Volume 218, number 3 PHYSICS LETTERS B 23 February 1989

D ~ n AS A T E S T O F FLAVOR A N D N O N E T S Y M M E T R Y IN C H A R M E D M E S O N DECAYS

S.P. ROSEN T-Division, Los Alamos National Laboratory, Los Alamos, NM 8 7545, USA

Received 21 November 1988; revised manuscript received 12 December 1988

We show that the standard model for Cabibbo-allowed decays of charmed mesons plus the nonet symmetry model for vector mesons lead to a small branching ratio for the mode Ds--,~ +. It follows that the observation of a large branching ratio would be a clear signal for a breakdown of nonet symmetry, or possibly for large flavor SU ( 3 ) breaking effects in charm decay.

The s tandard model gives rise to a specific set of flavor selection rules for the Cabibbo-a l lowed decays of charmed part icles [ 1 ]~. In isospace the interac- t ion behaves as an isovector, in U-spin space it trans- forms as a vector, and in the space of f lavor SU (3) it t ransforms as a l inear combina t ion of the 6* and 15 representations. The isospin selection rule should hold to a high degree of accuracy, being broken only by effects of order of the electromagnetic interaction; and in the the light of our experience with nonleptonic hyperon decays ~2, the SU (3) selection rules should hold at the level of 10%.

Nonet symmetry is an addi t ional assumption about vector and pseudoscalar mesons, inspi red by the al- most ideal mixing of the ~ and co mesons [4] . It has been assumed in all flavor SU (3) analyses of charmed meson decays with which this au thor is famil iar [ 1 ], and it has the effect of relating the coupling constants associated with SU (3) singlet mesons to those asso- ciated with the corresponding octets. Thus it leads to fewer independent ampl i tudes than are encountered in the most general S U ( 3 ) descr ip t ion of charm decay.

In this note, we point out that the decay mode Ds--,Qn serves to test the val id i ty of nonet symmetry, and possibly of the SU (3) rules themselves, for all relat ive admixtures of the 6* and the 15 in the effec- t ive decay hamil tonian. Our argument is based upon

~ For reviews of contemporary approaches to charm decay see ref. [2].

"-' For a review of nonleptonic hyperon decay see ref. [ 3 ].

a sum rule between the ampl i tudes for Ds--*~, and the I~*°r~ + and I~°p + decay modes of the D + meson. Since the branching ratios for the lat ter two modes are known, and since the ratio of the widths of the O + and Ds mesons is also known, we can use the sum rule to set an upper bound on the branching ratio of the Ds decay mode. Although this branching rat io has not been measured, we find that our upper bound is much smaller than the values discussed in the litera- ture [ 5 ]. Should this f inding be confirmed, then we will have to conclude that nonet symmetry is not valid for charmed part icle decay.

To obta in the sum rule, we construct the effective decay hami l ton ian by first coupling the final state pseudoscalar ( P / ) and ( V / ) meson nonets ( i , j = 1,

2, 3 ) to specific representat ions of SU (3) , and then coupling these representat ions with the charmed me- son tr iplet to form tensors which behave like the 6* and 15 representat ions. In the nonet model, the prod- uct of two nonets yields the same set of representa- t ions as the product of two octets, namely one singlet, two octets, one 10, and 10", and one 27. Because of the s tandard model selection rules for isospin and hy- percharge, the 10" final state can contr ibute to the decay, but the 10 cannot.

The specific forms of the relevant tensors are

[ 8s]j'= PaiVja w pjaVa i - ~O/(PV) ,

(PV) = PabVh" ,

[8A] f = p a i V j a - p j a v a i , (1)

0370-2693 /89 /$ 03.50 © Elsevier Science Publishers B.V. ( Nor th -Hol land Physics Publishing Divis ion )

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Volume 218, number 3 PHYSICS LETTERS B 23 February 1989

[10*] i f f '= r~ ~V b 16 at [8A]jb+ ( P ) V / ' - P / ' ( V ) )

+ ( a ~ b ) ] - (i*-*y),

(P) =Pk k ,

(V) =Vk ~ ,

[27 ],j"~'= [ P, a V / ' - ½6fl( [ 8s]jh+ ( P ) V / % P / ' ( V )

- ~ d / ' ( ( P ) ( V ) - ~ ( P V ) ) ) +aob] + ( i~ j ) , ( 1 cont 'd)

where numbers are used to denote representations and the subscripts S and A refer to the symmetric and anti- symmetric octet products of the P and V nonets re- spectively. The effective hamiltonian in the nonet symmetry case is given by

HNor~ET =$616"; 8s] +A616"; 8A] +T616"; 10"]

+S,5[ 15; 8s] +A~5[15; 8A] +T ,5 [ 15; 2 7 ] , (2)

where the notation [X; Y] denotes the overall rep- resentation X constructed from the charmed meson triplet and the representation Y of the PV final state. The (S, A, T)6, 15 are the coupling constants for the relevant terms in the effective hamiltonian.

