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