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Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982
A UNIFIED DESCRIPTION OF THE CHARMED AND STRANGE MESON
WEAK NON-LEPTONIC PROCESSES
T. TANUMA and S. ONEDA Center for Theoretical Physics, University of Maryland, College Park, MD 20 742, USA
and
K. TERASAKI Research Institute for Theoretical Physics, Hiroshima University, Takehara, Hiroshima-Ken, Japan
Received 31 August 1981 Revised manuscript received 10 December 1981
In a non-perturbative theoretical framework of asymptotic SU(4) symmetry and asymptotic level realization of flavor symmetry, asymptotic selection rules are obtained which produce a unified explanation of the characteristics of the charm- ed as well as the strange meson non-leptonic weak interactions.
Recently two of us (S.O. and T.T.) proposed [1,2] an explanation for the presence and also the small vio- lation of the [A/I = 1/2 rule in the non-leptonic K- meson decays based on a new non-perturbative ap- proach. The purpose of this paper is to show that its straightforward extension to the charmed mesons al- so explains the characteristics of the presently known D-meson non-leptonic processes rather well.
The approach confronts the problem of confined quarks and gluons explicitly asfollows; the Hilbertl space with which it deals consists only of observable hadrons. (For a rigorous formulation of such type of theories, see, for example, ref. [3] .) However, this world of hadrons submits itself to a subtle control of underlying quarks and gluons. First, hadrons obey a certain level scheme in its infinite momentum frame, i.e., the scheme of (mainly q~t and qqq) constituent quarks. Second, hadrons are severely constrained by the presence of chiral algebras involving, especially, the observable weak quark currents and their charges.
In the (non-perturbative) theoretical framework of asymptotic flavor SU(N) symmetry, a single dynam- ical ansatz, which places a prominent role on the con- cept of levels of hadrons, is introduced. It requires that, in the realization of certain class of algebras in-
volving axial-charges among asymptotic hadronic states, the flavor SU(N) symmetry should be secured level-wise. This ansatz imposes non-perturbative dy- namical constraints (level-wise) upon the asymptotic single-particle hadron matrix elements of vector and axial-vector quark currents and their charges, which are in good agreement [2] with experiment.
By considering the algebras which now involve the non-leptonic weak quark hamiltonian H, the same an- satz produces similar dynamical constraints level-wise, among the single-particle asymptotic hadronic matrix elements of H. By using the algebras involving the strangeness changing weak non-leptonic hamiltonian, H (0,-) - H ( A C = 0, AS = -1 ) , i.e., for example,
[[H(0'-),A +] ,At_ ] = [[H (0'-), V÷], I I _ ] , (1)
[[H (0'-), At_ ] , A t . ] = [[H (0'-), V~r-], V÷], (2)
it was discovered [ 1,2] that the asymptotic ground- state meson matrix elements of H (°,-), (n[H]K(p)), (p[H[K*~)), (n[HIK*(p)) and (plHIK(p)) with p -+ 0% do obey, among others, the strict [A/I = 1/2 rule. [We, of course, assume exact SU(2) symmetry.] In partic- ular, we have derived
260
V o l u m e 1 1 0 B , n u m b e r 3,4 P H Y S I C S L E T T E R S 1 A p r i l 1 9 8 2
<n01H(°'-)IKOGo)) + <rr+iH(0'-)lK+~)) = 0,
p + ~ . ( 3 )
Eq. (3) immediately implies [1,2] that the K -+ 27r matrix elements (nnlH(0,-)[IO satisfy the strict IMI = 1/2 rule in the soft pion limit. However, for phys- ical pions the rule is necessarily violated, providing a nice explanation why the rule is violated slightly.
