Pf, Pi, Pi, Pi,; VA + AV) 2 cos20c + 1 - 2 sin*8,

= -;

6

(

COSMIC - (1 - 2 sin*8,) 71 * T* +

6

T1’+ 7; X (trl.r2 - 37; 7;) - fsin;?e,---

x

2

i

S(Pf

+P: -PI

P:-P:--Pt+P;

Pf-P:.+PI-P: +i(o,xa,).

h-a*)*

2m,

2m,

i

-Pi)

I

’

(13)

where pi, pi on the one hand, and pf, pi on the other hand refer to momenta of particles described by wave functions sitting respectively on the right and on the left of the operator when calculating some matrix element. As in cl), a summation over colors is implicit. Our notation corresponds to a common convention, but nothing prevents the writing of the initial and final states respectively on the left and on the right of the operator, the hermiticity of V ensuring that the final results for observables be unchanged. The above precision about our notation is especially relevant for the term involving the spin operator, ia, X u2, whose properties under time-reversal are opposite to those of spin operators, u1 - uz, or u1 + u2 (this one, absent in (131, would be associated with isospin operators, (TV- T*)* or (pi X r21z). In the r-space description of the potential, one does not have to worry about the above precision. Any ambiguity is removed by the position of the operator p with respect to the S(r) function representing the propagation of an infinitely massive particle (to be replaced by a Yukawa function m* epm’/4nr for a finite mass). The factors containing spin operators and momenta in (13) are thus replaced as follows: P: -Pi

PC-PC-Pt+P;

+Pf -Pi, +i(u,

(v--~*)*

Xu,).

25

+i( u1 Xu,)

. [!!L$,

25

!

a( ‘:‘:‘;‘)~(r~2-r;,

(14)

614

B. Desplanques / Nuclear Physics A586 (1995) 607-648

Fig. 1. Representation definitions. The bubble gluons (g).

of the effective weak quark-quark contains W- and Z-meson exchange

interaction together with some accompanied by an undetermined

kinematical number of

where r12 = rl - r2. Throughout this paper, the difference of momenta (or coordinates) of particles 1 and 2 entering the potential will always appear in the combination p1 - p2 (rl - r2), which represents the most frequent choice. The opposite convention is used in some papers [1,22] with the potential however,

the relative

momentum

accordingly

(coordinate)

corrected

is denoted

known to which convention is referred [15]. Compared to usual potential models, expression

in other parts. Sometimes, in such a way that it is not

(14) contains

two extra functions,

which are usually integrated over. One of them, 6(i(r, + r21f - i
invariance

generalizes

to the case of a non-local

corresponding representation

interaction,

V(&,

ri,>. The expression

of IQ9

to Eq. (13) for any K may be found in the appendix, while a schematic of the interaction together with some kinematics is shown in Fig. 1.

For the quark-antiquark interaction, which is relevant for the parity admixture of p and at-mesons, or o and f, (or even p with f, or o with a, for the isovector part of HP,,),

one gets, using a form similar to that used in Ref. [15], V,%(P:,

p:,

pi, P;; VA+AV)

=3[n+(l)b+(Z)]g

X

2

c0s2ec

--

6 c0s2e, -

+

+

3

2

- 2 (1 7;1,

- &in’@,

3 111, + 71. T2

71 . 72

1 - 2 sin28,

+

sin20,)

6

(37; 7; -

71

.72)(i

+

1)

- 1,722 2

(-:+1

P: -P:.+Pi X(a,+a,). 25

)] -Pi

S(pf

+P:-P;

-p;)tb(+(l)l,

(15)

B. Desplanques / Nuclear Physics A586 (I 995) 607-648

where the indices anticommutation parentheses

1 and 2 respectively of the fields

refer to the quark and the antiquark.

introducing

the order of the creation

a sign (--)

and destruction

in (151, we indicate

expression,

V,4?i(p:,

P:,

With the between

operators which (15) corresponds

to. Furthermore these appear as uncolored quark-antiquark a factor (6)” = 3 has been factorized out. An alternative

615

pairs normalized

closer to the original hamiltonian,

to 1, so that

might be

pi, P;; VA+AV)

=3[o+(l)b+(Z)]r$ 2 c0s2ec + 1 - 2 sin*O, X

6 c0?e, -

(1

+

- 7f. 7i

3 l,li 7r’ Ti

+

6

- 2

sin*O,) (Tr.Ti-37;7F)(1+f)

6

T;li - $sin*t$,

+ lr7; ( 1+; -)I

2

Pf -Pi

X(-i)(UfXUi)'

+Pt -Pi,

6(Pf

25

+Pi-PI

-P~)[b(2)a(1)li

(16) where, this time, indices i and f refer to the initial and final mesons. Rearrangements quarks

appearing

accounted

in the two currents

for, but the corresponding

comprised

contribution

in the interaction

(6) have

of been

has been singled out in (16). Spin and

isospin operators or and or (TV and ai) have to be appropriately saturated with the spin-isospin wave functions of the mesons. The relationship between (16) and (15) may be checked by making in (16) replacements 7f. Ti + i(T, i(3

- T*) . (1 -l+(T,

- 7r. Ti) -+ i[3(1

$(T;+

7;)

+=;;;;:;X2)Z(l = ;(T;

-ia,

x cri -+ -i$(

In our opinion,

-P’)

expression

such as - T*) = $(3 + 71. Tz),

- i(7i

-P’)

- 7*).

+ $(l

(1 -P’)&

-PT)(r,

- 72)]

- 7 2 )‘I

- T;), CT, - cr2) x (1 -P”)+(

(T, - fQ> = U1 + G2.

(16) makes the check of the overall

sign of the PNC NN

interaction resulting from meson parity mixing easier (and safer) than expression (15) does. Indeed, the calculation of the PNC NN force then involves factors such as I(Mlrrof lo>l* or I(“17ioi lO>l*, whose sign is always positive. This feature avoids the worry about the intrinsic sign in defining the meson state IM). Extensions of Eqs. (15) and (16) to any value of K may be found in the appendix.

B. Desplanques / Nuclear Physics AS86 (I 995) 607-648

616

?-J-J---T

qh!z&q

p,“w

PJW

(a)

(b)

Fig. 2. Different types of contributions showing how a quark and antiquark produced by the weak interaction combine to make a meson. (a) corresponds to the earlier factorization approximation while (b) supposes some rearrangement (Fierz transformation).

As it is interesting to compare a quark-based approach of the PNC NN interaction to the standard one, we give the expression of the usual PNC NN force where PNC p- and W-NN couplings are calculated in the factorization approximation while only the part of the quark interaction (6) symmetrical in A and V currents is retained. In view of the discussion to be made in the next 2 sections, we separate two contributions as we did in (16). The first one corresponds to the earlier factorization approximation [l] where currents were identified to vector mesons [2] (the sign of the original VP has been corrected in papers published later on). The second one arises in the modified factorization approximation [22,23]. In this contribution, where the quark substructure is accounted for, quarks belonging to different currents may be combined (rearranged) together to make a meson. The two types of contributions are pictured in Fig. 2. The PNC NN potential calculated in the factorization approximation thus reads I$!,(

r.12; VA+AV;

= --

GF

2 cos*tI, + 1 - 2 sin*O, $1 * 72

fi +

p+o)

(I-

6

+)

7;1, + 117; 2

+ $( T1.T2 - 371’. T;) cos*O,- (1 - 2 sin*B,

) ( I + -)] 1

6

2 cos2ec + 1 - 2 sin*O,

3

B. Desplanques / Nuclear Physics A586 (1995) 607448

. P;;P2

) fw(r)])

+ (5.3

_ 1) Tf12;

l17; (-

617

$+(I

+ f)

[ X(a,+a,)* where fp(r> = f,(r)

w r f( 4)

(17)

*

mp"eCmpr/47rr, mp = m,.

=

A more general some similarity

t

PI -P2 7’

expression

between

for K # 1 may be found in the appendix.

the coefficients

From now on,

relative to the isospin factors appearing

in (17)

and in (16) may be noticed. In view of preparing the discussion to come, the gi(gi’ factor has been replaced by its non-relativistic value 2 (1). We also separated the quark-model ratio, g,,,/g,,, = 3. The quantities x, and x, represent the tensor couplings of the p and w-mesons to nucleons. The above expression does not account for form factors at the weak and strong meson-nucleon couplings. Although this effect is not really relevant uncertainties

at present for a comparison

(coupling

may be useful

constants,

here. Anticipating

short-range

to the experiment

correlations,

the use of a substructure

due to many other

higher-order

corrections),

of nucleons

non-relativistic constituent quarks moving in a harmonic-oscillator potential, gives rise to an intrinsic form factor, e-q2b7/6, one obtains a PNC potential, expression m2

is similar to (17), but with a radial form given by the following

it

in terms of which whose

replacement:

e-mr

($*r

-+l(r,

=

I

ml =/

dr’

X_/ dr’-3l-[exp(

i$)

-exp(

-is)]

Xm2b2 eCmr’exp In the limit where the meson mass goes to infinity,

l( r,

(-J--E)’

m>,+m =

exp( -i$).

