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Volume 85B, number 1 PHYSICS LETTERS 30 July 1979 NONLEPTONIC DECAYS OF CHARMED MESONS AND DETERMINATION OF WEAK INTERACTION MIXING ANGLES * Mahiko SUZUKI Department of Physics and Lawrence Berkeley Laboratory, Untversity of California, Berkeley, CA 94720, USA Received 21 May 1979 By the U-spin property of SU(3), we find one important constraint on the generalized Cabibbo angles by the decay branching rattos of DO~ K-~r ÷, K-K+, and n-n*. A substantial deviation of I'(D° ~ K-K+)/I'(D ° ~ 7r-Tr +) from unity would have a significant implication on the coupling of the charmed quark to heavier quarks of charge -1/3. It is of the utmost interest how many heavy quarks will appear in the energy range that we are going to probe in the future. Determination of mixing angles in the charged weak current has a crucial relevance to this question. The semileptonic decays of charmed particles will be able to fix the angles most directly, but there is a severe lim- itation on their accuracy for the moment. The nonleptonic decays have been measured more accurately for the Cabibbo-aUowed processes. When they are combined with the Cabibbo-forbidden processes, we can determine, without any dynamical assumption on strong interactions beyond SU(3), one of the weak mixing angles. In par- ticular, we focus on the two-pseudoscalar meson decays of the charmed mesons for this purpose. The nonleptonic weak hamiltonian is written as itw = C J", J. = .... ) Yr. , i, L (1) where U is the unitary matrix of quark mixing and qL = (1 - 75)q. In the following we write the upper left 2 × 2 part of U as o0u n0u :1 U = -sin Ocd cos Ocs , and therefore J. = K3'.(d cos 0ud + s sin 0us)L +~3,u(-d sin 0cd + s cos 0cs)L + .... (2) In the four-quark model, cos 0ud = cos 0cs = cos 0 C and sin 0us = sin 0ca = sin 0 C with the Cabibbo angle 0 C given by sin 20 C -~ 0.05. In the six-quark model, they are written by the Kobayashi-Maskawa angles as [I] * Work supported by the National Science Foundation under Contract Number PHY-77-23512 and in part by the US Department of Energyunder Contract Number W-7405-ENG-48. 91

Nonleptonic decays of charmed mesons and determination of weak interaction mixing angles

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Volume 85B, number 1 PHYSICS LETTERS 30 July 1979

NONLEPTONIC DECAYS OF CHARMED MESONS

AND DETERMINATION OF WEAK INTERACTION MIXING ANGLES *

Mahiko SUZUKI Department of Physics and Lawrence Berkeley Laboratory, Untversity of California, Berkeley, CA 94720, USA

Received 21 May 1979

By the U-spin property of SU(3), we find one important constraint on the generalized Cabibbo angles by the decay branching rattos of D O ~ K-~r ÷, K-K +, and n-n*. A substantial deviation of I'(D ° ~ K-K+)/I'(D ° ~ 7r-Tr +) from unity would have a significant implication on the coupling of the charmed quark to heavier quarks of charge -1/3.

It is of the utmost interest how many heavy quarks will appear in the energy range that we are going to probe in the future. Determination of mixing angles in the charged weak current has a crucial relevance to this question. The semileptonic decays of charmed particles will be able to fix the angles most directly, but there is a severe lim- i tat ion on their accuracy for the moment . The nonleptonic decays have been measured more accurately for the Cabibbo-aUowed processes. When they are combined with the Cabibbo-forbidden processes, we can determine, without any dynamical assumption on strong interactions beyond SU(3), one of the weak mixing angles. In par- ticular, we focus on the two-pseudoscalar meson decays of the charmed mesons for this purpose.

The nonleptonic weak hamiltonian is written as

i tw = C J", J . = . . . . ) Y r . ,

i , L (1)

where U is the unitary matrix of quark mixing and qL = (1 - 75)q. In the following we write the upper left 2 × 2

part of U as

o0u n0u :1 U = -sin Ocd cos Ocs ,

and therefore

J . = K3'.(d cos 0ud + s sin 0us)L +~3,u( -d sin 0cd + s cos 0cs)L + . . . . (2)

In the four-quark model, cos 0ud = cos 0cs = cos 0 C and sin 0us = sin 0ca = sin 0 C with the Cabibbo angle 0 C given by sin 20 C -~ 0.05. In the six-quark model, they are writ ten by the Kobayashi -Maskawa angles as [ I ]

* Work supported by the National Science Foundation under Contract Number PHY-77-23512 and in part by the US Department of Energy under Contract Number W-7405-ENG-48.

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Volume 85B, number 1 PHYSICS LETTERS 30 July 1979

COS 0ud =COS01 , COS 0cs= COS 01 cos02 cos 03 - s i n 0 2 sin03 e 1~ ,

s in0us=Sln01 cos 03 , sin 0cd = Sin 01 cos 02 .

