Parity-violating nucleon one-meson exchange potentials in a current-current quark model

Volume 44B, number 4 PHYSICS LETTERS 14 May 1973

PARITY-VIOLATING NUCLEON ONE-MESON EXCHANGE

POTENTIALS IN A CURRENT-CURRENT QUARK MODEL

J.G. KijRNER

Institu t fiir Theoretische Physik der Universitiit Heidelberg, Germany

Received 23 March 1973

We calculate the parity-violating coupling constants ANN, and h&(w) in the framework of a current-current quark model and determine from these the nucleon one-meson-exchange potentials. Compared to the previously used potentials we obtain a much weaker n-exchange and a somewhat stronger p(w)-exchange potential. In contrast to the isospin-violating p-exchange potential resulting from the standard factorization approach (i.e. AI = 0,2) our p(w)-exchange is isospin-conserving.

The recent observations of parity-violating (p.v.) phenomena in nuclear transitions has led to consider- able interest in calculating the p.v. potential that in- duces such transitions*. The single meson (n, p, o)- exchange mechanisms depicted in fig. 1 give important contributions to the p.v. potential. One of the vertices involved contains the strong coupling constants

gNNrr3 gNNp and gNNw which are assumed to be known. The p.v. couplings that appear in the other vertex are usually calculated with the help of the current-current weak interaction Hamiltonian involving the product of charged Cabibbo currents. There are contributions from the product of strangeness conserving currents (-cos28) that are evaluated in the so called factorization approximation [3-51 and, for the rr, a term arising from the product of strangeness changing currents (-sin28) which is calculated by using soft pion techni- ques [6, 71 or by an SU(3) analysis [8].

In this note we would like to report on the results of a calculation of the p.v. weak vertex in fig. 1 by means of the quark model based on the current-current interaction of charged quark currents. Our main con- clusions are the following:

(i) the factorization contributions (-cos20) are also present in the quark model approach

(ii) sin20-contributions are absent due to the absence of h-quarks in N, 71; p and o.

* For a comprehensive discussion of the theoretical and experimental aspects of p.v. phenomena in nuclei we refer the rea- der to the two recent review articles by Fischbach and Tadic [l] and by Gari [2] . These articles also contain a complete list of references.

N

1 4 N

TP.W _ _ _ _ - _ _ _

IN IN

Fig. 1. One-meson-exchange contribution to the p.v. nucleon potential. The p.v. weak vertex and the strong vertex are de- noted by a shaded square and a shaded circle, respectively.

(iii) there is a term proportional c0Ge similar to the factorization contribution that couples neutral vector mesons. This term affects the isospin transfor- mation property of the p.v. nuclear potential

(iv) there is an additional contribution coupling the

p-meson which changes the strength and phase of the p-exchange potential relative to the factorization contribution.

The four lowest order diagrams that can contribute

to the weak vertex in fig. 1 in a current-current interaction ansatz are drawn in figs. 2 and 3.

The same diagrams also give rise to a quark model description of the (AY = -1) non-leptonic hyperon decays. Such a model was investigated in detail in ref.

[9] and has lead to a reasonable fit of the experimental

decay parameters of the octet baryons and the W-particle. In particular, one can show how the various dia-

grams correspond to the equal time commutator and meson and baryon pole terms of the conventional current-algebra treatment [ lo] .

In the present case one can similarly identify the contribution of fig. 2a as corresponding to the factori-

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Figs. 2 and 3. Lowest order contributions to the p.v. vertex for a quark current-current interaction. The wiggly lines in figs. 2a and 2b are drawn for illustrative reasons so as to indicate how the quark current transitions act. The results of calculating the contributions of the diagrams have been indicated in the labelling of the meson lines, i.e. fig. 2a contributes to the p.v. coupling of charged mesons, fig. 2b to neutral vector mesons, figs. 3a and 3b to the p-meson, and fig. 3c does not contribute to the p.v. couplings.

zation contribution in the conventional treatment. To this end we evaluate the contribution of fig. 2a using the methods outlined in ref. [9] :

Anpn- = (1) G Hl

fi 3m;fin cos28 urn -m l(2m2 -Iq2)ap

P n N2 npn- T

h,&- = (2)

The amplitudes A and h* are invariant amplitudes de-

fined by the standard decomposition of the matrix ele-

ments [ 1, 21

(27r)g~2(NnB!Hp~v~(0)lN) = -i(2n)3(42-mz) (3)

Xjd” x exp(i4x)(NIT(~“(xyi,,v.(0))l~ z ~ANN,~ ,

and

(277)g/2(Np, E&,.(O)tN> = -ieE(2n)3(q2-m$ (4)

X .I- d4x exp(~x)<NIT(~~(x)~Hp~v~(0))IN) z P(q2)ez ,

P’(q2) = ii y5(hA(q2)yfi + hE(q2)uUYq,,+ hp(q2)q’“)u .