(18) the above expression

becomes

(19)

This expression has the same functional form as the one given by Grach and Shmatikov [14] for the quark-quark contribution. This is also true for the full expression l(r, m) (last line of Eq. (1811, which has to be compared to their meson-mediated quark-quark interaction (direct-term contribution: see appendix of Ref. [24]). The above observation indicates that the two pictures may have some relationship. Due to differences in signs,

B. Desplanques/Nuclear

618

in factors or in the full expression not, however, make any statement arguments to have it disproved. Instead of gaussian

Physics A586 (1995) 607-648

of the PNC NN force, Grach and Shmatikov about this relationship;

form factors, one may introduce

on the contrary,

could

they had

for the axial current some form

factor supposed to be given by the dominance of the axial-vector current by the a,-meson 111, while for the vector current one may consider that the functional form, mi,,

n-r,

e-mpJ/4

be obtained

already accounts

by the following

e --mpr

eernpr

rnimf,

m2’ 4Tr

+

mf, -m;

This is the expression close relationship

for it. The appropriate

functional

form would then

modification:

i

z

e-m,,r -

- 4Tr

(20)

i .

given in Ref. [15], which again shows that there may be some

between

the standard

description

one based on PNC meson mixing. The difference

of the PNC NN interaction

and the

in sign found by the authors prevented

them from drawing such a conclusion. Again, it could support that the mechanism of parity admixture of mesons was not accounted for in the standard derivation of PNC NN forces. In absence

of short-range

correlations,

the strength of PNC effects at low energy is

determined by the volume integral of the potential given in (18)-(20). In all cases, this integral is equal to 1. The different radial expressions of the potentials given by Eqs. (18)~(20)

are shown in Fig. 3, after multiplication

appropriate Yukawa

r-space representation

is also given. The value associated

r2

by the space factor

of their contribution.

to get an

The result for the usual single

with the gaussian

nucleon

form factor is

f(rV (fm4) t

0.0

1.0

20

rifm)

Fig. 3. Radial representation of different types of potentials; continuous line for the pure p exchange, dot-long-dashed line for the p-a, exchanges (Eq. (2011, dotted line for the gaussian potential (Eq. (19)) and the short-dashed line for the p exchange together with gaussian form factors (Eq. (18)).

B. Desplanques /Nuclear

taken to be b = 0.5 fm while

Physics A586 (1995) 607-648

619

mp = 770 MeV and maI = 1270 MeV. There are large

differences at quite short distances or at large distances, where the contribution is small in any case. There are also differences around the maximum, in the range 0.2-1.2 fm, which

may show up in very accurate

measurements.

At present,

observables

at low

energy are only sensitive to an average over this region and are much less sensitive to the detailed radial description of the potential than to the PNC meson-nucleon coupling constants

themselves.

3. PNC NN force directly derived from a quark description of the nucleon Here we consider

the PNC force between

nucleons

resulting

from the quark force

given by (13). For our purpose, we describe the nucleon wave function with momentum k by assuming that the quarks are moving in a confining harmonic-oscillator potential. It reads IclkN(% =

r27

r3)

I xC(l

3)) I xU’(l

2

2 3)) + r2 +

rl

X

/

dr,

Xexp

exp(iK.r,)6

(r2-rA)2

2b2

-

The factor 1 ~‘(1 2 3)) 1 ~~‘(1 function. Only its well-known explicit

expression

- ‘A)( b617:3’,:)1’2

3

(r1-rA)2

-

r3

2b2

(r3-rA)2

-

2b2

(214

’

2 3)) represents the color spin-isospin part of the wave symmetry properties are needed in the following; its

is not required.

As to the normalization

of the wave function,

it is

given by /

dr,

dr,

=I

dr,

@(r,,

rZr 4*

+:(rr,

d( y)&exp[

x

d /i

-$(

2r, -r, G

= (271-)~6( ki -k,).

y)‘]

-r2 )&e.p[

rl + r2 f r3 3

r2, r3)

i(ki-k,).

-$(

2r3;

-y*]

rl + r2 + r3 3 (21b)

The parameter b, already mentioned, is often given values from 0.5 fm to 0.8 fm. The first one would be fixed by the spectrum of the nucleon excitations (lfiw, = 500 MeV, see Ref. [25] where the definition of b2 is twice the one used here). The second one

620

B. Desplanques /Nuclear

Physics A586 (1995) 607-648

would correspond to the mean-square charge radius of the proton (( rzh)p = b* = 0.74 fm*). In our opinion, the first value or even a smaller one is preferable, a large part of the charge radius being provided by the coupling of the photon to p and o-mesons, these giving the contribution

(22) Another part may be due to the contribution of the pion cloud not included in the zero-width p exchange (low-energy part of the spectral function and p width). The neutron mean-square charge radius indicates that this part may be as large as ( r,2h)2p,= 0.10 fm*. Finally, some contribution may be due to the Darwin-Foldy terms at the quark level, ( rz, ) nW z 3/2E,(m, + Es) (= 0.11 fm* for m, = 300 MeV, E, =\i* * = 600 MeV). It is not sure that the different contributions to (I,‘,) should add together, but it seems that not much room is left for a value of b larger than 0.5 fm. This value, which characterizes the quark core radius of the nucleon, is important here because it determines the range of the distances between nucleons where quark-antisymmetrization effects may show up, a small value tending to suppress them. The wave function of the NN system may be built by antisymmetrizing the usual wave function where nucleons are considered as 3-quark clusters: &js=d

Ix’(1

=))I

2 3)) I ~‘(4 5 6)) I ~~‘(4 5 6))

~~‘(1

i

rr + r2 + r3 + r, + r5 + r, 6

r1 + r* + r3 I(+

3

x&exp[-&(yr-&(

r4 + r5 + r, -

3

2r3%-r2)

2:‘(r4-J2_

&(

2re;-rs

*

,

(23)

Hi

where JZ’represents the antisymmetrization operator while K is the total momentum of the NN system. +(r) describes the relative motion of the 2 nucleons. A different expression, closer to the notations employed by Grach and Shmatikov and more appropriate to the derivation of an effective potential involving the relative coordinates of the 2 nucleons, r, and rB, is &,=d

1xc@ 23))l ~“(1 xexp

iK*

I ~‘(4 5 6))

I ~~(4

5 6))

r1 + r2 + r3 + r4 + rs + r6 6

( 1

X- 27b’2T6

2 3))

exp

I(

~[(r1-r*)*+(r2-rA)*+(r~--TA)*

11

d( r, - ru) +(r.., - ru)

B. Desplanques / Nuclear Physics A586 (1995) 407-648

X6

rt +

r2

+

r3

3

-rA

I( 6

621

r, + rs + r, .

3

(24)

In principle the wave function describing the relative motion of the 2 clusters A and B should be calculated by accounting for the quark substructure of nucleons and in particular for the interaction effects it produces. Only such a consistent calculation can avoid the problem of the non-orthogonality of different energy states which occurs when the full wave function (23) is employed together with NN wave functions calculated from usual potential models. Furthermore, when the nucleons are getting close to each other, they get polarized, leading, e.g., to NA(1230 MeV) or AA components into (23) beside the NN one. The resulting 6-quark component may involve those introduced in different works within the present context [12,13]. There also, a consistent calculation is required to determine them, ensuring in particular the necessary continuity with the 2-cluster (Znucleon) component that is likely to dominate at moderate distances (0.8 fm). For our purpose, which is to compare some quark description of the NN force within a quark model to a meson-mediated one, it will be largely sufficient to start from (231, with even a truncation in the antisymmetrization operator. As will be seen, this is already very instructive. Obviously, for a comparison with experiment, it would be desirable to perform the above consistent calculation, including, however, some qq content in the NN wave function in order to get the right sign for the PNC asymmetry measured in low-energy pp scattering. Inserting (23) into the calculation of the matrix element of the PNC quark-quark force given by (13) implies the consideration of many diagrams. Referring to notations of Ref. [14], we will consider in particular two of them, which are depicted in Fig. 4. They are the direct term (4a) and the exchange term (4b) involving the quarks 3 and 6 which are weakly interacting together. The other terms lead to non-local forces, often of shorter range [14]. They have no counterpart in standard calculations of the PNC NN force and, according to Ref. 1141,have altogether a rather minor contribution compared

(a) Fig. 4. Two quark diagrams, nucleons.

