Volume 228, number 4 PHYSICS LETTERS B 28 September 1989

FLAVOR S E L E C T I O N R U L E S A N D T W O - B O D Y DECAYS O F C H A R M E D M E S O N S I N T O P S E U D O S C A L A R P L U S VECTOR M E S O N S

S.P. ROSEN Theory Division. Los Alamos National Laboratory. Los Alamos, NM 87545, USA

Received 27 June 1989

Wc analyse the measured branching ratios for the Cabibbo-allowed decays of charmed mesons into pseudoscalar plus vector mesons in terms of the flavor SU(3) selection rules of the standard model plus nonet symmetr)'. Our fit to the invariant ampli- tudes for the 6* and 15 interactions indicates that the effective hamiltonian involves comparable admixtures of the two SU(3) representations. The branching ratio for D~--,~n + is found to be 2.5% and that for D° -,l(*°q is close to its upper limit of 2.9%; the branching ratios for the K*+I( ° and p+r I decay modes of the D~ are predicted to be 4.3% and 5.8% respectively.

F rom the v iewpoint o f the f lavor SU (3) symmetry associated with u, d, and s quarks, it is interest ing to ask whether the Cabibbo-a l lowed decays of charmed part icles follow the same pat tern as strange pan ic le decays and are domina ted by a single representat ion o f SU (3) [ 1 ]. The dominance o f one representat ion, the octet in the case of strange particles, usually leads to s imple relat ionships between the decay modes o f part icles within the same mult iplet , and these rela- t ionships can be directly tested in exper iment --t. In this paper, we shall examine the quest ion for the spe- cific case of charmed meson decays into pseudosca- lar plus vector mesons ( D ~ P V ) .

Candidates for dominan t representation in the case of Cabibbo-a l lowed charm decay are [ 1 ] the 6* and 15. In general the s tandard model requires the inter- action to be a l inear combina t ion of the two, and we shall be asking whether the invar iant ampl i tudes as- sociated with one of them are significantly larger than those o f the other. There is a hint from inclusive de- cays that this might indeed be the case: the l ifetimes [ 3 ] of the D O and the D~ are approximate ly equal to one another , and they arc both significantly shorter than the l ifet ime for D ÷. Dominance of the 6* could explain both of these features, while dominance o f thc 15 is consistent with the near equali ty o f the first two lifetimes.

~ See ref. [2] for a review ofnonleptonic hypcron decay.

The question, then, is whether the hind carries over to exclusive decays. It is possible that, as in the case o f strange particles [2] , the same representat ion dominates each o f the various decay channels. There are, however, many more channels avai lable for charm decay than for strange decay, and the pat tern of dominance for some channels may be quite differ- ent from thc pat tern for others, in which case the proper ty of the l ifetimes may be an accidental feature rather than a clue to underlying characterist ics of the interaction. Obviously we have to study this aspect o f charm decay on a channel-by-channel basis.

Besides flavor SU (3) selection rules, we make use of nonet symmetry in this discussion. Nonet sym- metry, inspired by the almost ideal mixing o f the and to mesons [4] , has been assumed in previous fla- vor S U ( 3 ) analyses [1,5] and it relates coupling constants associated with S U ( 3 ) singlet mesons to those associated with the corresponding octets. While nonet symmet ry is most appropr ia te for vector me- sons, we shall use it as well for pseudoscalar mesons unless we find a clear contradic t ion with experiment . As a result, we have four fewer independent ampli- tudes than would bc encountered in the most general SU (3) descr ipt ion of charm decay [ 5 ].

It should be noted that in an earl ier paper [ 5 ] we used the decay mode D~--,@n to argue against the va- l idi ty o f nonet symmetry, and possibly against the S U ( 3 ) rules themselves. Our argument was based

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upon a sum rule between the amplitudes for D~--,¢n and D +--,I~*°n + and D ÷ ~I ( °p + which implied that the branching ratio for the D~ decay mode must be less than 0.6%. Although this branching ratio has not been measured, it is generally believed [6] to lie be- tween 2% and 4% and hence either nonet symmetry or the SU(3) selection rules would have to fail. For- tunately, Kohara [ 7 ] has observed that one invariant amplitude was omitted from our analysis, and that when it is included, the sum rule is no longer valid. Thus the paradox of the earlier paper is resolved and we are still able to use nonet symmetry. The original argument will be still useful in discussing the general question of a dominant representation, and the par- ticular Ds decay mode will play an important role in the choice of phenomenological parameters describ- ing D ~ P V decays.

