P H Y S I C A L R E V I E W D V O L U M E 1 2 , N U M B E R 5 1 S E P T E M B E R 1 9 7 5

Charmed nonleptonic decay in asymptotically free theories*

John F. Donoghue and Barry R. Holstein Department of Physics and Astronomy, University of Massachusetts, Amherst, Massachusetts 01002

(Received 10 February 1975)

The nanleptonic decay of charmed states is studied by means of renormalization-group techniques in a class of asymptotically free models. An SU(3) sextet rule is shown to result. The enhanced piece of the Hamiltonian belongs to a 20-dimensional representation of SU(4). which distinguishes the renormalization-group approach from some alternate octet-enhancement schemes.

1. INTRODUCTION

The recent discovery of q(3105) a t SLAC and BNL has promoted renewed interest in the possible existence of a charmed partner-@'-for the con- ventional quark triplet-@, 37, X .' In this picture the fundamental symmetry i s SU(4) and the pseudo- sca la r and vector mesons constitute a pentade- cuplet. The SU(3) decomposition of this represen- tation i s

s o that in addition to the usual octet (nonet) there a r e predicted to exis t a charmed tr iplet and anti- triplet for both 0- and 1- s ta tes . Nonleptonic weak decay will presumably be the dominant decay channel f o r these s ta tes and it is our purpose here to study such processes .

[I. ENHANCEMENT EFFECTS

Nonleptonic decays have always been enigmatic in that the s imple Cabibbo picture cannot account for the empir ical ly observed A l = h ru le in AS = 1 processes. ' Recently, a promising new approach- the renormalization-group-has been applied to these c a s e s with some (though not total) ~ u c c e s s . ~ In this note we shall study nonleptonic charmed decay via these methods and will l a te r point out s t r iking differences with some al ternate enhance- ment approaches.

We work in a n asymptotically f r e e model with colored SU(4). The s trong interactions a r e me- diated by m a s s l e s s , non-Abelian neutral gauge fields with the gauge group being SU(N). The weak interactions a r e described by a Weinberg-Salam type gauge the01-y.~ The s trangeness-conserving and strangeness-changing charged cur ren ts a r e given by

J A S = O = (@i c o s ~ - @ i sinO)yA(l + y 5 ) l , , t

(1)

J A S = I , A (61, s i n ~ + g ; cosO)yA(l +y,)Xi . I

Here 0 i s the Cabibbo angle and i = 1 , 2 , . . . , N designates the color degree of freedom s o that both c u r r e n t s a r e color singlets. Writing

J - j A S = O + j M = 1 A - A X 3

the effective nonleptonic Hamiltonian i s written a s 5

where f i s the fundamental weak coupling constant and D,(x, M,') represen ts the propagator fo r the W bosons which mediate the weak interaction.

In the absence of s t rong interaction renormal - ization effects the charm-changing Hamiltonian consis ts of th ree pieces

where we have written ~ . l f i = f 2 / L 2 ~ , 2 . The domi- and change in charm a r e identical. F o r example, nant term-not suppressed by fac tors of tane- represent ing the charmed pseudoscalar t r iplet favors decays in which the change in s t rangeness a s (do, dt , s + ) the two-body channels

12 - C H A R M E D N O N L E P T O N I C D E C A Y I N A S Y M P T O T I C A L L Y F R E E . . . 1455

proceed without suppression, while r a t e s fo r p rocesses

wherein s t rangeness i s conserved but c h a r m i s not a r e down by a factor tan2@-& and r a t e s f o r channels such a s

wherein the s t rangeness change i s opposite to the charm change a r e suppressed by tan4@-&. Via renormalization-group techniques we can examine the effects of the s t rong interactions upon these conclusions. Working in the ' t Hooft gauge, and neglecting the effects of Higgs ~ c a l a r s , ~ we find a Wilson expansion of the t ime-ordered product in Eq. (2),6