(a) direct and (b) exchange,

(bl contributing

to the weak interaction

between

622

B. Desplanques / Nuclear Physics A586 (1995) 607-648

to the direct one. From now on, some topological similarity of weak processes represented in Figs. 4a and b with those in Figs. 2a and b, respectively, may be noticed. The calculation of the matrix element of the quark-quark force can be performed in many different ways. Results should be independent useful check, especially

of the choice, thus providing

some

in dealing with the motion of each cluster. For the commutator

part? [Pa -& f(rs,)l ( arising from the second term of the last line of Eq. (13) for its momentum-space expression), it is known that it can be written as the derivative of the function

f(r):

[ P3 -P6,

@36)]

=

-2iv3f(r36)’

The contribution of this term can be calculated also by having the operator p3 -p6 act on the initial or final states, according to whether it appears on the left or on the right of the function

f(r).

This is possible

because

the difference

between

the two approaches

involves a surface term which cancels due to the fact that the interaction vanishes when the quarks are far apart. As to the operator, p3 (or p,), acting on the initial or final state, it can involve the motion of the quark 3 within the cluster A, but also the relative motion of the 2 clusters. This can be seen in Eq. (23) of the 6-quark wave function, in which case the derivative of the wave function describing the relative motion of the two nucleons, alternative r3)

-

rA)

especially

I,/J(+(~, + r2 + r3) - i(r,

+ r5 + r6))

appears.

This is not so obvious

in the

expression (24). In this case the derivatives of the functions S(i(rt + r2 + and S($(r, + r5 + r6) - rB) have to be considered. These derivative terms are important

for the anticommutator

part of the interaction,

{(p, -p6),

f(r)>

(arising from the first term of the last line of Eq. (13) for its momentum-space expression). They have been omitted by Grach and Shmatikov, perhaps due to the absence

of an explicit

consideration

of the

S-function

imposing

that the nucleon

coordinate is equal to one third of the sum of the quark coordinates. After having emphasized the points that deserve some caution, we can now enter into the details of the calculations under consideration

here as

where the exchange

operator,

Pij = P;P,yP;P;?. The different exchange isospin and space. The which separately involve calculation of the matrix effect of the interaction

of the matrix elements,

Pij, can be factorized

which can be written for the part

as (27)

operators entering in (27) respectively refer to color, spin, fact that the nucleon wave function (21) factorizes into parts color, spin and isospin and, finally, space greatly simplifies the element (26), which can thus be performed by considering the in the different spaces. Assuming that

B. Desplanques / Nuclear Physics AS86 (1995) 607-648

623

allows us to write ~xclo,E,lxc~=~xcIo~61xc~,

(29)

~~“~Io:ipI~sp~=~~s~lo~~l~s~~,

(30)

(,uT10~71 x”‘>=(

x”‘Io;;I

y>=$(,wTI

c iEA,jE

071 x”‘)

(31)

B

and (I

C

V,jI>=(XcIo~~l

Xc>(XuTI

C

OrI Xv’>

isA,jEB

iGA,jeB

xww;pgIdm.

(32)

A similar result holds for the part of the matrix element (26) involving operator P36, which keeps the structure given by (28). The various matrix elements

appearing

in (32) are reviewed.

the exchange

For the color part, one

has

I xc> = 1, (xCl~A~h~I x”>=O, ( XC I1

(x’I~~~Ixc~=f> ( xc I cA;AgP-&

I xc> = y.

(33)

C

To get some of the above results, we used the relation

h?AC p;=++

CLL. c

For the spin-isospin between

(34)

2 degrees

of freedom,

one has the following

operators acting at the quark and nucleon C

(“i_Oj)

correspondence

levels:

+3(mA-uB)?

iEA,jEB C iEA,jEB

c

(ai-uj)(Ti.7j)

( ui -

+$(“*-u~)(T,&‘T~)Y

U,)(Tiz + T,“) --) 5(

UA7A”-

u,7;>

+ ( aAT; - U,TJ)

(35)

i~A,jtB

C

iu,Xu,-+iu,Xu,,

iEA,jGB C

iu,X~~(7~.7~)-,~iu~Xu~(~~.~~),

iEA,jEB C

iuiXuj(7i.~j-37i2~~)~~iu~Xu8(7A.78-37AZ~~),

iEA,jEB C iSA,jEB

iu, X u,(7f

+ T,‘) + $iu,

X a,(

7: + T;),

(36)

B. Desplanques / Nuclear Physics A586 (1995) 607-648

624

( ui - uj)PiyP;

$a* x a,(1 + $T*. Q),

+ -

iEA,jEB c

-~iuAxuB(3-~7A.TB),

(ai-uj)(Ti.Tj)PiyJipa

isA,jEB ( ui

c

-

Uj)

( Ti

. Tj - 37&y P,yP;

icA,jEB +

-~~UAXU,(~A.~B-~~A~~),

c

(ai-uj)(7:+7jz)P,~P~~

-$,,xfl,(~,2+T;),

(37)

iEA,jEB

c

iu, X ujPiyP;

C

iuiXuj(~i~~j)Pi~P~+

+ - :( 0, - a,)(3

+ 5~~. rB),

iEA,jEB

-$(u~-u~)(~-$T~.T~),

icA,jeB

i ui X aj( 7i * 7j - 3r~~~‘)P~7P~

C ieA,jaB +

-$(a,-

uB)(~A~TB-3T~T;),

iu, X

C

uj(7: + T~‘)P,~P~

iEA,jEB -+

-

[5(

U*TA

-

U,TgJ

+

( OATi

-

uBT,)]

*

In order to get the above results, we used the following Pi~=~(l+ui~uj),

(38) relations:

P;=+(1+Ti*7j),

(ai-uj)Piy=

-iuiXuj,

(39a)

i(uiXuj)Piy=

-(ai-uj),

(39b)

( Ti . Tj - 37; Tjr ) P; = ( Ti . Tj - 37; 7;) )

(39c)

(T: + T;)P;

(394

= 7: + 7;.

While the above relations

are easy to establish,

it turns out that the relation

(39b) is

given with the opposite sign in the pioneering work of Michel [l], leading to a wrong sign for the single-particle PNC force he derived. In dealing with the spatial part of the matrix element,

4 types of quantities

involving

the anticommutator or commutator terms in (131, with or without the exchange operator P,“!‘,have to be considered. In this respect, it is useful to recall the following relation [l]: {(Pi-P,),

G(rij>}=

[(Pi-Pj),

s(‘ij>]p~p~

(40)

which assumes that wave functions have a regular behavior at the origin (constant for an s-wave, r for a p-wave). In such conditions, only the term in the commutator or anticommutator with the operator p acting on a p-wave is different from zero, hence the equality of their matrix elements up to a sign, which is compensated by the presence of the exchange operator Psp in (40). Note that the combined effect of this spatial exchange operator together with the spin-one on the spin-plus-space part in the quark potential fulfills what is expected from a Fierz transformation on a VA + AV combina-

B. Desplanques / Nuclear Physics A586 (1995) 607-648

625

tion of currents. In the following, we will not use the above relation, but it is clear that results should be consistent with it. Furthermore, expressions will be derived for the more general case where the potential between quarks has a finite extension, represented by a function f(r) instead of having a zero range, as given by a S(r) function. This may be useful for some discussion to be made in the following. Although it is somewhat long, we give the complete expression of a spatial matrix element for the interaction between quarks 3 and 6. This is done because the order of operators and their dependence on different variables matters. fdir.(exc.)=

/

dRf dR’ exp(-iKf.R’+iK’.R’)S(R’-R’)

Xdrf

dri

Xexp

+*(rf)+(ri)

--&(r,

d(

-r2)2-

r1+;+r3

$

--&ire+ r2 + r3 3

-

2

2

1

11

9

r, + r5 + r, 3

r4+;+r6)

+ r2

(

-&(r4-r5)2rl

r, -

r3 -

[

XS

_

’

-ri. i

(41)

The function S(Rf -R’) arises from the conservation of the center of mass at the quark level (function iS(i(r: + r:) - i
B. Desplanques / Nuclear Physics A586 (1995) 607-648

626

been integrated derivatives

out, it would

of the functions

introduce

I,LI* (rf)

the extra variables

two combinations:

or $(r’>

so clear that the final result

hermitic

(relative

terms in the potential

sign + > and commutator

and anti-hermitic

from the definition

contains

(see later on). This is an other reason to

rf and ri. The derivative

anticommutator

which are respectively which originates

not have been

(accounting

of the p operator:

(relative

appear in sign - >,

for the front factor

-i

p = - i d ). The space exchange

operator Pig, only relevant for exchange matrix elements, operates on what follows in the expression and exchanges coordinates r3 and r,. This concerns wave functions describing the motion of quarks within the nucleon as well as the &function defining ri in terms of coordinates rr, r2, r3, r4, r5 and r6. The variable ri, being an independent one, is not affected by itself in the present approach; this would not have been the case if an integration over the variable, i(r, + r2 + r3) - i(r, + r5 + r,>, had been previously performed. Many of the integrations

in (41) are straightforward.

one has to recall that, beside operating quarks within the nucleons,

they also operate on the S-functions

rf or ri, with the following potential

In dealing with the derivatives,

on the wave functions

result for the second

describing

the motion of

defining the coordinates

term in the bracket

containing

the

in line six of (41):

rl + r2 + r3 r, + rs + r6 3 3 -ri

(

(a,-d,)6 = $1

rl + r2 + r3 r4 + r5 + r6 3 3 -ri

(

i

1

rl + r2 + r3 r, + r5 + r6 3 3 -ri.

