PACS numbers: 13.30.Eg, 14.20.Kp OCR Output

algebra should be possible.places where good discrimination between the pole model and currentable data, compare it with the results of current algebra and indicateis also given. We apply the full pole model to the description of availof parity violating amplitudes for decays with vector meson productiontion contribution, interfering with it destructively. Symmetry structuresmaller and of opposite sign and (ii) a term proportional to the factorizaterm proportional to the standard current algebra expression but muchbaryon and an octet pseudoscalar meson consists of two pieces: aof parity violating amplitudes of charmed baryon decays into an octetdifference). We find that in the pole model the symmetry structure(Ac - charm-noncharm mass difference, Aw - 1/2' — 1/2+ masscussion of departures from current algebra for any values of Ac/Awexcited 1/2* baryons of given charm. The technique permits easy dismediate state is applied to sum the contributions from all intermediatecurrent algebra. to the case of flavour symmetry breaking in the interof charmed baryons. A simple technique generalizing the expressions ofin parity violating amplitudes of Cabibbo-favoured nonleptonic decays

We study effects of pole-model-induced S U (4)-symmetry breaking

Abstract

December 13, 1993

Radzikowskiego 152, Krakow, PolandH. Niewodniczafiski Institute of Nuclear Physics

Dept. of Theor. Physics

P. Zenczykowski

AmplitudesSymmetry Properties of Parity ViolatingNonleptonic Charmed-Baryon Decays:

than the ground·state charmed baryons under consideration. In other words, OCR Output

the intermediate excited 1/2‘ baryons were implicitly assumed much heavier

amplitudes were treated in both references [1] and [4] in similar ways. That is,

Contrary to the case of parity conserving amplitudes, the parity violating

respect to the scheme of ref.[1].

constituted the main difference of the quark diagram approach of ref. [4] with

prescription is opposite to the one inherent in the approach of ref.[1]. This

of quark diagrams are to be added: the sign resulting from the pole model

the relative sign with which the spin-flavour factors corresponding to two types

to the propagation of ground-state baryons in the intermediate state affects

scheme. It was shown that the presence of energy denominators corresponding

of the standard pole model is to be generated in a genuine quark—diagram

the elucidation of how the symmetry structure of parity conserving amplitudes

diagrams has been presented. An essential part of that paper was devoted to

In a recent paper [4] a new analysis formulated in the language of quark

in the framework of the pole model.

Tseng [2] and Xu and Kamal [3] consists in carrying out explicit calculations

tempted by Korner and collaborators The second, followed by Cheng and

first, based on quark diagrams and symmetry principles, was originally at

become possible. Two main theoretical approaches are being studied. The

crimination between competing theoretical descriptions of these processes will

mulated, we are slowly approaching the moment when phenomenological dis—

As more and more data on nonleptonic decays of charmed baryons is accum

1 Introduction

The main advantage of our approach is that we gain a much better insight OCR Output

results obtained previously in conventional S U (4)-breaking pole models [2, 6].

In Section 2 using a recently developed simple technique [5] we reproduce

of this paper which are of a group-theoretical nature.

consideration of momentum dependence could qualitatively change the results

like the bag model considered in ref.[2, 6]. However, we do not think that

of the momentum dependence needs the introduction of more specific models,

whose nonzero values also question the applicability of current algebra. Study

the pole model only. We will not consider the dependence on meson momenta

be interested in the S U (4) breaking effects arising from energy denominators of

are evaluated by the (S U (4)·breaking) 1/2` pole model prescription. We will

diagram approach of ref.[4] to the case when the parity violating amplitudes

It is the purpose of this paper to discuss the modifications of the quark

algebra.

expects the pole model predictions to be totally different from that of current

baryon decays (m,——m,,_d_,)/Aw w 2.510.5 (see Section 2). Consequently, one

between CA a.nd pole model predictions may be observed [7, 5]. For charmed

imation to that of the pole model, although even there significant differences

CA prescription for parity violating amplitudes may constitute a fair approx

For hyperon decays, where (vn, —m,,_d)/Aw ¤. 1 /3 and m, —m,,_d N m,,, the

of the PCAC/CA approach is that the emitted pseudoscalar meson is soft.