The coefficients ap and uv are quark model coefficients whti are listed in table 1. H, is a convolution over quark wave functions and has been evaluated in ref. [9] through a fit to the non-leptonic hyperon decays. The result was HI = -2.77 X 10e3 [GeV] 3/2. This leads to the values (for q2- 0)

A,,- = 0.24 X lo-* (0.14 X lo-*)

hkp,,- = 1.0 X 1OPj (1.4 X 10-6). (9

For comparison we have added in brackets the coupling values resulting from the factorization approximation [3-S].

The approximate agreement of the respective coupling values in eq. (5) is a consequence of the above mentioned correspondence between the two methods. This can be explicitly seen by writing the quark model result in a factorized form [9] , namely,

(27r)9/2(pn- IHp v In) = . .

(2~)3~pIV~+(0)ln)(2.rr)3&r-L4~(O)IO),

(2n)g~2(pp-iEip v In) = . .

(2n)3(pL4~+(0)ln~2n)3/2~-iVJO)10), (6)

where P and Ap are the usual quark currents that compose Hp.,. [9]. The equivalence of two calculations can also be understood by comparing fig. 2a with fig. 4, where we have drawn the diagrammatic representation of the factorization contribution [e.g. 1 l] . The slight deviations of the quark model values from the factorization values in (5) result frcm the combined effect of HI and GA being calculated too large in the quark model, and fP (normalizing (01 VP I@) being predicted too small in the quark model (“van Royen-Weisskopf para- dox” [ 12, 131).

The absence of sin20-contributions to the weak p.v. COUphgSANNn, hiNp and hiNw mentioned in point (ii) can be easily understood from the fact that the particles involved do not contain any X-quarks. In particular this means that there is no equivalent to the

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Volume 44B, number 4 PHYSICS LETTERS

Table 1

14 May 1973

Quark model coefficients for the p.v. vertices. In order to obtain phases corresponding to the Condon-Shortley convention one

has to change (p+, n+) to c-p+, -n+).

pw+ npp- PPPO nnpo PPw nnw PPV nn9

av 80 80 4ofi -4oJz -24fi -244 64tg% -16tg%

bv 96 96 484 -48,/? 0 0 0 0

pnn+ npn-

a P -48 48

sin28-contribution toANNn that arises in the conventional PCAC and CA calculation of this coupling value, which can be written as [6, 71

ANNn _ (1 t fld) sin28 , (7)

where f/d is the f/d-ratio associated with the one particle matrix elements (B’INp,c.lB). In the current-current quark model one obtainsf/d = -1 for this ratio [ 14-161, and thus the absence of a sin20-contribution in our model is not inconsistent with the relation eq. (7) following from PCAC and current algebra. One can check that the only contribution toANNn in our

model is the one given by eq. (l), which corresponds to the factorization result and thus we obtain the same value for ANNn as Schulke [5] .

We now complete our discussion of the contribu-

tion of the diagram fig. 2. As indicated in fig. 2, diagram 2b couples only to neutral vector mesons, which can be readily seen by following the quark lines in the diagram*. By introducing charged quark currents, one thus obtains an effective neutral current coupling of the vector mesons. In table 1 we have listed the resulting contributions in terms of the quark model coefficients defined in eq. (2).

From table 1 it is clear that the coefficients uv satis- fy a AZ = 0 rule which is the equivalent of the AZ = l/2 rule in the non-leptonic hyperon decays. These isospin selection rules follow from the use of the symmetric quark model with Bose-type quarks [ 16, 171 or from the model introduced in ref [9, 1.51 which uses para- quarks and a modified current-current quark Hamilto-

* The coupling to neutral pseudo-scalar mesons is forbidden

by charge conservation [l] .

nian. For other quark schemes, for example those dis- cussed in ref. [ 181, one would obtain different isospin selection rules**. However, we feel that the AI = l/2 rule in the non-leptonic decays is so firmly established that these other possibilies should be ruled out.