(

= -g,is

(42)

1

By an integration by parts, this last derivative may be transformed into a term involving the derivative of the wave function $(r’), describing the NN initial state. Similar results hold for the first term in the bracket containing the potential, for which, however, we prefer to have it operating on the left, allowing us to keep a symmetrical treatment

of initial and final states.

A more compact expression Idi’.= (~v)~S(

Kf - Ki)/

x(-i)[-$[f(y)

X

il

-$i+$*(r’)

for the direct matrix element is drf dr’ dy

exp[--$(y

$(ri)

+ i$*(r’)

*[-~$‘(rf)(y-~)~(ri)+@*(ri)-;41~(ri)]~.

-

yr\S(rf-ri) ( y-

f$f)*(ri)]

(43)

B. Desplanques / Nuclear Physics A586 (1995) 607-648

The relevance

of the derivative

the anticommutator

627

terms omitted in Ref. [14] can be seen as follows.

For

term (sign +>, they are the only terms that survive, the other terms

cancelling each other. For the commutator represent one third of the total contribution.

term (sign -), they also are relevant and This can be seen by an integration by parts

which allows us to write the corresponding

contribution

Izi’.= (2~7)~6(K~

-IL’)/

as

drf dr’ dy

x~yr’)(-+$f(Y)(~

+

3

[--$=)3 exp[--$(

y-

qj’]

6( rf - 2) +( ri).

X

(44)

The numbers 3 and $ in (44) respectively represent the contributions of the derivative and non-derivative terms in (43). On the other hand, the extra contribution due to the derivative terms allows us to get a result consistent with what is expected from a calculation implying an alternative expression of the commutator term, which involves the derivative of the function f(y), multiplied by a factor - 2i (see Eq. (25)). In (43), the integration performed,

taking advantage

doing it, this function does not generally

over one of the two variables of the presence of the function

being a particular

(

-$tYrf+*(rf)

+(ri)

-$8,f+*(rf)

but of

may be written as

drf dr’ dy(-i)f(y)6(y-

- y+*(rf)+(ri))

X

We refrain from

a local interaction

survive beyond the direct term. This is proven by an examination

1exc.=(2x)36(Kf-Ki)/

X

6(rf - r’).

case which characterizes

term I”““, whose expression

the exchange

rf and ri could be easily

:(r’-ri))

-+i+!~*(r’)t!l,,(c,(r~)

(for the anticommutator)

+(ri)

+flc,*(rf)d,,#(ri)

(

+2 y$*(r’)

y!r(ri))

(for the commutator).

(45)

The function exp[ - (3/4 b*>(rf - ri)‘] in (45) accounts for the part of the above-mentioned non-locality; though it has a short-range character, it cannot be reduced to the function 6(rf - r’>.

628

B. Desplanques / Nuclear Physics A586 (1995) 607-648

Removing from the matrix element the wave functions as well as the momentum-conservation factor allows us to get the expressions to be entered in an effective NN potential. They are v:r.=

Vcdjr.= (p; -#)Vdir.( yc.=

rf, ri)

J-[(PL _$)vdir.(

(pi

+ Vd”(

rf, 8) - vy

-pi)Vexc.(yf,

rf, 2)

r’)(pi

ri) +Vexc,(rf,

-;[(r;-f;)vyff,

- pi)]

)

rf, fi)( p; -pi),

ri) - Vexc.(yf,

Vcexc.= +[(p~-pf,)VexC~(rf,

(pi

-pi),

r’)(pA-pi)]

2) -PC.(rf,

ri)(r;-rg)],

(46)

where rf*i = (rA - r,)‘~‘, and

vdirQf, ri) = / dyffy)

1

Xexp[--$(y-T)

Vexc.(rf,

ri) = / dyf(y)S(y-

6(rf--r’),

Wa)

exp( --&(rf-ri)‘).

(47b)

i(rf-ri))

Xexp[--&(

y)‘]

Expressions (47) obey the volume normalization: d,.f

d,.i

vdir.(exc.)

/

(rf,

8)

=

/

dyf(y)

= 1.

(48)

They generalize the Yukawa (or gaussian) potentials appearing in the NN interaction: e m*~ij(r’ -r’)[or

[&)‘exp(

--$r2)S,rf-2)).

In the case where f(y) = 6(y), bir.(rf, ri) and Verc.(rf, ri) are identical to Eq. (19) (up to the factor 6(r’- ri). In the case where f(y) = m2 eemy/4ny, Vdirfrf, ri) is identical to (18) (up to a factor S(r’- ri)). The expressions of V:‘. and I(:“. given in (46) may be put into a form evidencing that they involve the derivative of the function f(y). More important, however, is the relationship between the different terms given by (46). In the limit where f(y) = 6(y), which is appropriate for the quark interaction given by (13) together with (14), one gets v:r.=

vy.=

[p&Is,

ydir.= a

yexc., c

+{pA

-PB?

f&f, fG(rfy

ri)], ri)}7

(49a) (49b)

B. Desplanques / Nuclear Physics A586 (I 995) 607648

629

where fo(yf,

#) = (--J$r

exp[ ---$-(

q,‘]S(rf--ri).

In the above expressions, it should be understood that the operator p refers to coordinate rf or ri, depending on if it operates on the final or initial state. The above relations have their origin in the relation (40) between commutator and anticommutator terms at the quark level. Notice that, due to the exchange operator P;i in (41), a term that has a commutator character at the quark level can get an anticommutator character at the nucleon level (and vice versa). As to the factor f present in Vadi’.and Vcexc.(Eq. (49b)), which concerns terms that have an anticommutator character at the nucleon level (always in the limit f(y) + 6(y)), it has its origin in the fact that the part of the quark momentum which matters in this case is given by its average contribution to the momentum of the nucleon to which it belongs, namely one third of it. In the case of terms which have a commutator character at the nucleon level (Eq. (49a)), what matters is the transfer of the quark momentum from the initial to the final state, which is obviously identical to the nucleon momentum transfer (the 2 other quarks are spectators). Gathering the results given in (331, (3%(381, (461, (491, it is now possible to get the potential between nucleons resulting directly from the quark-quark interaction (13). For K= 1, it reads V,“=“,((rf, r’; VA + AV; b)

= --

G, fi

2 cos2t$ + 1 - sin2f?, 5 TT*. 7n(l6

$)

2 cos2/3,+ 1 - sin%, +3

(0 + +)

6

$$-y

+i(u*xu,)[

*

9

fo(rf, ri)

II

+ cos2f3,- (1 - sin2BW) $( TA. T* - 3rA’7;)(1+ 6

$)

630

B. Desplanques / Nuclear Physics A586 (1995) 607-648

+5i( a* x a,)

e,

* [

fo(rf, ri)

.

4

T/f + 7; - +sin”t?, - 2 (1+f)

Ii

x13(~~-u.).(~,jg(lf,ri))

e,

+5i(u*xu,).

fG(rf, ri) 9

.

I

T; - 7;

- +sin’8, .2 2(1

+ +)( a,

II

+ un) .

e,

&(rf,

II

ri)

(50)

.