We break down the charm-changing part o f H w m the following way:

H w =HW(AISI = 1) +HW(AS = 0 ) ,

HW(AjSI = 1) = ((fid)(gc)cos 0ud COS 0cs -- (~s)(dc)sin 0us sin 0cd) + h.c. ,

Hw(2~S = 0) = ((:s)(gc)sin 0us cos 0es - (~d)(dc)cos 0ud sm 0cd) + h.c. ,

(3)

(4) (5)

(6)

where the Darac "),-structure IS suppressed. The Cabibbo-suppressed part is split further into two parts, one sym- metric and the other antisymmetric under the interchange of s and d,

Hw(AS = 0) = H w - ( A S = 0) + Hw+(AxS = 0 ) , (7)

Hw+-(AS = 0) = ½ ((Es)(gc) + (Ed)(dc)) (sin 0us cos 0cs -Y- cos 0ud sin 0cd ) . (8)

We can derive an important sum rule among the decay amplitudes for D O ~ K - n +, K - K +, and zr-n + from the structure exhibited above. Hw-+(AS = 0) is even (odd) under interchange of s and d, and therefore under a ~7 ro- tation by 180 °. Note that in the four-quark model Hw+(&S = 0) = 0 and therefore - M ( D 0 ~ K-K+)/M(D 0 -+ 7r-n +) = 1 [2] because K -+ -~ 7r -+ under a ~7 rotation by 180 °. In any model of more than four quarks, however, both H W + ( ~ = 0) exist and the ratio is not unity any longer. Although this ratio depends on strong interaction dynamics, the difference of the amphtudes M(D 0 ~ K - K +) - M(D 0 ~ 7r- lr +) is determined only by H w- (AS = 0). Then, it can be shown that the Cabibbo-allowed decay amphtude M(D 0 - K-Tr +) is simply related to this differ- ence by the U-spin invanance" Make a k7 rotation by 90 ° for the difference

AM = (K- K + - 7r - 7r+ I H w- (AS = 0)}D °) c~ ( K - K + - 7r - rr + I(fis)(~c) - (~d),(dc)lD0) •

We obtain

AM c~ (Tr-K + + K-Tr + I(Es) (dc)+ (~d)(~c)lD 0)

= (rr+K - IC6d)(~c)lD 0) + (K+rr - I(~s)(dc)lD 0) = 2(Tr+K - I(ffd)(gc)lD0), (9)

where we have made a k 7 rotation by 180 ° in the second term after the first equal sign o feq . (9). We have thus proved the sum rule

M(D0 ~ K - K + ) - M ( D 0 '* 7r-Tr+) = M(D0 ~ K - n + ) (10) cos 0cs sin 0us + cos 0ud sin 0cd cos 0ud cos 0cs

Though no positive detection has been made so far, D O -~ K+Tr - is possible with a severe suppression of its rate by 2 2 a factor of ~sin 0us sin 0cd. This mode is related to (10) as

(10) = - M(DO ~ K+Tr-) (11) sm 0us sin 0cd

The same U-spin property o f H w leads to many other, less important, sum rules by the SU(2) algebra. One could also use for this purpose the tensor representation

M ( A ~ S : O ) = : (al(M'~Ma 3 1 - - M~M2a)D 1 +a2(Ma3M~ _ MaM2)D2 1 a + a3(M]Mla _ M2M1)D a +a4(M~D 3 _M2Da 2)Ma}l

× (cos 0cs sin 0us + cos 0ud sin 0cd )

1 , a 3 a 2 1 ' 3 1 2 1 a , 3 1 2 1 a a~I(M]D 3 +M~D2)M 1) + : ~ a I ( M 3 M a + M 2 M a ) D +a2(MaM 3 +MaM2)D +a3(M3M a + M 2 M a ) D +

X (cos 0 cs sin 0us - cos 0ud sin 0 cd) + a5 Mg M a b D 1 + a 6 Mg MID b , (12)

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Volume 85B, number 1 PHYSICS LETTERS 30 July 1979

M(AIS[ 1)= a 3 1 3 1 a 3 1 a+a4M~M1D3 ) 0udCOS = { a l M 2 M a D + a 2 M a M 2 D + a 3 M 2 M a D cos 0cs

a 2 1 2 1 a + a 3 M 2 M 1 D a a 1 2 - {a 1M3MaD + a2MaM3D + a4M3MaD } sin 0us sin 0 c d , (13)

where M is the (3 × 3) pseudoscalar octe t and D a is the charmed meson triplet (D 0, D - , F - ) . The results are tab-

ulated in table 1. The sum rules, eqs. (10) and (11), can be readily read o f f there.