As before [5], we construct the effective decay hamiltonian by first coupling the final state pseudo- scalar (P/) and vector (Vi i) meson nonets ( i , j= 1, 2, 3 ) to specific representations of SU (3), and then coupling these representations with the charmed me- son triplet to form tensors which behave like the 6* and 15 representations. In the nonet model, the prod- uct of two nonets yields the same set of representa- tions as the product of two octets, namely one singlet, two octets, one 10, one 10", and one 27. The 10" cou- ples with the triplet to form a 6* but not a 15, while the 10 an 27 can only form 15 tensors.

The spccific forms of the relcvant final state ten- sors are given in our earlier paper [5] with the sole exception of the 10, which takes the form [ 7 ]:

[ 10],j~b= {PY ~ - ~OY( [8Al l + (P) V/'--Pjb(V) )

- ( a ~ b ) } + (i,--,j),

(P)=Pk k, (V)=Vk k, ( l )

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

H,o.~, =$616"; 8s] +A616"; 8A] +76[6*; 10"]

+Sj5115; 8s] +A,5115; 8A] q-T',5115; 10]

+I'j5115; 27] , (2)

where the notation [X; Y] dcnotes the overall repre-

scntation X constructed from the charmed meson triplet and the representation Yof the PV final state. The (S, A, T, T')6.~5 are the invariant amplitudes (coupling constants) for the relevant terms in the ef- fective hamiltonian.

The contributions of all the terms in the effective hamiltonian, eq. (2), to specific PV final states are shown in table 1. Besides having a larger number of channels open to it, charm decay also differs from strange-particle decay in the matter of final state in- teractions. In strange decay final state interactions tend to be small and can be neglected, whereas in charm decay they become important because the masses of charmed particles fall in a region occupied by many meson resonances. This has already become apparent in two-body decays of charmed mesons, for which large phases have to be invoked [8] in order to fit the observed branching ratios to the AT= 1 sum rule of the standard model [ 9 ]. Accordingly, we shall treat all decay amplitudes and the invariant ampli- tudes ofcq. ( 1 ) as complex numbers; and we attrib- ute the phases to final state intcractions rather than a breakdown of time-reversal or CP invariance. Such phases will play a key role in our analysis.

Let us now consider the relationship between Ds--,(~n and single representation dominance. Be- cause of the standard model selection rules for iso- spin, charm, and hypercharge ( A T = A T 3 = - A C = - -AY= 1; A= [final] - [initial] ) and the properties of triangular [ 10 ] SU ( 3 ) representations, the 10* fi- nal state can contribute to the decay of the D +, but the 10 cannot. Both contribute to D~(~n, and hence the new terms in eqs. ( 1 ) and (2) invalidatc the sum rule discussed above. Moreovcr, in order to avoid a small branching ratio for the Ds decay mode, we ex- pect the new invariant amplitude, T'~5 in eq. (2), to be relatively large. This immediately rules out the possibility of 6* dominance for D--.PV decays and leaves us with either 15 dominance, or a roughly equal admixture of the two representations as the only pos- sibilities for the effective interaction.

To investigate these possibilities we need to study the amplitudes for the D + and the D5 decay modes more carefully. Many of the branching ratios for the D + and D ° decay modes have been measured to a reasonable degree of accuracy and so we can use this data to extract the invariant amplitudes of table 1, in particular 7"6, T~5 and 1"~5. In general there will be a

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Table 1 Nonet symmetry amplitudes for D--,PV