necessary to reduce pole model to current algebra (CA). Another ingredient

excited-state baryons. Such an assumption constitutes one of the conditions

compared to the splitting Aw between the 1/2+ ground-state and the 1/2'

in refs.[1, 4] quark mass differences mc — m,,_d_, were treated as negligible when

and pole model should be possible. OCR Output

other decay channels where a cleaner discrimination between current algebra

how one can alleviate the problem of the Aj —-> E+1r° asymmetry and indicate

negative numbers) the agreement with the data is good. We discuss briefly

the pole model predicts a small positive value, while experiment and CA yield

data. With the possible exception of the Aj -+ E+1r° asymmetry (for which

In Section 3 predictions of our scheme are compared with the most recent

amplitudes on charmed quark mass can also easily be made.

comparison with the approach of Korner [1] as regards the dependence of the

By casting the structure of pole model into the language of quark diagrams a

in this case no term proportional to the factorization contribution is present.

of two pieces (separately for transverse and longitudinal vector mesons) but

meson) the baryon—pole contribution to the parity violating amplitudes consists

which it interferes dcstructively. Similarly, for the B., —» BV decays (V - vector

opposite sign), the other is proportional to the factorisation contribution with

proportional to the current algebra prescription (though much smaller and of

amplitudes consists of two pieces of familiar symmetry structure: the first is

octet pseudoscalar meson) the baryon-pole contribution to the parity violating

BP decays (B., - charmed baryon from the antitriplet, B - octet baryon, P

deviations from the predictions of current algebra. It appears that for the B., —+

into the symmetry structure of the pole model and the general tendency of its

baryons is contained in the W-exchange diagrams (bl) and (b2). In ref.[4] the OCR Output

grams (d) vanishes. The pole model contribution from the intermediate 1/2'

amplitudes. For the parity violating amplitudes the contribution from dia

are shown in Fig.1. Diagrams (0,) and (a') correspond to the factorization

The quark diagrams relevant for the nonleptonic charmed baryon decays

baryons.

apply the basic idea of ref.[5] to the case of nonleptonic decays of charmed

denominators of the pole model is to be considered. In the present paper we

the pole model) are to be modified when S U (3) symmetry breaking in energy

commutator A°H""· - H '·"·A° (which correspond to appropriate two terms in

In ref.[5] it was shown how the relative weights of the two terms of the CA

recently been developed in ref. [5] for the specific case of weak hyperon decays.

the pole model. Such an approach phrased in terms of quark diagrams has

of current algebra and yet discuss flavour symmetry breaking characteristic of

CA formulas. It turns out, however, that one can easily maintain this virtue

troublesome 1/2" poles - their summed-over contribution is hidden in the final

of current algebra is that it bypasses the need to know anything about the

state baryons relevant for the latter. One of the reasons of the attractiveness

constants of 1/2` excited baryons of which we know much less than of ground

since the former involves weak transition matrix elements and strong coupling

is usually regarded as less reliable than that of parity conserving amplitudes

Description of parity violating amplitudes in the framework of the pole model

plitudes

2 Symmetry structure of parity violating am

Thus, the relative signs of various individual contributions are taken care of OCR Output

from intermediate 1/ 2` baryons of given charm are already summed over.

violating transition with B; and B,. Please note that in Eq.1 contributions

masses of corresponding hyperons connected through the c d —-> s u weak parity

1/2" excited charmed baryons, respectively. Similarly, B, and B: denote the

B; denote the masses of the decaying charmed baryon and the intermediate

In Eq.1 baryon masses have been denoted in a self·explanatory way: Bc,

B., + B; -B, + B;b1(i) : b,(i)

proportional to

given decay i, the pole model expression for the parity violating amplitude is

Denoting by b1(i) and b2(i) the spin-flavour weights corresponding to a

factorization diagrams.