With the branching rauos defined as

= r(Do -, K- K +) {P. 1 e = r(D° - " "-"+) {PK ] _ r(Do -, K +. - ) r r ( D 0 ~ T r _ T r + ) \ p K / , F ( D 0 ~ K _ T r + ) \ p , r / , co P ( D 0 ~ K - n + )

eqs. (10) and (11) become

I = _+ Vce(x/-r- + 1) _ + ~

COS 0ud COS 0cs COS 0cs sin 0us + cos 0ud sin 0cd sin 0us sin 0cd '

(14)

(15)

ignoring the relative phase o f the final-state interact ions. I f sin 0cd/COS 0cs = tan 0 C, the plus sign is to be chosen

in x / ~ - + 1 o f e q . (15).

Table 1 Decay amplitudes in terms o f a i 0 = 1 . . . . . 6) and at (i = 1 . . . . . 4). The tabulated numbers are the coefficients of the invariant am- plitudes. For a l , a2, a3, and a4, multiply the (AS = 0), (AS = - 1), and (AS = + 1) amplitudes of (D ÷, D °, F ÷) decays with -(cos Ocs

i r t I

X sin 0tt s + sin 0cdCOS 0ud), cos 0udCOS ecs, and -sin 0ussm 0cd, respectively. Note that the a l , a2, a3, and a 4 amplitudes are not completely independent of the other amplitudes in the full SU(3) because of their definitions as seen in (12). The factor x / ~ , which arises from the final phase space integral in the ~r°u ° and r/rl modes, is already incorporated in the coefficients m the table.

Decay modes a I a 2 a 3 a4 a~ a~ a~ a~ a s a6

D O ~ n-n + 1 1 0 0 ~o~o x / l ~ 0 - . , / ~ 0

~°n -4i7~ o , / ~ o ~ -4i7~ o , / ~ o K-K + -1 -1 0 0 Y.°K° 0 0 0 0

D+--,~°,~ ÷ o - 4 ~ - , / ~ o n~ ÷ o , / ~ 4 ~ 4 ~ g,°K+ 0 -1 0 1

F+~ K0~ "+ 0 1 0 -1 K% 0 0 0 - x / ' ~ - x / ~ K+,~ o , / ~ , / ~ , / ~

D O ~ K-lr + 2 2 0 0 ~o,~o - . , /~ 0 ~ 0

~% - 4 ~ o , / ~ o D + -+ K.°n+ 0 2 2 0 F +~ I~.°K + 0 -4/w~" 0 4/,v/6"

nTr + 0 0 2 2

D O --+ n-K + 2 2 0 0 ~°K0 - x / 2 0 x~- 0 ~K 0 - x / ~ 0 x / ~ 0

D + ~ K°~ + 0 0 2 2 K+~ "0 0 - x / 2 0 K+~ 0 x / ~ 0 - x / ~

F + ~ K+K ° 0 2 2 0

1 1 0 0 1 1 o - , . ~ o ~ , / ~

- , / ~ o - , / ~ o o 5x/2-/6 0 - x / ~ 6 0 x / ~ x /~6 1 1 0 0 1 1 2 0 0 0 1 0 o - , / i ~ - , / i ~ o o o o , / ~ - 4 ~ 427~ o , / ~ 0 1 0 1 0 1 0 1 0 1 0 1 o o - 4 i ~ 4V~ o 4 i ~ o - , / ~ - , / ~ -4i76 o - 4 ~

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Volume 85B, number 1 PHYSICS LETTERS 30 July 1979

To conclude this short note, we give briefly a qualitative account of the implication of eq. (15) on the structure of the weak current involving quarks heavier than the charmed quark. The orthonormal relations on the weak an- gles are gwen as

[cos 0cs? + [sin 0cd 12 + ~lUcil 2 = 1 , (16) 1

- c o s 0ud sin 0cd + sin 0us cos 0cs + ~UuiU~c, = 0 , (17) l

where Uut (Uc~) are the mixing coefficients of u (c) with heavy quarks of charge - 1 / 3 . If the branching ratio r de- viates largely from unity and the ratio of F(AS = 0)/F(AS = 1) given by e is of the order of 1/20, it may happen that sin 0cd differs significantly, more precisely speaking, by a factor substantially different from unity, from sin 0us (= sin 0C). In such a case, (16) and (17) cannot be satisfied simultaneously with values of cos 0cs close to unity; if K = s m 0cd/sin 0us is smaller than unity, orthogonality requires that cos 0cs be ~ ~cos 0ud since I Uutl 2 are much smaller than sm 2 0us. Then, the normalization condition must be saturated largely by ~t] Uct [2" That is to say, one or more of the couplings of the charmed quark with the heavier quarks must be large and lifetimes of charmed particles would be K -2 times longer than in the case of coS20cs ~ 1. It is not likely that K > 1 is realized with (16) and (17). In this way the measurement of B(D 0 ~ K - n +), B(D 0 ~ K - K +) has a profound implication on the structure of the weak current involving quarks heavier than the charmed quark.

References

[1] M. Kobayashi and K. Maskawa, Prog. Theor Phys. 49 (1973) 652. [2] M.B Einhorn and C. Quigg, Phys. Rev. D12 (1975) 2015.

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