PHYSICS LETTERS B 28 September 1989

Mode $6 A6 T6 Si5 AI5 Tt5 T]5

D+ ~l~*°n * 0 0 - 2 0 0 2 0 D°-)I~*-~ + I 1 - 2 / 3 1 1 4/5 - 2 / 3 D°~I~*°n ° - 1 / , ~ - l / , f2 -~/8/3 - 1 / , / 2 - 1/,/2 3d2/5 ,f213 D + --, l~°p + 0 0 2 0 0 2 0 D°--, 1~- p + 1 - 1 2/3 1 - 1 4/5 2/3 DO , opo 3, /5 D~--* K + I~ *° - 1 + 1 - 2 / 3 1 - 1 4/5 2/3 D,~K**I~ ° - 1 - 1 2/3 1 1 4/5 -2 /3 D,--,p+rls - ~ 2 / 3 0 - , , 2/~/3 \ : / ~ 0 -2 , / 6 /5 - v 2/-~/3 Dc-~p+rh - 2/x/~ 0 0 2/,,F3 0 0 0 D ~ n + 0 0 0 - 2 / 3 0 0 4/5 --2/3 D,--,n+to - , / 5 0 -,/-_2/3 , ,~ 0 -2x/2/5 V~_/3

D°-, I~.*°rh 2/x/~ 0 0 2/(/5 0 0 0 D°--,K°to llx/2 -l lxi~ o 11,/~ - l l , f2 ~/215 -,i513 D°-,R% 1 1 0 1 1 - 2 / 5 213

_ ,i] _v~_2/3 D,-->rt°p "" 0 ,4/~2 / 3 0 . ~ 0 D,--)~+p ° 0 - - / 2 / 3 0 V 2 0 ,/;2/3

family of solut ions with differing predic t ions for the Ds--,~n branching ratio. By limiting ourselves to those solutions that give a large branching ratio, we may hope to elucidate the SU (3) behav iour o f the effec- t ive interact ion.

We choose units for the ampl i tude A ( D ~ X Y ) by expressing the branching rat io for the decay mode in the form

BR ( D - + X Y ) = I A ( D ~ X Y ) 1 2 × F ( X Y ) / F ( D ) , (3)

where F ( X Y ) is a phase space factor propor t ional to the cube of the center-of-mass momen tum and F ( D ) is the total width of the parent D-meson. In view of the empir ical relation [ 3 ]

F ( D °) = F ( D , ) = 2 . 5 F ( D + ) , (4 )

we can express all ampl i tudes as mult iples o f Go, the square-root of the D O width:

G o = x f ~ D ° ) . (5)

Measured branching rat ios and phase space factors are shown in table 2. For our purposcs, it will be suf- ficient to use the central values of the branching ratios.

To establish the family of solut ions for the invar- iant ampli tudes , we begin with the two decay modes of the D ÷ meson (see table l ):

A ( D + --, I~° p + ) =2T6 +2Tt5 = A ~ ,

A (D + ~ I~*°Tt + ) = - 2 T 6 +2T,5 = A 3 . (6)

Both decay modes have pure isospin 3 in the final state and this is indicated by the nota t ion A3 and A~ at the end of cq. (6) . Now the branching ratios for these two modes are almost equal to one another (ta- ble 2): therefore the corresponding ampl i tudes are almost equal in magnitude, but may differ in phase. Adopt ing a convent ion in which the first ampl i tude ofeq . (6) is real, we use the informat ion in table 2 to write

A~ =2T6 + 2 T l s =0 .26G0 ,

A3 = - 2 T 6 +2T15 =0 .23 exp (i~u)Go. (7)

The phase ~u defines a one-parameter family o f solu- t ions and it has a clear physical meaning: when ~, is close to 0 °, T) 5 must be >> than 76; when q/is close to 180", T) 5 << T6; and when ~u is 90 o, the two ampli- tudes are approximate ly equal in magnitude, but 90 ° out o f phase.

Our next step is to de termine the isospin ½ ampli- tudes, A'~ and A~ from the I~p and I~*n decay modes o f the D o meson. Using the s tandard analysis of the A T = 1 rule [9] and the branching ratios in table 2, we obta in

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Volume 228, number 4 PHYSICS LETTERS B 28 September 1989

Table 2 Measured branching ratios and phase space factors for D- ,PV. (Central values of branching ratios arc quoted and error bars are not taken into account.)