for the B., -—> BP decays (bl + b2, P) and (2) the contributions from the

from ref.[4] and added to Table 1: (1) the sums of the spin-flavour factors

For better clarity of the exposition two additional columns have been taken

B., ——> BP, Bc —+ BV; and B, —» BM') in Table 1 (in columns denoted b1—b2).

the differences only. We have gathered all of them (appropriate for transitions

the sums have already been tabulated in ref.[4] it would suffice here to give

to present their sums and differences rather than the individual factors. Since

(b1) and (b2). For the sake of the ensuing discussion it is more appropriate

breaking requires knowledge of spin-flavour factors of the individual diagrams

factors corresponding to these diagrams. Consideration of S U (4) symmetry

quark—model technique of refs. [8, 9] was used to determine sums of spin·flavour

b is the reduced matrix element extracted from hyperon decays (see ref.[4]). OCR Output

ing the overall factor of 1/ Aw by la = bcot Go = -22.2 (in units of 10°'7) where

its multiplication by appropriate coupling constants what amounts to replac

The full parity violating amplitude of the pole model is obtained from Eq.6 by

.... lbn(*) + b2(*)l + [b1(¤) · b2(*)l (6)A 1 £} i‘j(%‘F1 E {

Eq.1 may be rewritten (without making any approximation) as

more.

Other estimates of the ratio Ac/Aw give values in the range 2 to 3 or even

(5)Ac/Aw = (mc — m,)/Aw M 2.4

and

m, — m, M 1125MeV (4)

Aw M 475MeV

we obtain (effective) values

(3)B:——B,, M m,-{-Aw——m,,

B;—B, M my}-Aw-m,

Since

(2)B: — B., M 1700 — 2350 = -650 MeV

B; — B, M 2750 — 1150 = 1600 MeV

automatically. We estimate

will extract therefore an incorrect (i.e. reduced) value for the factorization OCR Output

experimental data performed in the current algebra + factorization framework

for Ac/Aw = 2.5 ;i: 0.5. The corresponding number in ref.[2] is -0.32. Fits to

diction is given by the multiplicativc factor of 1/(1 — (Ac/Aw)2) w —0.19;i:3j?2

hence the whole difference between the pole model and current algebra pre

est. For example for Aj —> E°1r+ there is no factorization contribution and

1.95/(-5.40) = -0.36. Similar agreement is found in other cases of inter

“°··the Aj —» A1r+ parity violating amplitude of ref.[2], where A*’°l'/A·f

pared with the pole model correction to the factorization contribution in

i.e. a substantial destructive contribution. The above number is best com

If one uses g z 4.5 from ref.[4] one gets ( = —0.39;i;gj22 for Ac/Aw = 2.5 i 0.5

C: (7)1 — (Ab/Aw)’;

assume g re: —g')

this pole—model induced contribution relative to the factorization term is (we

factorization contribution (sextet—dominance requires g = —g’). The size of

doscalar meson production this term is proportional to the sextet part of the

factors b1(i) — b,(i). One can see from Table 1 that for flavour octet pseu

bution has symmetry structure determined by the difference of spin-flavour

of Ac/Aw. We might call this term an anti-CA piece. The second contri

1/ [1 — (Ac/Aw)’] changes from 1 at Ac : 0 to around -0.2 for realistic values

from Eq.6 in the limit Ac —> 0). Its size and sign are, however, different:

has the symmetry structure of the original current algebra term (obtainable

parity violating amplitudes. The first is proportional to b1(i) + b,(i) i.e. it

From Eq.6 we see that in the pole model there are two contributions to the

mass of the pseudoscalar meson). OCR Output

if one approximates the mass of the noncharmed final baryon by 2Aw (mp

2 w

<8>1%- Am Ac

of ref.[1] may be cast in a form easily comparable with Eq.6:

butions from diagrams (bl) and (b2) to B, —> BP parity violating amplitudes

with that of ref.[1]. Up to an overall normalization factor the relative contri

quark mass is very simple permits an easy comparison [10] of this dependence

amplitude can be well approximated by Eq.6 where dependence on charmed

Aj —-> E+1r° asymmetry in the next section. The fact that the p0le·model

metry between ref.[2] and ref. We shall come back to the question of the

processes. For the Aj —> E‘*'1r° decay this explains the difference of asym

the sign and the size (in some cases significantly) of the asymmetries of such

the change from CA to pole model for parity violating amplitudes will change

contribution is estimated similar by different authors · see e.g. refs.[3, 2, 6])

tudes is in both cases described by the ground-state baryon pole terms (whose

Aj ——> E"’vr°. If one accepts that the structure of parity conserving ampli

one has to look at those processes where factorization cannot contribute, e.g.

simple current algebra and those of the S U (4)·breaking pole model. Namely,

In principle there is a simple way to distinguish between the predictions of

(recall that the destructive baryon pole terms do satisfy sextet—dominance).

(than actual) breaking of sextet dominance in the factorization contribution

in pole model than in current algebra, the CA fits would suggest a stronger

is correct. Because in such fits the "true" factorization contribution is bigger

contribution in the parity violating sector if the S U (4)—breaking pole model

10 OCR Output

cleanly between current algebra and pole model (see next Section).

contribution is important as it provides us with a possibility to discern more

ance of nonzero b1(i) — b,(i) terms for processes where there is no factorization

ality factors are \/E(1/ respectively (as indicated in Table 1). The appear

b1(i) + b,(i) terms for lQ|(V_L) mesons. The relevant (spin—related) proportion

that the b1(i) —- b2(i) terms for V_L(I{|) mesons have the symmetry structure of

algebra terms for pseudoscalars.) A comparison with Table 2 of ref.[4] shows

called CA-like terms since they are symmetry-related equivalents of current

bution either. (The b1(i) + b,(i) terms for vector mesons should perhaps be

b1(i) — b,(i) and b1(i) + b2(i) terms is proportional to the factorization contri

by the same formulas as for the B, ·-> BP decays). No linear combination of

apart from the size of the corresponding reduced matrix elements, are given

b2(i)-type terms are not proportional to the factorization contributions (which,

From Table 1 we also see that for the vector meson production the b1(i)

relative signs of the two contributions are identical.

of roughly 1.5 to 3 bigger than the contribution from diagram (b2), and the

case, however: in both models the contribution from diagram (bl) is a factor

significantly different. For the realistic case of Ac/Aw z 2.5 this is not the

models agree at Ac —> 0 their predictions at other values of Ac are in general

factorization contribution, though its sign is opposite.) Although the two

state baryon pole contribution to Bc ——> BP processes looks exactly like the

survives. (Thus, in the limit of very heavy charmed quark mass the ground

(Eq.6) it is the difference of the spin-flavour weights from both diagrams that

dominates over (b2) in the limit Ac -> oo. On the other hand in the pole model

From Eq.8 one can see that for the approach of ref.[1] the diagram (b1)

11 OCR Output

amplitude (see Table 2). The underlying reason is that all such cancellations

other hand no such cancellations are seen in our pole-model expression for this

cancellations among the contributions from individual baryon poles. On the

EQ —> E'*K') is quite uncertain because the final number is a result of strong

of the parity conserving amplitude of the Af —> E°K+ decay (as well as that of

In ref.[6] Cheng and Tseng have stressed that their pole-model prediction

matrix elements of VL and W sectors, see also ref.[4].)