We start with the effective non-leptonic weak hamil- tonian in the standard four-quark scheme which is given by [4],
Heft . G f d3x :j(+)(x)j,(_)(x) : 2x/~
where
(+) = cos 0 c ~Tu L d + sin 0 c gTu L s g
+ (--sin 0c)~3'uLd + cos 0 c ~3`~L s, (4)
j ( - ) = (j(+))t and 3' L = 7 1(1 + 3' ) As far as the # /a ~u gz 5 " transformation property of the quark fields under SU(4) in flavor space is concerned, the current may be denoted by
j(+) oc cOS0c n+ + sin0e K+ + (_sin 0c)D + + cOS0c F+.
Among various terms in H eft , we consider the follow- ing two parts;
H (0'-) - H ( & C = 0, AS = - 1 )
= sin 0 c cos 0c(Tr+K - - D+F- ) , (5)
H (- ' - ) = H(AC = &S = - 1 ) = coS20cTr+F - . (6)
The I&/] = 1/2 rule associated with the H (0,-) is al- ready discussed [1,2]. (The new term D+F - is now present in the four-quark scheme . )H( - , - ) is respon- sible for the Cabibbo favored non-leptonic decays of charmed mesons. Using the chiral SU(4) ® SU(4) charge-current algebras, we can bridge over the above two hamiltonians Simply by
[ H ( - ' - ) , VDo ] = cot 0cH(0'-), (7)
where VDO is the vector charge with the transforma- tion F 9 - iF10. In the spirit of current algebra [5], the equal-time commutator, eq. (7), is valid even with the large symmetry breaking of SU(4) flavor. In asymp- totic flavor SU(N) symmetry, the SU(N) transforma- tion maintains [6] its linearity even in broken sym- metry but only among the SU(N) multiplets with in-
finite momenta. Coupled with the presence of eq. (7), asymptotic SU(4) symmetry then enables us to derive broken SU(4) relations among the asymptotic hadron- ic matrix elements of H. (This is an extension of sim- ilar assertion previously made on the asymptotic ma- trix elements of the quark currents and charges in asymptotic SU(N) symmetry.) This result constitutes one of the major distinctions between the present ap- proach and the arguments based on the notion of ex- act SU(4) symmetry. In perturbative SU(4), it is cer- tainly hard to argue that the notions based on exact SU(4) remain intact in the real world.
We insert eq. (7), for example, between (rr+(p)l and IK+(p)) and also between @0(p)[ and [KO(p)) with p -+ ~, and use asymptotic SU(4). We then obtain
(Tr+IH ( - ' - ) [F+(p)) = cot 0e(a+l//(0'-)iK+(p)),
p - , oo, ( 8 )
(D01H(-'-)IK0(p)) = vr2 cot 0 c (~0IH(0'-)IK0(p)),
P -> oo. (9 )
By adding eqs. (8) and (9), we immediately notice that the strict IZk/l = 1/2 rule obtained in eq. (3) for the asymptotic matrix elements of H(°, - ) leads to the prediction,
(rr+[H(-'-)[F+(p)) + (D01//(- '-)lK0(p)) = 0,
p ~ o~. ( 1 0 )
Eq. (10) also reads equivalently (through CP-invariance),
<rr+IH(-'-)IF+(p)) + ff[0l H(- ' - ) ID0(p)) = 0,
p ->oo. (11)
Eq. (10) or (11) constitutes the charmed counterpart of the asymptotic IzX/I = 1/2 rule, eq. (3), and plays a fundamental role for the Cabibbo favored D-meson processes. They are valid in broken SU(4) and SU(3) symmetry. Although we have here chosen to derive eq. (10) from eq. (3) by an asymptotic SU(4) rota- tion, eq. (11) can also be obtained directly, by Using the level realization ansatz, from the algebras such as
[ [H(- ' - ) ,AK+ ] ,A,r_ ] = [[H ( - ' - ) , VK+], V ] etc.,
(12) exactly in the same way as eq. (3) was derived from the algebras eqs. (1) and (2). Thus, internal consisten-
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Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982
cy of the theory is quite satisfactory. We may remark here that eq. (10) or (11) can be
read as implying the asymptotic AV = 0 rule (conser- vation [7] of the V-spin and V3) for the asymptotic
lr + F + two-body weak matrix elements, since (K o) and (D o) or (DO) wi th infinite momenta may be regarded as the V-spin doublets. The inevitable violations of the asymptotic AV = 0 rule (in varying degrees) for the physical processes are caused by the soft-meson ex- trapolation we have to invoke, in order to connect the physical amplitudes with the two-body asymp- totic amplitudes satisfying the AV = 0 rule.