Mode Branching ratio Reference Phase space (%) F (XY)

D + --* I~*°~ + 6 D°-M~*-n + 4.8 D °--, l~*°n ° 2,1 D + ~I~°p + 6.6 D O. ~I(-p + 8.2

D °--, I~°p ° 0.7 D~-~K+I~ *° ~.01t Ds-,K*+I~ ° D ~ p + r l D~-~p+rl ' D~--,n+0 2-4 D~-n+co <0 .5 (¢n ) D O ~ I(*°~ < 2.9 DO-, l~*°W D O -* K°t9 4.2 D ° ~ g ° O 0.8 D~--. n°p + ~ 0 D ~ n + p ° ~ 0

(8) 0.45 (3) 0.45 (3) 0.45 (3) 0.39 (3) 0.39 (3) 0.39 (3) 0.33

0.33 0.4 0.11

(6) 0.37 13) 0.58 l l ) 0.24

0.002 11) 0.37 (3) 0.17 (3) 0.59 (3) 0.59

3 A, = -~(So +Ac, + S,5 +A,s)+½xf2 T,5-x/2 T'~5

=0.638 exp [i(~,+ 77)°] Go,

A', = -~2 ( S6-A6 + S,,-A~5) + ~x/~ TI, +,v/2 T'I,

=0.789Go. (8)

The sum of the two parts of eq. (8) gives us an expression for the combination ($6 + S~ 5 ) in terms of T~ 5, which in turn can be obtained from eq. (7); and the difference gives us an expression for a combina- tion of (A6+A15) and T]5. Since we need the latter amplitude in order to calculate the branching ratio for D ~ n , we must use at least one more piece of information.

We obtain this information from the measured branching ratios for D O into [ 11 ] I~°o~ and [3] I~°~. The values in table 2 yield the magnitudes of the cor- responding amplitudes; the phases are taken, for the moment, to be arbitrary. Thus we find:

A ( K ° c o ) = ~(36-A6+315-a15) r , , - r ' , ,

=0.337 exp(i0)Go, (9)

528

A ( I ~ ° 0 ) = (S6+A6+SIs+AIs) - - ~ T l 5 + ] T 1 5 2 "

=0.221 exp(i0' )Go • (9 cont'd)

Taking the sum of x/~ times the first part of eq. (9) plus the second part, we obtain a second expression for ($6+St5). Equating this to the expression ob- tained from eq. (8), we arrive at a consistency condition,

0.477 exp(i0) +0.221 exp(i0 ' )

=0.355+0.298 exp[i(~,+ 80) °] , (10)

which enables us to determine the angles 0 and 0' once the angle ~' has been fixed. Because of the magnitudes of the complex numbers in eq. (10), ~ is restricted to lie between - 2 1 3 ° and +53°; typical values of the three angles are shown in table 3.

Eqs. (9) and (10) now give us enough informa- tion to determine the amplitude T'js, and hence the amplitude for D ~ 0 n :

A(D~--* ~m) = [0.077 exp(ku)

+0.238 exp( i0) -0 .186] Go. (11 )

From this result and the values for 0 in table 3, we find that the branching ratio for D ~ Q n can only be

Volume 228, number 4 PHYSICS LETTERS B 28 September 1989

Table 3 Phases of the amplitudes for D°~ l~°oa, ¢ as a function of the phase for D + --* I~*°n +

(deg) 0 (deg) 0' (deg)

53 55 235 30 21 150 0 10 107

- 3 0 2 72 -60 - 6 43 -90 -18 28

- 120 -36 2 4

- 1 5 0 - 5 6 31

- 1 8 0 -63 55 -210 -71 88 -213 - 55 125

reasonably large when gt is close to - 180 °. When the angle is zero, the branching ratio is 0.6%, and when it is 90 °, the branching ratio is vanishingly small; but when ~ is - 180 °, the branching ratio turns out to be 2.5%, a value consistent with expectations [6] . We shall therefore work with this value o f ~, for the rest o f our analysis. Amongst other things it implies that T 6 is much greater than T~5.