(In Table 2 we have adopted quark-model relationships between the reduced

diagram expressions for the parity conserving amplitudes of the Aj decays.

amplitudes. For convenience in Table 2 we reproduce (from ref. the quark

sets therefore the scale of the W-exchange contribution to all parity conserving

and refs.[1, 3, The measured value of the Af —+ E°K+ branching ratio

decay only the parity conserving amplitude is different from zero (see Table 1

solely on the W-exchange processes. Moreover, in the case of Aj -—> E°K+

process Aj -> E+1r°. The reason why they are important is that they depend

for the Af ——> E°K+, Aj —» E+¢ decays as well as the asymmetry of the

quite significantly. These are the recently measured [11, 12] branching ratios

There are several new data points that restrict the freedom of our approach

the most recent measurements that have not been used in the fit of ref.[4].

description of the experimental data in the pole model - including results of

the fit of ref. [4] in an essential way. In this section we shall give an examplary

the symmetry structure of parity violating amplitudes would certainly change

differences existing between current algebra. and the pole model and concerning

In ref.[4] we have presented a CA fit to the experimental data. Significant

3 Description of Data and Discussion

12 OCR Output

factorization contribution (and keeping g = —— g' = 4.5 as in ref.[4]) finds then

spect to the size of the parity conserving part M (assuming M' = ——M) of the

(in units of 10'7), to be compared with B z -100 in ref.[4]. The fit with re

B m -140 (9)

the Aj -> E°K+ and Af —+ E+¢ decays is obtained then with

model we are interested in. Reasonable description of the branching ratios of

the available data obtained in this way preserves all qualitative features of the

and r : 1 we decided to present results obtained for r = 0.7. Description of

has to keep the value of ·r fixed. To give the reader a feeling for both r : 0.5

0.30. Thus, a very small value of r would be chosen by the fit. Therefore, one

small), in disagreement with recent experimental number [12], a = -0.43 :I:

E+1r° decay. The asymmetry predicted by the pole model is positive (but

model yield different predictions for the sign of the asymmetry of the Aj ->

as it has been mentioned in the previous section, current algebra and pole

performed in the pole model with respect to r is meaningless, however. Namely,

amounts to the replacement of fw by For reasons discussed below the fit

and the overlap factor r (see ref.[13]) to the available data (introduction of r

ref.[4] and fit the factorization contribution in parity conserving amplitudes

branching ratios, most notably that of Aj —> E+¢) we might proceed as in

amplitudes roughly fixed by the Aj ——> E°K'*` branching ratio (as well as other

Having the effective size of W-exchange contribution to parity conserving

also in ref.[4].

the cancellation of contributions from (d)-type diagrams (Fig.1) as discussed

carried out when obtaining entries of Table 2. This cancellation corresponds to

between contributions from different intermediate baryons have already been

13 OCR Output

by the bl — bg correction (recall that for pseudoscalar meson production this

where the anti-CA(-like) piece (i.e. the bl + bg term) is additionally enhanced

likely. Namely, one can pinpoint several decays with vector meson production

their sign?). Fortunately, there is a place where such a cancellation is not

anti-CA contributions from lowest 1/2‘ excited states (maybe even change

is larger than flavour symmetry breaking). Such terms might reduce the small

standard current algebra contribution (obtained when 1/2‘ excitation energy

ate states. These terms (apart from their small overall size) would behave like

of the pole model one might expect small contributions from heavier intermedi

butions from the lowest 1/2" excited states only. Within the general framework

Our pole model description of parity violating amplitudes includes contri

ratios of Aj —-> E°K+ and Aj —> E+q5 as well as other processes.

amplitude around -13.5. The size of the latter is determined by the branching

ity violating amplitude around -0.85 (in units of 10‘7) and parity conserving

our approach we obtain a similar description of that decay channel with par

plitudes may Hip the sign of the asymmetry. It is therefore interesting that in

Cheng and Tseng argued then that small corrections in parity violating am

parity conserving amplitude of same sign thus yielding a positive asymmetry.

model calculation gives a small parity violating amplitude and a fairly large

algebra, despite the disagreement in the sign of the asymmetry. Their pole

that the pole model provides a better description of the data than current

This problem has recently been discussed by Cheng and Tseng [6] who argued

the already mentioned problem of the asymmetry of the Aj —» E+1r° decay.