L

The result has a structure identical in every point to the usual NN potential derived in the factorization approximation (see (17)) provided that M = 3m,, x+,= 4, and x,, = 0, all assumptions nucleon

which

currents.

are currently

To them,

used

one should

in the non-relativistic add

gi = $ and

description

g,,,/g,,,

of the

= 3, already

anticipated in writing (17). Quite similar statements hold for the more general case where strong-interaction effects are turned on (K # 1, see appendix). We therefore conclude that the quark picture of the PNC NN interaction as considered in this paper and the meson-exchange one are essentially identical, including the signs. The only difference

concerns

the radial part: the Yukawa potential

is replaced by a gaussian

of shorter range (if b = 0.5 fm), which would not make any significant estimates

in absence of short-range

While the difference

correlations

concerning

difference

one, for

and at low energy.

the gaussian

potential

is not a real one (as already

mentioned, the introduction of gaussian form factors at meson-nucleon vertices would have produced it), the difference as to the Yukawa shape calls for some comments. This shape results from the quark-antiquark

correlations

in some meson channel.

in Fig. 5, there are two places where such correlations exchange character. The first one corresponds to the earlier factorization

(a) Fig. 5. Different ways to introduce meson-like

occur for processes approximation,

but at the quark

(b) correlations

As shown

that have an

between quarks.

B. Desplanques / Nuclear Physics A586 (1995) 607-648

level: the W- and Z-bosons

couple to p and w-particles,

631

which in turn couple to quarks.

The second one rather corresponds to the input of the modified factorization approximation. Physically, the two types of processes should be considered. They should not be simply

added, however.

diagrams relative

coincide, importance

expressions

It is clear that in the limit of infinite-mass

evidencing

a possible

of the above

of Vdirfrf,

double counting

correlations

mesons,

if the addition

may be appreciated

the two

is done. The

by comparing

r’) and Vexc.(rf, r’), given by (47) with j(y)

the

= m2 eemy/4ry.

By taking the limit of a small quark structure for the nucleon (b + 01, one can convince oneself that the first expression exchanged

leads to a force between

nucleons

meson, while the second one keeps a very short-range

with the range of the character (of the order

of b). Treating at the same time both types of correlation is probably manageable, but if some choice has to be made, it is preferable to retain those correlations giving rise to the longest-range force, the correlations of shorter range being undistinguishable from other short-range effects, mostly unknown. The above longest-range correlations (Fig. 5) may be accounted for by using PNC meson-quark coupling factorization approximation, quite similar to meson-nucleon retaining

only direct terms in the derivation

constants calculated coupling constants,

of the PNC NN interaction

in the and by

(Eqs. (43),(44).

The interaction so obtained is identical to (17), where the Yukawa potential would have been replaced according to (18). Adding without scrutiny the exchange term as in Ref. [24] would be inappropriate. In the above developments, we only considered quark antisymmetry effects for those quarks interacting weakly together (Fig. 4). There are other antisymmetry effects, whose contributions can be calculated by inserting the potentials V13, VI, or VI, in Eq. (41) of the matrix element between initial and final NN states. These further contributions may be part of a selected

contribution

involving

a 6-quark component

in the configuration

(s,,,)~ (or (s,,,)~(P,,,)~) retained in Refs. [12,13]. Following the work by Grach and Shmatikov [14,24], the present approach offers the advantage of establishing some continuity

between

one essentially inner

region

configuration. contribution reproduce

the contribution

has two nucleons, where

to PNC effects coming from the outer region where each of them being composed

the six quarks

This continuity of this component,

the experimental

mix together,

is of special

relevance

involving

of three quarks, and the in particular

in determining

a (s,,~)~

the sign of the

which in Refs. [12,13] has been introduced

one. Apart from the fact that this contribution

by hand to is probably

reduced in any case due to the small extension in space of the (s~,~)~ component, we checked that it has the same sign as the one given by the factorization approximation which underlies the present paper and, therefore, has a sign opposite to the experimental one.

4. About the contribution

of the meson parity mixture to the PNC NN interaction

We examine in this section the contribution of the parity admixture of mesons to the PNC NN interaction and its relationship to the PNC NN interaction derived in the

632

B. Desplanques / Nuclear Physics A586 (1995) 607448

factorization approximation. We first consider the effective hamiltonian description of such an admixture and its contribution to the NN interaction, for which two different approaches will be considered. The next development is devoted to the calculation of the meson admixture. The different approaches considered by Iqbal and Niskanen [15] will be examined. A close relationship between them will be established, allowing a check of their relevance. As in the previous sections, particular attention is given to the overall sign of this contribution, as well as to the dependence of intermediate results on the conventions. The effective hamiltonian describing the parity mixture of vector mesons may be written as

The coefficients have the dimension of mass squared factorized out, suggesting some possible relationship of the corresponding terms to usual mass terms in the hamiltonian. From now on, the Lorentz structure of the couplings in (51) should be noticed. Depending on if one looks at time or spatial components, a difference in sign appears and it is important to precisely define which component one refers to when speaking of the parity-mixture of spin-l mesons. Obviously, the most natural component to consider is the spatial one, since it is the only one to survive in the meson rest frame. It is also the component for which the calculation in a quark model offers less ambiguity from the viewpoint of the Lorentz covariance. As a result, and according to the metric we used, the signs of the H coefficients in (51) will be opposite to those determined by calculating matrix elements of the quark interaction, H%(K) (Eq. (611, between mesons at rest with a given spin. We now derive the PNC NN interaction induced by the coupling given by Eq. (51). There are two ways to do it. The first one amounts to computing the contribution of diagrams such as given in Fig. 6a. The second one supposes a redefinition of the spin-l fields, in such a way that the coupling given by (51) is eliminated. The PNC NN

Fig. 6. Representation of the parity admixture of p” and a: bare mesons to the PNC NN interaction and the corresponding diagrams in terms of eigenstates of the strong + weak interactions ( p and a,). The weak interaction is represented by a box while circles refer to the strong interaction.

B. Desplanques / Nuclear Physics A586 (1995) 607-648

interaction

then results from the exchange

6b and 6~). Whatever

the approach,

of mesons as in the standard approach (Figs.

we first need to define the hamiltonian

mesons and their coupling to nucleons. The hamiltonian describing free spin-l

where i refers to the different

633

mesons,

mesons may be written as

p, w, a, and f,, including

freedom. As to the effective hamiltonian

describing

describing

the coupling

the isospin degrees of of mesons to nucleons,

it may be written as H

+g,lNNNYuy5~a’7N+gflNNNY~Y~~~f’N.

and gwNN used here were denoted by g, and g, in DDH.

Note that the couplings g,,, They have to be distinguished

from the constants

which are two times larger (see below). between

the vacuum

and the spin-l

relating currents to vector-meson

The matrix elements

mf%’

g,,&

where, in terms of quarks, the currents VP = $4Y,V,

of the various

fields currents

mesons may be defined as

(olv,O I w(k)) =

(OIA,Ia,(k))=

(53)


-&

&L/G IL)

m’

(54)

g,,gG %’

VP and A, have the expression

VP0 = $Yw4,

A, = %‘,‘,y,Ym,A;

The mass factor in (54) can always meson being included

(55)

= f4y,y5q. be taken as the pmass,

in the factor g,

the dependence

on the

(M = p, w, a,, fr ). These factors determine

the

intrinsic phases of the mesons in terms of quarks, especially with respect to the arbitrariness in the choice of the sign of the y,-matrix. They can always be chosen to be positive, as we will do in the following. The dominance of the currents Vti and A,, respectively, the relations between couplings 1 _ ?=

gpNN

g,

’

llTgA-

appearing

by p and a,-mesons

g a,NN

--g,, 4, ’

where gi = + 1.26 ($ in a non-relativistic quark model). Weinberg with the KSRF relation [26] provide further relations [27]: gp=ga,,

m2a, =2m2 p’

gives

in (53) and (541, together with meson masses:

(56) sum rules together

(57)

B. Desplanques/Nuclear

634

Relations extension

Physics AS86 (1995) 607-648

(56) may be extended to zero-isospin mesons such as w and f,, whereas an of relations (57) is more doubtful because chiral symmetry in the correspond-

ing sector is not so good (masses

of the lightest zero-isospin

much larger than the pion mass). Relations

(56) are essential

pseudoscalar

mesons

are

in what follows, whereas

the relations .(57) may be mainly useful in checking the validity of a quark-model estimate of the parity mixture of p and a,-mesons. In this respect, we may add that using a simple quark model allows one to get relations

such as rni = rnz, g, = gp. The contribution of Fig. 6a to the PNC NN interaction can be calculated by relying on the standard Feynman rules or, more simply, but only for the spatial components of the currents,

by using second-order

diagrams. The expression thus reads

perturbation

of the potential

theory and considering

in momentum

all time-ordered

space, unsaturated

by spinors,

VP\: (meson-mixing) =2m~CgPvS(pf+p~-p~-pi,) P>”

goNN

gf,NN *,

+(

goNN

Y,

Y,

-

gf,NN Y,

+(

mi

+1++2,

+

~‘T,d’)lh~r)z

i

d+4*)(4,+4*)gald g,NN

g$pv~‘)l(sY5~*

&,NN *:G

+t

-

i

mZ,+q2)(mtI+q2)gf,d

4*)(4,

+

-

$+d’)l(wr)2

4’)gfld

1

(58)

where aW,,= (1/2i)(y,y, - ‘y&1, 4; = (pf -p’>l”. An alternative way to calculate the PNC NN interaction resulting from the parity mixture of mesons is to first eliminate the corresponding interaction (51) by performing a unitary transformation on the meson fields. Keeping first-order terms in the weak interaction,

this transformation

is given in terms of primed fields by

B. Desplanques / Nuclear Physics A586 (1995) 607-648

where upper indices,

i, refer to the isospin components,

The strong meson-nucleon

interaction

635

x, y or z.