Before we can write down the potentials corresponding to the one-meson exchanges we need to discuss the strong vertex part in the one-meson exchange diagram fig. 1. With the standardfld-values [ 191 that correspond to the vector dominance model and ideal w-cp mixing one obtainsgppp/g = l/3 (i.e.f-coupling) and

gNNw = 0 (i.e. f/d = l/3) !??the electric and magnetic coupling, respectively, and gNNV = 0, implying a vanish-

ing cp-contribution to the p.v. potential.

Using the standard nonrelativistic approximations [ 1, 21 one obtains from the contributions of figs. 2a and 2b the following one-meson exchange potentials

v, =

(8)

**This can be seen by considering the different numbers of

quark loops in diagrams 2a and 2b. The use of different sta-

tistics results in different predictions concerning the relative

phase and magnitude of charged (fig. 2a) and neutral (fig. 2b)

vector meson coupling.

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‘N

Fig. 4. Factorization approximation to the p.v. weak vertex.

where we have set m, rical values for ANNn

= mp fpr simplicity. The nurne- and h* m (8) and (9) are the ones

given in eq. (5) for the specific charge configuration n + pn-@-).

In the factorization approach the corresponding po-

tential due to charged p-exchange is proportional to

]1>21

$pzl I=0 t i [r1.z2 - +;I’=2 .

Thus by adding neutral p-exchange to the AI = 0 rule,

the AZ = 0 part fo the p-exchange potential is strength- ened by a factor 3/2. The interference between p- and o-exchange is destructive or constructive depending whether the nucleons are in a I3 = 5 1 or I3 = 0 state.

A further contribution to the p-coupling arises from the diagrams 3a and 3b, while fig. 3c gives a vanishing contribution. The result is the same as in eq. (2) with the quark model coefficients av replaced by bv (see table l), and the overlap integral H, replaced by H2. In ref. 191 H2 was estimated from a fit to the non- leptonic hyperon decays to be H2 = 8.8 X 1O-3 (GeV)3/2. Thus, the net final p-exchange contribution is obtained by multiplying the p-potential in eq. (9)

by factor - -2. We would like to mention that the same diagrams also contribute to hE in the covariant decomposition eq. (4). This invariant, however, does

not contribute to the potential in the limit of isospin

invariance. Henley has attempted to estimate off-mass-shell ef-

fects in the p.v. potential by including N*-intermediate states [20] , which involve the p.v. couplingsfN*Nn. In our model the couplingfA(1236)Ns turns out to be zero, which supports the conclusion reached in [20] that the A-resonance can be neglected in calculating such off- mass-shell effects.

We close by remarking that the exact values for the

coupling constants calculated in our model should be viewed with a little caution because of the approximations inherent to the quark model. However, experience with the many applications of the quark model has taught us that the quark model provides us with many important qualitative insights into the basic mechanism of particle interactions. Therefore, we try here to put more emphasis on the qualitative predictions resulting from the present calculation for the one-meson-exchange potentials. The important qualitative conclu- sions are the following: (i) The (hl = l)n-exchange can be neglected compared to the p(w)-exchange. (ii) The p-exchange potential may be somewhat larger than the value calculated in the factorization approach and has a phase opposite to the latter. A somewhat larger p-exchange contribution may be welcome in order to ex-

plain the large p.v. effect seen in the 482 keV transi- tion in l*lTa [2]. As to the significance of the phase of the p-exchange contribution we refer to the discus-

sion in ref. [ 11. (iii) The p(o)exchange potential obeys a pure AZ = 0 selection rule. A crucial test of the latter

aspect of our model would be to establish, whether the Al = 2 transitions suggested by the factorization model are absent in nuclear p.v. transitions. Unfortunately, most of the data on the p.v. transitions involve y-emis- sion, which raises some additional theoretical problems. However, even in this case, the maximal change of iso- topic spin in these transitions due to p(w)-exchange should be AI = 1, with possible AZ = 2 transitions resulting from the much weaker n-exchange.

The author would like to thank Prof. L. Schiilke for helpful discussions, Prof. D. Tadic and Prof. E. Fischbach for informative correspondence and Dr. N.S. Craigie for a critical reading of the manuscript.

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Parity-violating nucleon one-meson exchange potentials in a current-current quark model