Table 3. It is seen that there is a good agreement with the data apart from

in ref.[2]. Description of the data and predictions of the model are given in

M M 65 (in units of 10’7) to be compared with 45 in ref.[4] and M z 95 zh 20

14 OCR Output

question should be considered fairly reliable. An important case is the asym

considerably, our predictions for the approximate size of the asymmetries in

amplitudes. However, although the sizes of a and a' affect branching ratios

description. These terms dominate numerically the relevant parity violating

ent pair of reduced matrix elements ( a. and a.' , see ref.[4]) enters into the

torization terms can contribute should be treated with caution as a differ

Our description of those decays with vector meson production where fac

cancellation of contributions from diagrams

will occur in the vector sector as well) is reasonably well described by total

the cancellation in parity conserving amplitudes discussed in ref.{6] (which

the sign of asymmetries for these decays. The above argument assumes that

like contributions from heavier excited states should not be able to change

to large negative numbers (compare ref.[4]) of comparable size. Small CA

to +0.8) for all three decays listed in Eq.10 while current algebra corresponds

be reliable. The pole model predicts fairly large positive asymmetries (+0.5

therefore, the pole—model predictions for the signs of the asymmetries should

resulting from the 1/ (1 — (Ac/Aw)2) factor (see Eq.6). For these processes,

bz factors are identical and add coherently [14] lifting most of the suppresssion

For these decays there is no factorization contribution while the bl +bl and bl

AL··* $45 (10)" +

Aj __) E0K•+

29 —» 212*

ref. [4] one can see that the cleanest examples are provided by three decays:

interferes with them destructively). Upon examining Table 1 and Table 2 of

correction is present only in amplitudes with nonzero factorization terms and

15 OCR Output

(1979).

A.LeYaouanc, O.Pene, J .-C.Raynal, and L.O1ive1·, Nucl.Phys. B149, 321[7]

Phys.Rev.D1.

H.Y.Cheng and B.'I`seng, preprint ITP-SB-93-20, submitted to[6]

Phys.Rev.D1.

Breaking, preprint 1647/ PH IFJ Krakow, October 1993, submitted to

P.Zenczykowski, Weak Hypemn Decays: Quark Sea and S U { 3} Symmetry[5]

mitted to Phys.Rev.D1.

Charmed Baryons, preprint 1643/ PH IFJ Krakow, September 1993, sub

P.Zenczykowski, Quark and Pole Models of Nonleptonic Decays of[4]

Q.P.Xu and A.N.Kamal, Phys.Rev. D46, 270 (1992).[3]

H.Y.Cheng and B.Tseng, Phys.Rev.D46, 270 (1992).[2]

(1991).

(1992); J.G.K6rner and H.W.Siebert, Annu.Rev.Nucl.Part.Sci. 45, 511

Z.Phys.C2, 117 (1979); J.G.K5rner and M.Krimer, Z.P}1ys.C55, 659

J.G.Korner, G.Kramer, and J.Willrodt, Phys.Lett. 78B, 492 (1978);[1]

References

ref.

predicted positive and large, i.e. similar to the prediction of CA approach of

decays [15]. In our S U (4) breaking pole model approach this asymmetry is

mctry of the Af -—» pI?‘° decay which is of some interest in the analysis of Ab

18 OCR Output

Ar—»»F<<K> ll -;b;B+&¤Ml$B—;%¤sMl#§B+#zM-•0 ‘°

$512Aj —» z°K+(K·+) H $512

Aj ——> Avr+(p+) §M' $114* }M'

$5BAj —> E°1r+(p+) ¢B ;`%—B

Aj —» E+m(¢) $8 $5B mE

—éBAj —> E+T]g(w) $B

E§§BAj ——> E+1r°(p°) QB £7§B

process Byl BW

OCR OutputOCR OutputTable 2. Parity conserving weak amplitudes for Af —-> BM decays.

21 OCR Output

Fig.1. Quark diagrams for weak decays of charmed baryons.

(d1) (d2)

(b2)(b1)

(¤’)(¤)

OCR OutputOCR OutputJil L