When nonet symmetry breaks down, we pick up three additional representations in the direct product of the two nonets: a singlet from the product o f the singlet components of the nonets, and two octets from the product of the singlet in one nonet times the octet component of the other. Four additional terms must be included in the effective interaction

H;~n,:,~c~xc;=V6[6*; 8p, lv] +V~s[15; 8p, 1v]

+P616"; 1p, 8v ] -}-PI5 [ 15; 1p, 8v ] , (3)

where the notation indicates which terms are con- structed from the vector meson singlet ( I v = (V) ) , and which from the pseudoscalar singlet ( 1 p = (P ) ) . The contributions of all terms to the amplitudes for charmed meson decays into specific PV final states are shown in tables 1 and 2.

Now the final states of the D + decay modes under consideration both have negative strangeness and isospin 3/2, and hence they belong only to the 10" and 27 combinations of the PV system. In addition, the 0 meson is, in the nonet scheme, a pure strange- anti-strange quark-ant i -quark construct, and its cou- pling to the r~ + also only occurs in the 10" and 27 combinations [1] in the limit of nonet symmetry.

When we couple these two representations to the charmed meson triplet /34 ( i = l , 2, 3), we obtain only two contributions to the nonet symmetry ham- iltonian ofeq. ( 1 ), a 6* from the 10" and a 15 from the 27. Therefore, irrespective of the relative admix- tures of these two terms, the three amplitudes we are considering must obey a sum rule. From table 1 we find that sum rule to be

A (Ds- ,0~ + ) = ~A (D + ~K*°n + )

+ I A ( D + KO~+ . ) , (4)

where A ( D ~ X Y ) denotes the amplitude for the de- cay mode indicated in the brackets.

Eq. (4) holds only in the nonet symmetry limit. When the symmetry breaking terms of eq. (3) are included in the effective interaction, the amplitude for Ds--* 0~ picks up additional contributions from the V6 and V~5 terms (see table 2) and eq. (4) breaks down.

Care must be exercised in applying this sum rule because of the need to take final state interactions into account [ 6 ]. Since we are using effective amplitudes, the effect of final state interactions is to multiply them by a phase factor which depends on the phase shift appropriate for the particular final state. The sum rule should therefore be regarded as a triangular relation in the Argand diagram, rather than an algebraic one on a line; and without a knowledge of the phase shifts, we can only extract bounds on one of the amplitudes from the sum rule.

The branching ratios for the two D + decay modes in eq. (4) have been measured [7]

B R ( D + --, I(*°~ + ) = ( 5.9 _+ 1.9 _+ 2.5 )%,

BR ( D + - . I~°p + ) -- (6.9_+ 0.8_+ 2 .3)%, (5)

and the lifetime ratio for the two charmed mesons is ~3

r ( D + ) = 2 . 5 r ( D s ) . (6)

The phase space for PV decay varies as the third power of the center-or-mass momentum, and so the relevant ratios are

PS (I~*°Tr + ) : PS (I(°p + ) : PS (07r + )

=0 .45 :0 .39 :0 .37 . (7)

~3 For general reviews of the data on charm decay, see ref. [ 7 ].

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Table 1 Nonet symmetry amplitudes for D ~ PV.

PHYSICS LETTERS B 23 February 1989

Mode $6 A6 T6 S~s At5 Tl5

D+~I~*°rt + 0 0 - 2 0 0 2 D°--* I(*-/t + 1 1 - 2 / 3 1 1 4/5 D ° ~ l~*°rt° - 1/ , /2 - 1 / ~/~ - .f8 / 3 - 1 / V'5 - 1 / v/2 3 v/2/5 D+--,I~°p + 0 0 2 0 0 2 D°--,I~- p + 1 - 1 2/3 1 - 1 4/5 D° ~. I?,.°p + - 1 Iv/2 1/,,,/'2 .~8/3 - 1/v/2 1/x/~ 3~//2/5 Ds--,K+K *° - 1 1 - 2 / 3 1 - 1 4/5 D~-,K*+I~ ° - 1 - 1 2/3 1 1 4/5 Ds ~. p +q8 - x, 2 / ~ 0 - V/2/3 v/2/~ 0 - 2V/6/5 D~-,p+lh - 2 / . / 3 0 0 2/v./3 0 0 D ~ n + 0 0 0 - 2 / 3 0 0 4/5 D,--,n+co - , / 2 0 V/2/3 V/2 0 -2.v/2/5 D°.K*°q8 - 1/v/6 v 3/~/2 0 - 1 IV/'6 x/'3/'~ x//6/5 D °--> K*°11, 2 / ,//3 0 0 2 / x/"3 0 0 D°--+ K°o) 1/,//2 - 1/x.,'~ 0 1/.//'2 - 1/~./2 . ,d/~ D ° i~oO 1 1 0 1 1 - 2 / 5

Table 2 Nonet symmetry breaking amplitudes for D-, PV.