(53) now has to be reexpressed

new fields defined by (59). This generates parity-non-conserving plings that are described by the following hamiltonian:

in terms of the

meson-nucleus

cou-

HMNN PNC

(60) Notice that HEcN contains PNC pNN and wNN couplings, which are usually considered, but also PNC a,NN and f,NN couplings. In the notation of Ref. [6], the first ones correspond

h:, =

to

2m;

(m;, _

gf,NN

2) mp

g,,d

B. Desplanques / Nuclear Physics A586 (1995) 607448

636

4,

2m;&,NN

hZ=

mt-mtHO’

(mf,-m:)gf,gi*‘O=

hZ,=

2m,2ga,tai

(61)

H; =

(m:, - mZ)g,,gi

The last equality on each line results from using the relation (56) and a similar one for the f,-meson. The PNC NN interaction corresponding to the hamiltonian (60) can be obtained according to the standard Feynman rules. Two contributions appear, involving the exchange of vector mesons p and o on the one hand, and of axial-vector mesons a, and f, on the other hand. As they differ by the meson propagator and the sign, they can be gathered two by two. The resulting interaction, written explicitly for only one type of term, thus reads VPic (meson-mixing) =2m~CgPvG(pf+p,f-p: cL,v

-pi)

(62) Using the relation 1 mtl - m3 ( w

1

1

1

- mf, + q2 i = (rni + q2)( mt, + q2) ’

(63)

it is easy to check that the two ways to incorporate the effect of meson parity mixture lead to PNC NN interactions, (58) and (62), that are identical. This result is not surprising since, as is well known, there is no physics in a unitary transformation. In the one case, one is dealing with mesons that are eigenstates of the strong interaction and therefore of parity. Parity non-conservation in the NN interaction is then directly related to the parity mixture of mesons. In the other case, one is dealing with mesons that are eigenstates of the total interaction (strong + weak). Parity non-conservation in the NN interaction is then due to the fact that the coupling of these mesons to nucleons does not conserve parity, in analogy with the standard picture of the PNC NN interaction. These couplings may be compared to the usual ones. We now turn to the calculation of the parity admixture of mesons described by the effective hamiltonian (51). The first approach is based on the factorization approxima-

B. Desplanques / Nuclear Physics A586 (1995) 607-648

637

tion at the meson level. In the simplest case of a product of currents gp’“V’AI and using relations (541, this approximation leads to the following result:

where 8,” and $1 represent the p and a, polarizations. Accounting for the quark substructure of the currents provides further contributions which can be calculated with the help of a Fierz transformation. In the case K = 1, the constants H appearing in the effective hamiltonian (51) which can be calculated from the quark interaction thus read Hp”=

G=m;

1

2 cos20c+ (1 - 2 sin28,)

,EgA

HP’ -=--fi

3

(I-

i),

, cos20C- (1 - 2 sin2w,) (I + f), 3 gpgA

2Gr m; fi

G, mp’ o sin*w, HP’= -,gpg, 3 (I++), 2 cos2& + (1 - 2 sin2w,) 3 H:=-zg,&

G, mp’ 1 sin2w, 3 (I++).

(0 + 3)

(65)

The last factor in the expression of the H’s contains two terms. The first one corresponds to the factorization approximation in its simplest form (as in (64)) while the second one accounts for quark rearrangement (first considered in Refs. [22,23]). The complete results for K # 1 are given in the appendix. Inserting results for H given by (65) into the expression of the PNC NN interaction in terms of these H, (581, allows us to get the PNC NN interaction due to the parity mixture of mesons. The result so obtained may be compared to the PNC NN interaction calculated from the exchange of mesons in the factorization approximation (Eq. (17)). Taking into account relations (561, the comparison shows that the two potentials have exactly the same structure and, more importantly, the same sign. The only difference is in the following replacement of the vector o-meson propagator:

mf m2,+q2

mf mt+q2

mi+q2’

(66)

where V and A here stand for vector and axial-vector mesons. The second factor, rni/(rni + q2), once introduced in calculations of the PNC NN interactions [l], is generally neglected in low-energy processes where q2 is small. It may, however, show up at high energy. It is also instructive to look at the effective meson-nucleon PNC couplings which result from the concept of parity mixture and are given by (61). In the way they have

B. Desplanques / Nuclear Physics A586 (I 995) 607-648

638

been defined,

the constants

HP”, Hi’, HP”, H,” and Hj

are equal to the meson-nucleon

PNC coupling h;, h:, h2P’ ho0J and h: when these are calculated in the usual factorization approximation. The difference between the vector meson-nucleon coupling calculated here and those calculated in this factorization approximation therefore resides in a - m$), as can be seen from (61). The value of this factor, around + 2, factor mi/(rni does not imply any change in sign. As to the enhancement of the effects one may expect from it, it is largely cancelled by the existence of a companion contribution due to the exchange relation:

of the axial-vector

mA2

meson. This cancellation

9 2

mV2

4 -~

2 mA-mV

2

m2,+q2

occurs according

2 mA

=

m2,+q2

mi+q2

to the following

(67)

mi+q2’

We therefore conclude that the PNC NN interaction arising from the parity mixture of spin-l mesons is essentially identical to the conventional one obtained in the factorization approximation

and that it does not really contain new physics, contrary to what was

claimed by Iqbal and Niskanen. In their work, Iqbal and Niskanen

also considered

a quark-model

approach

to

calculate the meson parity mixture, here represented by (51). The matrix element for the p-a, mixing in this approach was found to be -0.38 MeV2 instead of -0.17 MeV2 using the factorization

approximation

seems to suggest that the underlying sion, we just consider the isoscalar

at the meson level as in (64). The discrepancy physics

p-a,

is significantly

and o-f,

mixings,

different.

For our discus-

as in Ref. [15], and use the

same inputs concerning the weak interactions. This corresponds to taking cos20, = 1, effect). sin2B, = 0 and K = 1 (no strong-interaction We first recall the results obtained in either approach for the parity-mixing matrix elements. Specializing to the spatial components of the mesons (supposed with the spin s along the quantification axis), one gets

to be at rest

( ps I f&NC 1a,,) = -

(68) Results for the quark model, which can be recalculated from Eq. (151, have been given by Iqbal and Niskanen. With our conventions, which will be specified below, we get 12Gr a2 ( ps 1Hmc 1a,,) = - JZ -T3/2mq

(wJH~&&~)=

---

c1

-

:j7

(69

B. Desplanques/Nuclear

where we separated

which have a “direct”

contributions

suppose some rearrangement (16) of the quark-antiquark

Physics A586 (1995) 607-648

character

639

from those which

of quarks in the original hamiltonian (see Eqs. (15) and interaction). At this point, one already sees a strong

similarity between the two approaches to calculate matrix elements, (V, I HpNc 1A, >, since relative coefficients for different contributions are the same. The only difference is in the overall factors, rnz rni

1

1

--

2s,g,$K

/K on the other hand. In fact, there is a complete

on one hand, 12a2/mq/rr3/2

identity

between the two approaches, as we will show below. Indeed, the quark model used to calculate the matrix elements, (V, I H,,, 1A,), can as well be used to calculate the matrix elements of currents between the vacuum and the mesons,

allowing

axial-vector

a check of the relations

(54). Wave

functions

of the vector

and

mesons we used in getting results (69) are given by

I A,) = a 1’z(~)3’4+~(rl-r2)2)(l where 1 x’)

and 1 xc)

represent

x”)Xr),I~T)Ixc),

the isospin and color wave functions

(70) whereas

I x “)

represents the spin wave function of a quark-antiquark system with S = 1. What is of particular relevance in (70) is the relative sign of the two types of wave functions. Inserting

(69) into (54) allows us to get the following (y

m; g,fi=

+I

g,\ii2m,

= +1

3/4

77

(Y

mz

’

3/4

2J;; (71)

- mq ’

rr

The phases of the wave functions in accordance

relations:

in (70) have been chosen so that the relations (71) are

with our convention that g, (V = p, w) and g, (A = a,, f,) be positive. it is due to the color structure of the qq wave function. Multiplying

As to the factor 6,

now, member to member, the two equations gives the following relation: 2Gr Jz

2

mp' g,\lzm,

gA&

12G, = Jz

(71) (together

with a factor 2G,/fi),

CY= r312rn,’

(72)

This result indicates that the overall factors appearing in the two different estimates (68) and (69) of the matrix elements (V, I HpNc I A,) should be equal, hence the equality of the matrix elements themselves.