T(J:(x)J~(o)) = C c,(x)o,(o) . ( 4 ) i

Here { O i l i s a complete s e t of operators carrying the appropriate quantum numbers while Ci(x) a r e c-number coefficients which contain the space- time dependence. Opera tors of lowest dimension will dominate the resul t . F o r m s in which only two-fermion operators appear a r e purely A1 =

but can always be transformed away by the m a s s and wave-function renormalizations of the quark fields.' The operators of lowest dimension which survive a r e four-quark t e r m s of dimension s ix,

Oi - : qy;i,(1 +y5)q?7yX(1 +y5)q : . (5)

The corresponding coefficients C,(x) obey the Callan-Symanzik relation5 and, following Lee and Gaillard,3 a r e found to be

Here g i s the color gluon coupling constant, p i s related to the subtraction point and specifies the

m a s s sca le , b = (11N- 8)/3, where -bg3/16n2 i s the lowest-order approximation of the P(g) func- tion in the Callan-Symanzik equation,' and di is a parameter related to the anomalous dimension of operator 0,.

Treat ing f i r s t the A S = AC operator we f o r m op- e r a t o r s of definite dimension by separatingg

H A S = AC = H t S = AC +Has= AC , ( 7 4

where

To lowest o rder in g2 both H+ and H - a r e renor - malized multiplicatively and

Then using p = 1 GeV, .V, - 100 GeV, g2/4i7 "1 we find

H B = A C - ~-9(N-l) /N(l lN-8)~?= Ac eff

+ ~ ( N + I ) / N ( ~ ~ N - ~ ) H A s = - AC I

with

F o r N = 3

.AS= AC , (7) - 0 . ~ 4 ~ ; ~ =ac + (7)o.4eHas= AC . off

Thus H - i s enhanced and H + i s suppressed by the sanlefactors a s in the calculation of L e e and Gaillard for the charm-conserving s trangeness- changing case.3 T h e r e a r e two fea tures of non- leptonic K decay which a r e difficult to understand in the conventional Cabibbo model:

( i ) the suppression of the charged mode K+- n'no with respec t to the neutral Ko- n'n- by a factor of about 500;

(ii) the apparent lack of suppression by c o s Q s i n 6 of the nonleptonic amplitude.

The renormalization effects of the s t rong inter- actions a s s i s t on both s c o r e s , but in neither case i s the effect l a rge enough to be consistent with experiment. T h e r e a r e two ways out of th i s di lem- ma. One i s the realization that in other models- e.g. that of Lee, Prentki , and Zuminolo-wherein there a r e a l a rge number of basic fermion quarks, b i s reduced and enhancement effects become la rger . The other is the fact that renormalization- group ideas give only the effective nonleptonic Hamiltonian opera tor . Matrix elements must s t i l l be taken, and a t this point previously discussed

1456 J O H N F . D O N O G H U E AND B A R R Y R. H O L S T E I N 12

octet-enhancement ideas—current algebra, tadpole dominance, etc.— come into play. The A/ = | en­hancement actually observed presumably involves some combination of these mechanisms.

We shall take the view here that however the further enhancement ar ises in the case of the AS = 1, AC =0 decay modes, effects of a similar size appear in charm changing decays. What then are the consequences?

F i r s t we shall look on the SU(2) level. Both #f s = AC a n d # A S = AC carry 7 = 1, 7, = 1. Thus, even without enhancement effects we automatically have a A7= 1 rule. For the doublet pseudoscalar meson decays this reads

A(d+-ir+K°) =A(d°-Tr+K-)+S2A(d0-~ir°K0) .

(ID

The SU(3) consequences are more interesting. The original Hamiltonian J7AS = AC was composed of a combination of the SU(3) representations T5 + 15, 6 + 6. The enhanced operator_7f^ e^ is composed, however, solely of 6 + 6 while the suppressed operator #^ S = AC i s purely a member

UAS = 0,AC = ±1 _ , jAS = 0,AC= ±1 , rjAS = 0,AC= ±1

of the 15 + 15 representation. Thus we predict a sextet rule in the case of AS = AC decays. Using such a rule we may relate, for example, the various charmed pseudoscalar modes

(§)i/2A(s+- 77V) = A(s+- K°K+)

= A(d°~ir+K-)

= -f2A(d0-ir0K°)

= -S6A(d0-Ti0K0),

A(d+-ir+K°)=0. (12)

Thus the d+ cannot decay via a two-body channel in the pure sextet scheme. This is clear since -n+K° ca r r ies isospin f and must be part of a 27 representation which is not contained in J x 6 . Realistically, as in the case of K+~* 7T+7r°, we might expect the d+ decay to occur at the 10% level com­pared to the neutral channels.