Mode V6 V L 5 P6 P~ s

D~ --'P +qt 0 0 - 1 1

D~-~+, - 1 / \,,,'~ 1/v"3 o o D ~ + ¢ o -V /2 /3 ,£'2/3 0 0 D (~ ~ l~*°rl, 0 0 1 1 D°--+ 17,2°co V/2/3 v./2/3 0 0 D°-+ I~° 0 1/.¢/3 1/VI3 0 0

Using all o f this i n f o r m a t i o n in eq. (4) , we ob ta in a b o u n d on the Ds decay mode:

Br(D~ --,0n + ) ~< 0 .6%. (8)

This b o u n d is cons iderab ly smal ler t han the gen- erally expected [ 5 ], though as yet u n m e a s u r e d , va lue of ( 2 - 4 ) % . The only way in which it can be increased while preserv ing the n o n e t s y m m e t r y s u m rule is to increase greatly the D + b r a n ch i n g ratios, especially

that for the I~*°rt + decay mode . Since this is unl ikely, we conc lude that i f the expected value o f the b ranch- ing rat io is i ndeed correct, t hen there mus t be large none t symmet ry ( a n d possibly f lavor S U ( 3 ) ) break- ing effects in cha rm decay.

The s imples t way o f b reak ing n o n e t s y m m e t r y is to

inc lude the V6 and V~s terms o f e q . (3) in the effec- t ive in terac t ion . These terms are associated with the SU ( 3 ) singlet c o m b i n a t i o n (x/2c0 + ~ ) / x f 3 of vector mesons a n d , given the large difference be tween the b o u n d in eq. (8) and the expected b r anch ing rat io o f a few percent , they mus t make the d o m i n a n t contr i - b u t i o n to the ampl i t ude for Ds - , 0n . They also make a s ignif icant c o n t r i b u t i o n to Ds-,~ort but , because of con t r i bu t ions form octet f inal states, they m ay not d o m i n a t e this amp l i t ude [ 9 ].

We can check none t symmet ry for pseudoscalar mesons by m eans of the analogue of eq. (4 ) for

D- - ,PP decays:

A (Ds - , r t + (rh - v /2qs ) /x f l3 ) = ~A(D + -- ,z+I~°) •

(9)

Taking in to account the compl ica t ions of q - q ' mix- ing [10] , f inal state in teract ions , and phase space factors, we ob t a in a re la t ion be tween the b ranch ing rat ios for the Ds decay modes:

11.6 e i ~ / B R ( D s ~ r t + q ' ) - ~ /BR(D~ ~ t + q ) I

=0.8, (lO)

where o¢ is a f inal state in te rac t ion phase and we have taken the D +--, rt + I~ ° b ranch ing ratio to be 3.2% [ 11 ]. It appears f rom a recent expe r imen t [ 12 ] that bo th b r anch ing rat ios in eq. (10) are m u c h greater t han

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Volume 2 18, number 3 PHYSICS LETTERS B 23 February 1989

1% and that the n’ final state is much more copious than the rl one; thus it is unlikely that eq. (10) will be satisfied and that nonet symmetry is obeyed in D+PP decays.

Our bound on the branching ratio for D,+@t has definite implications for quark-diagram models of charmed meson decay [ 2,13,14 1. Generally speaking these models predict branching ratios of a few per- cent, and hence exceed the bound of eq. (8) by a sig- nificant amount. They must therefore violate nonet symmetry.

We can demonstrate this explicitly by using the general quark-diagram amplitudes given by Chau and Cheng [ 21. We find that all the sum rules implied by the AT= 1 and AU= 1 selection rules of the standard model [ 1 ] for D+PV and D+PP decays are satis- fied in the quark-diagram model, but that relations like eq. (4), which depend explicitly upon the addi- tional constraint of nonet symmetry, are not satis- fied. In terms of independent amplitudes. the quark- diagram model lies part way between nonet symme- try and simple flavor SU (3 ). For PV decay modes, nonet symmetry leads to six independent amplitudes (see eq. (2 ) ), and flavor SU (3 ) alone to ten (see eqs. (2) plus (3)); the quark-diagram model leads to eight. Similarly for PP decay modes, nonet sym- metry yields three independent amplitudes, SU (3 ) alone yields five, and quark-diagrams four. There- fore some trace of nonet symmetry will show up in the quark dynamics approach, but not in the form of

eq. (4).

The author thanks L.L. Chau, V. Novikov, I. Bigi, and M. Witherell for stimulating discussions and for their interest in this work. He is also grateful to R. Schindler for supplying him with the latest data on charm decay. Part of this work was carried out at the

Aspen Center for Physics and the author would like to thank L.M. Simmons Jr. and Paul Fishbane for the hospitality of the Center.

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