B. Desplanques / Nuclear Physics A586 (I 995) 607-648

640

The absence of equality in the results by Iqbal and Niskanen

is due to the fact that the

relations (711, which can be considered as self-consistency relations, are not verified with the choice they made for the parameters UJ and 11~s.The two members of the first equation

of (71), relative to the meson

2650 MeV312

and

equal to

3029 MeV312,

while for the second equation, 1705 MeV3i2

p, are respectively

and

relative

(73)

to the axial-vector

meson, a,, they are equal to

4453 MeV312.

(74) In order to get these numbers, we used the following values: mP = 770 MeV, ma, = 1270 MeV, gp = 5.7, g,, = 6.9, m, = 350 MeV and (Y= 1.7 fme2. This last value has been taken from Ref. [15]. We do not take it for granted, however. Having two unknown quantities (a and mq), the two equations by an appropriate

choice of these parameters.

game here, because the physics underlying do it. It may be noticed for instance current,

going beyond

(71) may be fulfilled

We do not want, however,

the quark model is too simple to allow us to

that a more realistic description one (p//w

the non-relativistic

to enter this

of the quark axial

instead of p/m,>, would

roughly lead to a change of the factor mq in (71) by the factor remove a large part of the discrepancy

between

m4 + 4o. This would r the two estimates given in (74) and, at

the same time, would greatly help to verify the relation

gp = gal (see Eq. (5711, which is

expected from chiral invariance. The identity between the two approaches can also be seen directly by comparing the two expressions of the quark-antiquark interaction given by Eqs. (15) and (16). While the first one does not a priori show a possible factorization of the matrix element into factors involving second

expression

entering

respectively

the initial and final meson (as in Eq. (64) for instance),

does. This

is straightforward

for the spin and isospin

Eq. (16). For the spatial part, one has to consider

where it is given by the anticommutator any problem,

with the product

of the p1 -p2

of &functions,

its full expression

S($
+ ri,))6(ri,

evidencing

tion of the spatial part of the operator into quantities

depending

any significant

difference,

in r-space

operator, which does not raise

(14)). This one can also be written as 6(r:,)6(ri2), and final states. As it does not show

the

operators

- ri2)

(see Eq.

the expected factorizaseparately

the above

on the initial

comparison

of two

approaches to calculate parity-mixing matrix elements for mesons tends to support our earlier conclusion that the concept of meson parity mixture does not carry real new physics with respect to what the modified factorization approximation was predicting. As the factorization approximation used to derive the PNC NN interaction gives the wrong sign to the PNC asymmetry observed in pp scattering at low energy, while Iqbal and Niskanen got the right one, we now discuss possible sources of sign uncertainty in their work. Their convention for the y,-matrix (see second paper of Ref. [15]), which has a sign opposite to ours, cannot in principle provide an explanation. The resulting change in sign in the non-relativistic expansion of the current product: i( pf)y,u(

Pi) u( P:)Y,YSu(

Pi) + (1 ++ 21,

(75)

641

B. Desplanques / Nuclear Physics A586 (I 995) 607-648

is compensated (conventions

by a change

concerning

in the sign

the couplings

of the matrix

gMNN, g,,

g,

element

(V, 1H,,,

I A,)

being the same). This change

simply results from the currents which are now of the V - A type instead of V + A. In these conditions, the fact that they got a sign identical to ours for the matrix element involving

the spatial components

of the meson is surprising.

In their approach based on

the factorization approximation at the meson level, it seems that their sign rather corresponds to the time component (whose sign is opposite to that for spatial ones). This is suggested

by examination

of Eq. (9) of the first paper of Ref. 1151. In their approach

based on a quark model, the most likely source for a difference in sign concerns the relative intrinsic phase of p and at-mesons. Their choice (Eq. (6) of their first paper) may be inconsistent

with the fact that they assume

ga,NN, gp and

are apparently

important

g,,)

positive.

that all strong couplings

The absence

of a discussion

point in the paper suggests that the authors overlooked

(g,,,,

about this

the problem.

5. Conclusion

In this paper, we looked at several studies that considered PNC effects in the pp system and assumed they could be directly described from a quark description or’ the nucleon.

We also looked at a model where parity non-conservation

between nucleons

is

supposed to arise from the parity mixture of mesons. A common point in all these studies is their reported ability to reproduce the sign of the PNC asymmetry measured in pp scattering

at 15 and 45 MeV.

Beside contributions calculated by Grach and Shmatikov, we considered contributions implying the derivative of the NN wave function. The direct contribution to the PNC NN interaction

which we obtained

in this way has a structure and a sign identical

to what was derived using the earlier factorization approximation (where currents are identified with mesons). The exchange contribution which involves the weak interaction between the exchanged quarks has a structure quite similar to the extra contribution brought about by the “modified”

factorization

approximation,

where rearrangements

of

quarks belonging to different currents are made. The only difference concerns the radial part of the interaction which accounts, not surprisingly, for a spatial extension of the nucleon quark core. We have not studied the effect of a full quark antisymmetrization

of the NN wave

function

considered

in looking

at PNC effects in the pp system. We nevertheless

what

could be the specific effect of an (s16 component in the ‘S, pp state, noticing that there should be some continuity between the contribution of this component, mainly located at short distances, and the NN one, dominating at medium and large distances. Such contributions were considered by Obukhowski et al. on the one hand and Kisslinger and Miller on the other hand. It was found that this contribution is identical in sign to that calculated in the factorization approximation, its magnitude depending obviously on the size of the spatial volume occupied by the component.

642

B. Desplanques / Nuclear Physics A586 (1995) 607-648

We finally admixture

considered

of mesons

the contribution

studied

by Iqbal

to the PNC NN interaction and Niskanen.

After

arrangement

effects, we found that their approach to calculate

factorization

approximation

at the meson level was leading

due to the parity

incorporating

quark-re-

the parity mixture with a to a PNC NN interaction

having again a structure and a sign identical to the standard one calculated in the factorization approximation. It was also found that self-consistency would require that their alternative

quark calculation

of this parity mixture would give the same result. The

only difference with the standard calculation concerns a modification of the radial part, already considered in earlier times, but generally neglected in most estimates due to its relatively

minor effect.

As the factorization approximation pp scattering at low energy opposite

is known to give a sign for the PNC asymmetry in to the experimentally determined one, it turns out

that the works we reviewed here made a wrong prediction for the sign of the above asymmetry. Contrary to what the authors may have thought, there is not much new physics in these works. At best, it may be said that the physics looks different. This is a typical property of unitary transformations. They do not change observable quantities but can considerably change the way one describes them, with possible double counting if no caution is exercised. This is specially illustrated here by the contribution of the parity mixture

of mesons

where the weak interaction states being eigenstates

(one may also add baryons). is switched

off at times

of parity. In a formalism

It corresponds

to a formalism

t = - 00 and t = + 00, asymptotic

where the weak interaction

is on the

contrary switched on at t = - CCand t = + ~0,for which incoming and outgoing particles are eigenstates of the total hamiltonian, including the PNC part, the same quark interaction manifests itself in PNC meson-nucleon couplings. Although there is no clear relationship to a unitary transformation, the duality between quark-exchange and quarkrearrangements effects (rather characteristic of s- and t-channels, respectively) is another interesting example to mention. In the field of parity non-conservation, it is common to look for asymmetries. The prediction of their sign matters at least as much as the prediction of their magnitude. Often, these signs are well determined theoretically. The practical determination, however, supposes to correctly deal with several quantities whose sign is convention-dependent, and the consistency of the inputs has to be carefully checked. We therefore developed at length everything concerned with these conventions, hoping that it will be useful for future work and that . . . we have not been misled ourselves by one of these delicate points. While many of the recent developments dealing with parity non-conservation in low-energy pp scattering have been shown to give the wrong sign for the presently measured asymmetry, it is appropriate to recall what is the present understanding of its sign. Beside contributions to the NN interaction coming from the factorization approximation, two extra contributions giving the right sign have been identified (see DDH [9], contributions b and c, Figs. lb and lc of Ref [28], and especially Fig. 2b2 of Ref. [9]). They are more complicated than the previous one and both involve the admixture of a quark-antiquark component into the nucleon. As a counterpart of their complexity,

B. Desplanques / Nuclear Physics A586 (1995) 607-648

however,

estimating

them accurately

is more difficult,

643

necessitating

some guess at the

present time.