Having fixed the parameter ~v, we are able to make predictions about the remaining decays o f the D o , namely D°-o I~*°rl,~ '. In the latter case the phase space is so small as to make the decay mode virtually unob- servable; in the former, we predict a branching ratio

BR (D°-oI~*°rl) = 3.7% f o r ( - 10°) ,

=2 .8% f o r ( - 2 0 ° ) , (12)

for the "qs, q~ mixing angles [ 12] shown in parenthe- ses. The ( - 10 ° ) value is slightly higher than the re- cent upper bound [ I0] o f 2.9%, but the ( - 2 0 ° ) one is just within it. Thus we anticipate that D°--,I~*°lq must be on the verge o f being detected. Should the decay mode not be seen at the level indicated by eq. (12), then we will have to conclude that nonet sym- metry for the pseudoscalar mesons is not valid and add appropriate terms to the effective interaction [ 5 ].

Except for the mode already discussed, the decays o f D~ depend on the differences o f the invariant am- plitudes (86, S15 ) and (A6, Ais) rather than their sums, and so we need additional information in or- der to incorporate them into our analysis. The data we shall use arc [ 3,13 ]

BR(D~-,I~+p °) ~ 0 ,

BR(D~ ~ K + I ~ *°) ~ BR(D~ ~ 0 ~ ) ,

BR (D --,Tr+o~) ~< 0.5BR (D~--, 07t). (13)

The first part o f eq. (13) allows us to relate the difference o f the A-amplitudes to the T-amplitudes

3 ( A 6 - A I s ) = - T6 + Tqs . (14)

The second relates the corresponding amplitudes up to an unknown phase o:.

A (D~ ~ K + I ~ *°) = 1.06 exp( ia )A ( D s ~ ~ ) ; (15)

and the third part restricts ~ to a narrow range of an- gles in the third quadrant:

215° ~<~<236 ° . (16)

From these conditions we then find an expression for the difference of the S-amplitudes:

($6-S~5) = - [1 .5+ 1.06 exp( i a ) ]A (D~ ~ 0n)

+0 .23 exp(i~,)Go. (17)

where the amplitude A (D~-- .~) is given in eq. ( 11 ). We have enough information at this point to deter-

mine all the invariant amplitudes, and to make pre- dictions about the remaining D~ decay modes. Tak- ing a to be in the middle of its range, namely 225 °, we predict that

BR(D~--.K*+ I~ °) = 1.7BR(Ds ~ 0 n ) = 4 . 3 % ,

BR(D5 ~p+TI) = 2.3BR(Ds--,¢~) = 5 . 8 % ,

BR (D~ ~p+rl ' ) = (0 .1 -0 .2 )%. (18)

where we have taken B R ( D ~ ¢ ~ ) to bc 2.5%. The first of these predictions is a test of honer symmetry for vector mesons, and the second two test nonet symmetry for pseudoscalars; should the latter fail, we can add appropriate terms to the effective hamilto- nian [5].

The magnitudes and phases of the invariant ampli- tudes corresponding to ~ = - 180 ° and tr as in eq. (16) are

$6=0.11 e x p ( - i 2 8 ° ) , S t s = 0 . 1 1 e x p ( - i 6 1 ° ) ,

A6=0.11 e x p ( - i l 3 0 ° ) ,

A~5 =0.073 exp( - i 1 9 7 ° ) , (19)

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Volume 228, number 4 PHYSICS LETTERS B 28 September 1989

T6=0.12, T~5=0.008, (19cont 'd)

and

T]5 =0.34 exp(i70 ° ) . (20)

Clearly, T'~5 is the largest single amplitude, but there are significant contributions from all three 6* ampli- tudes and from S~5 as well. Thus we cannot conclude that one single representation dominates the PV de- cays of D-mesons; instead it appears that the 6* and the 15 are comparable with one another in the PV channel of charm decay. It will be interesting to see how other channels behave.

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[2] S.P. Rosen and S. Pakvasa, in: Advances in Particle Physics, eds. R. Cool and R. Marshak, Vol. 2., (Interscience, New York, 1968) p. 473.

[3 ] Particle Data Group, G.P. Yost el al., Review of particle properties, Phys. Lett. B 204 ( 1988 ) 1.

[4] S. Okubo, Phys. Lett. 5 (1963) 105; S. Gasiorowicz, Elementary particle physics (Wiley, New York, 1966) p. 326-327.

[5] S.P. Rosen, Phys. Lett. B 218 (1989) 353. [6] Y. Kohara, Phys. Left. B 228 (1989) 523. [7] A. Chen ct al., Phys. Rex,. Left. 51 (1983) 634;

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