We now discuss the D -+ Krr decays. In the present treatment of flavor SU(N) symmetry, it can be shown that an important symmetry property of the matrix elements involving two or more bosons belonging to the same SU(N) multiplet remains to be preserved, even when SU(N) symmetry is broken. For example, one can show in the limit k 1 ~ ~ and k 2 + oo Co =
k 1 + k2),
(I~0(k 1 ), 7r+(k2)tH(-'-) ID+(p))
= (I~0(k2), 7r+(kl )It/(-'-) ID+(p)). (13) Eq. (13) implies that the on-mass-shell invariant am- plitude of (I~ 0, 7r+IHID+), which is a scalar function of (kl . k2), k~ and k~, must be symmetric in k 1 and k 2 even in broken SU(3) symmetry. This implies that it must be symmetric with respect to the masses, m~ and m~. As in the K ~ 21r decays [1,2], we now exe- cute a soft-meson limit k 1 ~ 0 in the asymptotic frame k2 = p ~ 0% paying close attention to the above in- herent symmetry of the invariant amplitudes. We thus define ,1,
(I~0 rr+ [H( - ' - ) ID+(P))Soft, sym,p_+ o ~
=g-1 lim [ (FZO(k l ) rr+(k2)[g( - ' - ) lD+(p) ) k l -~ O,p-~ ~
+ (k 1 +-+ k2) ] . (14)
We now carry out the standard soft-meson procedure, use the relation [ A s , H ( - , - ) ] = - [ V ~ , H ( - , - ) ] , and apply asymptotic SU(3). We then derive,
A ~ ~07r+lHID+(P))Soft, sym,p_~
1. -1 + + = ~ l [ f K (Tg IHIF (p)) +/-~(I~01HID0(p))], (15)
B - 07~°~'°IHID° (P))Soft, sym,p-~
= l i [ ( - -2-1/2fK1 +x/~f-1)(I~0lHlU0(p))] , (16)
C - (K- 7r+ IHID0(p))soft,sym,p~ ~
= ½i[(fK1 -- f ~ l ) ( ~ ( O I H I D O ( p ) ) + f K l O r + l H I F + ( p ) ) ] .
(17)
Eqs. (15)- (17) satisfy the isospin triangle relation C + x /2B - A = 0. From eq. (15), we see that in the soft-meson limit with fK = f~r, the asymptotic A V = 0 rule, eq. (1 1) , f o rb id s the D + ~ K07r+ decay, whereas the D O ~ KTr modes are all allowed, favoring qualita- tively the rather striking experimental fact that the lifetime of D + is longer than that of D 0. The triangle relation gives C = -Vr2B in the above limit. However, compared with the IA/] = 1/2 rule, these selection rules cannot be very strict. The rules are obtained not only in the soft-meson limit but also assumingf K = fTr" The effect of soft-meson extrapolation is certainly mini- mized, since we are led (through asymptotic symme- try) to take the limit in the frame p = k 2 ~ oo. Also, even though the soft-K-meson as well as the soft-pion are involved, the invariant amplitude is a s y m m e t r i c function o f m 2 and m 2 as noted before. In this pa- per, we, therefore, proceed to use fK ~f~r as a crude ,2 estimate of the degrees of violations of the selection rules. We then obtain from eqs. (15)- (17) using eq. (1 1), the following predictions on the branching ratios.
p(D0_+K_rr+)_ C 2 = I 2 ~ 71V~(fK/frr ) 2, R 1 ~- ~ - ~ ~0,n.0) (18)
R 2 r(D0-+K-rr +) = ~KK/f~ (19)
As mentioned before, in the limit fK = frr, R1 = 2 and R 2 = 0, respectively. For OeK/frr) "" 1.28,R 1 -~ 1.35
,1 We have omitted kinematical factors (1/2~r)3/2(2ko) -1/2, which cancel out in the ratios such as eqs. (18) and (19) in the limit kl--+0.