Acknowledgement We are very grateful to Dr. G. Blanpied

for carefully

reading the manuscript.

Appendix In this appendix,

we give further expressions

(QCD) effects, accounted

for the case where strong-interaction

for by the factor K (Eq. (511, are turned on. The aim is to

show that the relationship we found between some recent approaches to parity non-conservation in the NN system and the earlier factorization approximation is not limited to the case K = 1 (no strong-interaction

effect). The result is not a trivial one as it depends

on the case one uses; the quark-quark or quark-antiquark part of the full quark interaction (6a). It, however, holds within some approximation (no qq content in the nucleon, only the diagrams of Fig. 4 in the quark antisymmetrization process). In all cases, only the part of the quark interaction (6a) symmetrical in V and A currents is retained. The 7 and 8 coefficients appearing here are defined in Eq. (6b). The PNC quark-quark potential used in calculating the PNC NN potential (extension of Eq. (13)) is given by QqPf,

Pi,

Pi,; VA + AV)

Pi,

= - %s(p:

+p;

-pi,)

-pi

2 cos20, + 1 - 2 sin2B,

~0.48 + ~~-0.24

x

6 _~0.48

+

+K-0.24

4

3

71.72

cos26, hAhA 12

1 +

_~0.48

- 2 (1 6 sin%,)

(71 .72

-

37;79

+K-0.24

+

4

ApA:

+ 1,1,(y,,

+ S,,hpA;)

1

+- ‘; + ” [ y12 + yzl + (S,, + ~,,)A,A~,A] 2 P:-P:+Pt-P;

X

(@1-@2).

i

2m,

+i(a,Xo,).

P: -Pi

-Pi 2m,

+P;

I

’

(A.1)

B. Desplanques / Nuclear Physics A586 (1995) 607-648

644

The quark-antiquark interaction (uncolored part) appropriate to calculate the meson parity admixture (extension of Eqs. (15) and (16) is given by V,g’+:,

X

P:,P;, pi,;VA+AV)

2 cos2tIC+ 1 - 2 sin28,

--

6 _~0.48

+K-0.24

3 1,12 + TV. 72 K”.48 + 2K-0.24

16

4

+ cos2f?,+

2

3

I

- 2 (1 6 sin28,) (3 7; 7; - 71 '72)

K0.48 + ~~-0.24

X

3

+lll,(Yll

i+

_.0.48

(i+1)+

4

1112 -71 2

&F)+

+K-0.24

'T2_ Yll

7;1, - 1, . T; +

[(~12+i2,)(t+l)+(i,2+~21)(~+o)l)

2

Pf -P:.+Pi

-Pi,

6(Pf +p:. -Pf

Xh+~2)*

-P:)bwa)l~

2%

(A4 or also v;q(p:,

p:, pf, P;; VA+AV)

=3[a+(l)bi(2)]$ 2

c0s2ec +

X

+

1 - 2 sin28,

~0.48 + 2K-0.24 Tf

6 3 l,l, +

+

6

c0s2ec -

Tf * Ti

.

Ti

K”.48 + 2K-0.24

3 _,0.48

3 - 2 (1 6 sin28,) (V

+K-0.24

+

4 Ti -

16 .-

37; T;)

K”.48 +32K-0,24 (1 + f) + -K”,484+ K-o.24 o + ‘,” ( -)]

3

B. Desplanques / Nuclear Physics A586 (I 995) 607-648

+

lfli +2Tf.Ti lf1i3/11

tyll

3

+

$li

H)

8

+

64.5

11 9

+ lf7f

+

[ (712 + Y,,)(l

2

+ f) + (42 + %I)(0

+ 31 1

Pf-PS+PI-Pi,

X(-i)(UfXUi).

a(Pi +PZ-Pt

-Pi!)[b(2)a(1)li.

25

(‘4.3) The PNC NN interaction due to meson exchange with meson-nucleon couplings calculated in the factorization approximation (extension of Eq. (17)) is given by V,NN(r12; VA+AV;

p+w)

2 CO&~ + 1 - 2 sin*& K”.48+ 2K-0.24 = -- % $1 * T* Cl6 3 JT I [ i _~0.48 +

+

i)

+K-0.24

(o-1)]

4

+r,*(o+i)+),,(uif))

7;1, + 117; [f(%2+7*1)(l

2

+ $( 71 . T* - 371%;) _~0.48

+f)+f(~,,+~*J(O+~)]

COSMIC -

6

o+y

4

(

-)I)( [

2M

2 cos*t$ + 1 - 2 sin*& 6

(1+ 3)

‘pr f(

4

K”.48+ 2K-0.24 3 (0 + 3)

+K-0.24

(0 + +)I + %I(1 + f) + &I(0 + I))

4

+53

3

(?-a*).

Pl -P2

+i[l+&](a,Xo*).

_,0.48

[

+ ~-0.24

+

+

K”.48+ 2K-0.24

(1 - 2 sin*B,)

7;1, + 117; 2

[ 3(r,2

+ 3/2*)(1 +f)+f(&2+~21)(0+

Y)] i

(

x (a1 -u*).

(5,

+i(1 +&#)(a,

x u*).

f&9) PI -P2

[- 2M

‘Or f(

4

646

B. Desplanques / Nuclear Physics AS86 (1995) 607-648

Pl -P2 7j-p

(

X(a,+a,). The last term contains

(A4

&J r f( 41

p and w-exchange

contributions.

assuming fp(r) = f,,,(r).Some quantities, g:, non-relativistic values 3, 1 and 3, respectively.

gi,

They have been put together

gwNN/gpNN, have been given their

The PNC NN interaction calculated in a quark model from the quark-quark interaction (A.11 retaining the direct and exchange diagrams of Fig. 4 (extension of Eq. (50)) is given by VK”“( rf, r’; VA+AV; =

b)

GF --

2 cos20, + 1 - sin28,

K”.48 + 2K-0.24

6

3

Jz _~0.48

+K-0.24

+

(O-e)]

4

!$j$,

+5i( a* x UB) . [

ri)

fo(rf,

q

.

Ii

2 cos20, + 1 - sin2B,

K".48+ 2K-0.24

6

3

+3 i +

+i,,(0+b)+~11(0+~))

_K0.48

(0 + $)

+ ~~-0.24 (n+~)]+i~*(l+~)+I,,(o+B)i

x(,,:.

B'1

( e, .

+i(a*xa,).

fG(rf, ri)

fG(rf, ri)

e, [

i

9

.

9

cos20, - (1 - sin28,) +

6 _~0.48

~0.48 + ~~-0.24

X

II

3

(1+f)+

+ ~-0.24

4

( 0-t: -)]

(1 - +)

B. Desplanyues / Nuclear Physics A586 (1995) 607-648

f&f, ti

+ 54 a* x crB) . $jy [ *

+2-

‘A

z-

6 2

q

[(YIZ + Y,,)(l

Ii

+ $) + (&, + %I)(0 + G)]

1y--$, .

x(u*+uEJ.

ri)

f&f,

9

Examination

647

Ii .

(A.51

of (AS) shows that its structure is the same as (A.41, the Yukawa potential

being replaced by the gaussian

one, defined by Eq. (49), and assuming

take their value in a non-relativistic Expressions

of the constants

that M, x,, x,

model.

H, determining

sion of Eq. (64)) can be calculated

the parity admixture

of mesons (exten-

from Eq. (A.2) or (A.3). They are given by

2 cos20C + (1 - 2 sin2BW) K”.48 + 2K-0.24 3

i _

_K0.48

+

~-0.24

4

H,‘_

2GF

I

+>

(o+B)] +YI@+i)+1:1(0+~)i;

m;, cos2& -

(1 - 2 sin2B,) 3

J;s - -=gPgA

K”.48 +32K-o.24 (1 + ~) + -K+‘.‘sq+

H;=

(I-

3

K-o.24

,

+~~g~[(v,2+y2~)(l+;)+(B,2+82,)(0+4)], P

GF H,=

‘z:_g: -K

m; i +0.4s

+

+

4 H;=

2 cos20C + (1 - 2 sin28,) 3

K”.48 + 2K-0.24 3

(o++)

~-0.24

( o+; -)]

+ 2y,,(1+

+$7~[(~,2+721)(1+~)+(~12+~2,)(0+3]. lo

+> + 26,,(0

+ ;)

(A.61

648

B. Desplanques / Nuclear Physics A586 (1995) 607-648

The coefficients H can be compared to those entering the NN interaction obtained from meson exchange in the factorization approximation (Eq. (A.4)) with the result that they are the same.

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