We may perform a similar renormalization-group analysis for the strangeness-conserving, charm-changing Hamiltonian. Forming symmetric and antisymmetric combinations as before

(13a)

where

m i l = ^^cos^sin^rx(l+r5)(Pi(Pjrx(i+r5)xi±xirx(i+r5)^^rx(i+r5)^i

(13b)

We find after turning on the strong interactions the same enhancement (suppression) as before for the antisymmetric (symmetric) combinations

H AS = 0,AC= ± 1 r (7) - 0 .24 TT AS = 0 , AC = ±1

+ ^ \ 0 . 4 8 r r A S = 0,AC = ± l (14)

H e r e ^ s s 0 * A C = ± 1 is purely A7= | while #£S«O.AC«±I is a mixture of A7 = | , f. Thus on the SU(2) level we find that such decays are guided by a A7= | rule. Their rates a re , of course, still sup­pressed, however, by a factor of tan2# with r e ­spect to the AS = AC modes. In the case of the pseudoscalar two-body modes, this A7 = i rule imposes the restrict ions

f2A(s+-* K+ 77°) = A ( s + - # V ) ,

A{d+- T T V ) = 0 ,

A{d°-^Tl-)=A{d0-.TJ0T!0)y

A(d+~Ti+r)0)=j2A(d0-7i0ri0)-

(15)

In the SU(3) picture H AS = 0,AC= ±1 is again a mem­ber of the 6 + 6 representation while ^ +

A S s 0 ^ S 1 1

is part of a 15 + 15 dimensional representation. We envision a sextet rule then, which gives the predictions in Eq. (15) plus additional relations among the AS = 0, AC = ± l amplitudes themselves,

A(d0-* TT+TT) = - / 3 A ( d ° ~ TTV)

= f2A(s+-Tt0K+)

= -/6A(s+-n°r)

= -A(d°-71°710), (16)

and between strangeness conserving and noncon-serving decays, e.g.

A(s+~ TTV) =2 coteA(s+-K+r]0) . (17)

Finally, for completeness, we consider the AS = - A C channels, although these are suppressed with respect to AS = AC decays by a factor of tan40~4£o in the rate . We find

12 - C H A R M E D N O N L E P T O N I C D E C A Y IN A S Y M P T O T I C A L L Y F R E E . . . 1457

H,fy=&= + l Z (7)-0.24~;AS'Ac=+l + (7)0.48H~AS=AC= i 1 , ( 1 8 4

with

H ; ~ = ~ = * l = - -s in2~[Xiyx(l G +y5)&'ig;yX(1 +y5)x j * X ~ Y A ( ~ + Y 5 ) ~ i ~ ; Y X ( 1 + ~ 5 ) 6 j I + H . c . (18b) 2 0

Now the enhanced operator *' 1s '

purely A I = O , while the suppressed operator is a n isotopic vector. Thus a t the HIas= Ac= * l

SU(2) level we have a A I = 0 rule , which gives

A(s+-K+KO) = o , A(d+- n+KO) = a A ( d + - n°Kt),

(19) = -A(dO- li-K') =d?A(d0- T O K O ) ,

A(d'- K+qO) = -A(d O - KOqO) . In an SU(3) picture the enhanced (suppressed) op- e r a t o r belongs to the representation 6+6 (15 +=) s o that we predict a sextet rule , which gives the above predictions plus a n additional relation be- tween the Kq and Kn decay schemes

i o ~ ( d ' - K+qO) = -A(d+- K+nO) , (20)

and among these and the other charm noncon- serving modes, e.g.