,2 Strictly speaking, at the level of worrying about fK ~ f~r, the neglect of flavor SU(3) mixings between the ground state PS-mesons and their radially excited states may not be completely justified.
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Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982
and R 2 ~- 0.05. The value o f R 2 will be more sensitive to the effects neglected. The above values o f R 1 and R 2 are in reasonable agreement with present experi- ment [8].
We now briefly comment on the quasi two-body decays such as D O -~ ~0p0. For convenience, we con- sider the matrix elements in the crossed channel,
<pO(p)lH(-'-)iKO(kl )DO(k2)>.
The argument similar to eq. (13) implies that the in- variant amplitude should now be antisymmetric under k 1 ~ k 2. We thus use the same extrapolation k 1 -+ 0 for the invariant amplitudes which are now antisym- metrized in k l and k 2. Analogous to eqs. (15) - (17) , we then obtain (p --> oo),
D = (p0IHIK0, D0)soft = ½ i 2-1/2 [.fD1 (~.01HIK0(p))
- f K 1 (IT~*OIH[DO(p))], (20)
E = (p+IHIK +, D0)soft = ½i[OeD 1 +fK1)(p+IHIF+(p))
+ f K 1 (I~*01HID0(p))], (21)
F = (K*-- I n l n - , D0}soft
= ½i [fD l<F * - Inl r r - (p)) - f - 1 (I~*0[nlD0(p))], (22)
G = (I~*0[Hln 0, D0)soft
= ½i[(2-1/2fD1 + v r 2 f ~ l ) ~ * 0 1 n l D 0 ) ] , (23)
, + H - <p+IHIK 0 D )Soft = ½i[(fD1 + fK 1)<p+IHIF+(p))
+ fD 1 (I3*01HIK0(p))], (24)
I - ~ * 0 1 n l r r - , D+)soft = ½i[(fD 1 + f~ l )~*01n lD0(p )>
+ f D 1 (F* - In l r r - (p ) ) ] . (25)
Eqs. (20) - (25) satisfy the isospin triangle relations explicitly. Repeating exactly the same arguments now for the two-body asymptotic vector-meson-PS-meson weak vertices, our fundamental asymptotic constraints corresponding to eqs. (10) and (11) are ( H - H ( - , - ) ) ,
(p+[HlF+(p)) + (I~*°IHIK°(p)} = 0, p -~ o% (26)
(p+IHIF+(p)) + (I~*°lHIO°(p)) = O, p --> o o (27)
Furthermore, we also obtain the SU(8)-like asymp- totic constraints corresponding to the SU(6)-like ones [1,2] in SU(3),
(p+lHIF+(p)) = <rr+lHIF*+(p))
= (F*- IHl r r - (p ) ) etc., p -+ o0. (28)
We write the invariant amplitude A(V --* DP) as eu(p ) X (k 1 - k2)Ug , where g is a scalar function which is symmetric in k 1 and k 2. e u is the polarization four- vector of V. Then, eqs. (20) - (25) with the constraints, eqs. (26) - (28) , predict;
A(D 0 _+ pOlO) :A(D 0 _+ p+~) : A(D 0 _+ K * - n + )
: A(D 0 .+ ~.*OrrO ) : A(D + ..+ p+~,O) : A(D + ~ ~,.*Orr+)
= _ 2 - 1 / 2 0 e D 1 - - f K 1) : f D l : (fD 1 +fn - 1 )
: _2-1/2(fD1 + x / ~ f - 1 ) : f K 1 : ( _ f - l ) . (29)
We notice that the soft D-meson extrapolation is now involved. However, this is not a serious drawback, g is symmetric in k 1 and k 2 and should be a symmetric function of the two PS-meson masses involved. Fur- thermore, the soft-meson limit k I -+ 0 was taken in the frame p = k 2 -+ ~. The immediate very interesting simple prediction is that A(D 0 -+ o°K 0) vanishes in the limit fD = fK. This suppression of the D o + p0~0 mode relative to the D o -+ p + K - seems to be real. Recent experiment [8] indicates that the branching ratio of the D o "+ p + K - decay is (7.2 +3.0,o/~
- 3.1 ) ,v, whereas that of the O0g. 0 mode is (0.1 +0..6)% A more realis- -o.1 •
tic branching ratios may be obtained from eq:. (29) by using the physical values offn , fK and fD, although we have to be aware of the remark made in footnote two.
We add here a remark about the accuracy of asymp- totic SU(4) symmetry used. The crucial step in de- riving the main result from the algebra, eq. (7), is the selection of the intermediate states by the SU(4) charge VDO in the limit p -~ oo. By sandwiching an- other algebras involving VDo , i.e., [VDo , A~r- ] = [I~DO , At_ ] = 0, between (D*0(p)l and Irr+(p)) with
p -+ 0% exactly the same procedure leads to a mass formula D .2 - p2 = D 2 _ 7r2 which is reasonably well satisfied. This suggests that for the purpose of the present paper, we can neglect the leakage reasonably well, which takes place through the SU(4) mixings
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Volume 110B, number 3,4 PHYSICS LETTERS 1 April 1982
with the higher-lying states. For more detail, see H. Hallock et al. cited in ref. [6].
Generally speaking, our result is in qualitative agree- ment with the naive 20 representation [or 6 ~ 6* as- suming exact SU(3)] dominance [7] based on the no- tion of exact SU(4) symmetry. For instance, eqs. (18) and (19) do coincide with the predictions of naive 20 dominance (which, however, has to argue some- how that one can still use physical masses for the real decays), if we take fK = fTr and forget about the soft- meson effect. However, two crucial differences must be noted. First, as stressed, our sum rules are all valid in broken SU(4) and SU(3) symmetry. Second, our statement about the selection rules (or 20 dominance) is much more precise. Namely, only the two-body asymptotic matrix elements of H(0, - ) and H ( - , - ) sa- tisfy the strict I~fl/[ = 1/2 and AV = 0 rule, respective- ly. These selection rules have to be violated, in vary- ing degrees, for the physical processes, and the soft- meson procedures provide us with reasonable, if not perfect, estimates of the violations.
In deriving our asymptotic sum rules, the Ansatz of level realization of asymptotic flavor symmetry has played a crucial role. Therefore, we strongly suspect that the origin of these selection rules is non-perturba- rive in nature. Since the quark confinement mecha- nism is not yet explicitly formulated in the frame- work of QCD, subtlely is always present in converting the quark or gluon matrix elements into observable hadronic matrix elements. (For critical recent reviews, see, for example, refs. [9] and [10] .) The present ap- proach is free from this difficulty by design.
Under the circumstances, the present non-perturba- tive approach may provide an alternative. It yields a
simple unified description of the charmed as well as the strange meson decays. In other realms of particle physics, exactly the same Ansgtze have produced [2], in non-perturbative way, the quark-line rule and its violation, correct value ofgA(0 ) and the predict ion of kp = - k n for the nucleon anomalous magnetic mo- ments, etc. More detailed work will be published else- where.
References
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(1979). [4] Y. Katayama, K. Matumoto, S. Tanaka and E. Yamada,
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[5] M. Gell-Mann, Physics 1 (1964) 63 • [6] S. Oneda, H. Umezawa and Seisaku Matsuda, Phys. Rev.
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[8] R.H. Schindler et al., Phys. Rev. D24 (1981) 78. [ 9 ] Ling-Lie Chau Wang, Experimental meson spectroscopy -
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