A(d+- l<+vO) =cotBA(s+- Ktv') . (21)

If we examine nonleptonic decay f rom the stand- point of SU(4) symmetry , each of the enhanced operators

H h S = A C , H h S = ~ . 3 2 = ~ ~ H I M = A C = + l , H ~ S = + l . & = O 9

i s a member of a 20 dimensional representation of SU(4) while the suppressed operators

H ~ S = A C , H ~ S = o , M = t l , H ; & = U = + l H ~ S = i l . A C = O 9

reason we have in this note emphasized the SU(2) and SU(3) relat ions which follow in our model.

111. CONCLUSION

We now have sketched the complete picture of charmed nonleptonic decay. Although, fo r the sake of sanity, we quoted specific relat ions between decay amplitudes only f o r pseudoscalar two-body decay, s imi la r relat ions obtain f o r two-body vector-meson decay channels and, of course , the group s t ruc ture of the Hamiltonian must be the s a m e for any charmed decay mode.

One of the most interest ing fea tures of this r e - sult i s that the enhanced nonleptonic charm chang- ing operator belongs to a 20 dimensional r e p r e - sentation of SU(4). The tadpole method of octet enhancement and perhaps some other mechan- isms13 predict that the enhanced operator belongs to the adjoint o r 15 dimensional representat ion. Of course, the 15 i s not present in the original Hamiltonian but is produced when the l a r g e SU(4) breaking effects a r e inser ted. In this c a s e the operator responsible fo r charm-changing decay t rans forms like do, do and thus favors non-strange- ness-changing decay, whereas the renormaliza- tion-group enhancement method predicts a p re - dominance of AS= A C decay channels, a s would be expected f rom the naive Hamiltonian.14 Experi- mental study of such charmed decays thus might enable the differentiation between octet enhance- ment schemes-each of which would work perfectly

a r e part of an 84 dimensional representat ion. well a s long a s only AS= 1, A C = O channels can be

A generalization of Gell-Mann's K - nn (Ref. 11) studied. The exis tence of c h a r m then, while ex- theorem forbids any two-body pseudoscalar decays

panding the already crowded l i s t of par t ic le s ta tes , in a s t r i c t ~ ~ ( 4 / l i m i t . We can attempt to account

might enable some light to be shed on the long- for SU(4) breaking effects in a simple quark-model

standing Al= enigma. calculation, which g i ~ e s , ~ e.g. Note. After the completion of this manuscript,

A(K'- n'n-) MK2 - .Vf n2 (22)

we w e r e informed of a report by Cabibbo, Altarel- c o t 6

A(s+- K+KO) - M, - "dK2 ' l i , and ~ a i a n i ' ~ in which s imi la r resu l t s were obtained.

Note that both amplitudes vanish if pseudoscalar m a s s e s a r e s e t equal, in accord with the Gell- ACKNOWLEDGMENT Mann theorem. However, SU(4) breaking effects a r e considerable and by no means i s the relation One of u s (B. R. H.) would like to thank Professor in Eq. (22) necessar i ly trustworthy. F o r this L . Wolfenstein for severa l useful conversations.

1458 J O H N F . D O N O G H U E A N D B A R R Y R . H O L S T E I N - 1 2

*Work supported in part by the National Science Foun- dation.

IS. L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2, 1285 (1970).

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5 ~ e do not consider here the weak AS =AC =O Hamilton- ian which i s responsible for parity violation in nuclear physics, to which neutral currents contribute.

k. Wilson, Phys. Rev. 2, 1499 (1969). 7S. Weinberg, Phys. Rev. D 8, 4482 (1973). 8 ~ . B. Callan, Phys. Rev. D 2, 1541 (1970); K. Symanzik,

Commun. Math. Phys. 18, 227 (1970). e ere we have used the F ie rz identity to reduce the num-

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Y. Hara, in TheEightfold Way, edited by M. Gell-Mann and Y. Ne'eman (Benjamin, New York, 1964).

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