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Soft hadron interaction effects in charmed meson decays
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
SOFT HADRON INTERACTION EFFECTS IN CHARMED MESON DECAYS Y. IGARASHI 1 , S. KITAKADO 1,2 and M. KURODA Department of Theoretical Physics, University o f Bielefeld, Fed. Rep. Germany Received 21 January 1980
The effects of soft hadron interactions are considered in estimating the non-leptonic weak decays of charmed mesons. We find (i) the contribution from the usually disregarded annihilation-type interactions to be large and (ii) the two-body decay rates to become quite different from that of the standard model and to be in good agreement with the recent experimental results.
The conventional understanding [ 1 ] of the nonleptonic decays of hadrons is based on the simple-minded assumption that hadrons are made of an almost free quark-antiquark pair or of three quarks, and that soft gluons, which are as well contained in hadrons, can be neglected. Therefore, it is not so surprising to find that the recently measured D meson decay widths are in complete contradiction to the standard weak interaction predictions. In particular, the following two experimental results [2]: Ptot(DO)/Ptot(D +) ~ 5 - 1 0 ,
(1)
r ( D 0 ~ K - ~ r + ) / r ( D 0 ~ R%0) ~ 1 - 2 ,
(2)
cannot be explained by the conventional model, which predicts ~1 and ~ 4 0 for the ratios (1) and (2), respectively. The standard model [ 1 ] starts from the effective weak hamiltonian, Heft, where the strong interaction effects at short distances due to hard gluon exchange are already taken into account, H e f f = (G/N/r~) coS20c [1 (f+ + f _ ) (cs) (du) 1
+ ~ ([+ - f _ ) (~u) ( a s ) ] ,
(Es) = EaTu(1 - 3,5)Sa
(a = color index).
Including the short distance renormalization effects, the coefficients f± have been calculated in ref. [3] : ( 33 2Nf m 2 ] 12/(33-2Nf) f_= I + 121r as In-~2.15 kt2 / (4) -
f+ = f _ - l / 2 ..~ 0 . 6 8 .
When evaluating the matrix elements of neff, one usually assumes a rather naive picture of the valence quark approximation, which consists of the following two assumtions: (i) Of the possible two types (figs. la, b) of hard interactions induced by neff, one assumes the dominance of the charged quark decay type interaction c -+ sdu (fig. 1a) over that of the annihilation type cfi -'- sd (fig. lb), which is based on the assumption of factorization and helicity arguments (c -~ sdu disintegration dominance). (ii) When taking the matrix elements of the product of the two quark bilinears whose color structure is mis-
(3) s
where 1 Alexander von Humboldt Foundation Fellow. 2 Present address: Faculty of Physics, University of Wuppertal, 5600 Wuppertal, Fed. Rep. Germany. Permanent address: Institute of Physics, Faculty of General Education, University of Tokyo, Japan.
c
o(32
(a,,
~-~d)
( bJ
Fig. 1. Two possible types of weak interaction induced by Heft: c ~ sdu disintegration (a) and c~ ~ s] annihilation (b). 125
Volume 93B, n u m b e r 1, 2
PHYSICS LETTERS
matching with that of the participating hadrons, one uses the color Fierz identity, e.g.,
(eu) Os) = ~-(~s) Ou) + ~(~Xas)(d~au),
(5)
and completelydisregards the secondpart of this equation (color matching). The first assumption of the c -+ sdu disintegration dominance predicts Ptot(D0)/Ftot(D+) = I, which contradicts the experimentalresult (I). This strongly suggests the inadequacy of the assumption of c -+ sdu dominance and we have to take the annihilation type cfi -+ sd interaction into account,which is present in DOdecaysbut is absent in D+ decays.The breakdown of assumption (i) is related to the existence of gluons inside hadrons [4] and to the consequentbreakdown of factorizationat the weak interactionvertex. The same fact also invalidates assumption (ii), which leads to F(D0 -+ K-n+)/F(D 0 -+I(07r0) ~ 40. The soft gluon can always change the color nonsing]et combinations to color singlet configurations [5,6]. Abandoning these two assumptions, we present, in the following, arguments for the possible description of the nonleptonic decays of charmed mesons on the basis of soft hadron dynamical considerations and give some quantitative estimates. We first discuss assumption (i) which is mainly related to the problem of total decay rates. We note that out of the two hard interaction types of fig. 1 we can construct the six duality diagrams depicted in fig. 2, the imaginary part of which gives the total decay rates. Only the first two diagrams (figs. 2a, b), corresponding to the charm quark disintegration with a passive spectator quark, have the parton limit. The third and the
(a)
(b)
(c )
(d)
(e )
(f )
Fig. 2. Duality diagrams for D m e s o n decays. The cross in the diagrams stands for the current creating D mesons.
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2 June 1980
fourth diagrams (figs. 2c, d) have a similar soft behaviour to the first two diagrams, although they do not have the parton limit, being of non-diagonal type. The last two diagrams come from the annihilation type interaction. The conventional argument that disfavours the annihilation contribution is valid only for the last diagram 2f, while the contribution from diagram 2e, in which the annihilation occurs only at the final state interaction is not, in general, small, because the c~ system (and consequently the d~ system) does not remember its original jPC. In figs. 2 a - d , the direct channel (decay channel) intermediate state is an exotic qglq?t state, which corresponds to the Pomeron exchange in the t-channel while in fig. 2e the direct channel intermediate state is a non-exotic ds state and correspondingly the t-channel is described by the u~ reggeon exchange. The nonleptonic decay widths are then written in general as [3,9] FNL(DO)/F(DO~ evX) = ( 2 f 2 +f2)_ + ~r ,
(6a)
FNL(D+)/F(D + ~ e vX) = (2f+2 + f2_) + (2f2 _ f2)p, (6b) with * 1 = 9 [(1+ k)f+ - (1 - k ) f _ ] 2 .
(7)
Here k stands for a color matching factor which represents the relative importance of the color mismatching quark bilinears to that of the matching one. This point is discussed in detail in the subsequent paragraph in connection with the exclusive two-body decays of D mesons. The symbols r and p represent the relative importance from the final state interactions of the reggeon exchange type (fig. 2e) and of the non-diagonal Pomeron exchange type (figs. 2c, d) in comparison with that of figs. 2a and b, which can be regarded as the Pomeron exchange contribution. In the case of complete interference [7] we expect p = 1, although for the heavier meson decays p becomes smaller, leaving the spectator quark less active. The reggeon exchange contribution r of the annihilation diagram is estimated based on Regge pole phenomenology [8], which gives r = r 0 X (S/So)-l/2 with r 0 ~ 1.2. Setting r 0 = 1 as a typical
,1 Note that the annihilation contains an extra N c factor compared with the disintegration. This leads, besides to 7r° ~ % to another possibility of "directly measuring" the colour
factor in the weak interaction. .2 To be more explicit ro = (~3~rK + ~ K ) / ~ K
Volume 93B, number 1, 2
PHYSICS LETTERS
value, s o = 1 GeV2 as a hadronic energy scale and s = (D meson mass) 2, we have r = 0.55 for charm meson decays. Let us next turn to assumption (ii) which is mainly related to the exclusive two-body decay widths. The two-body nonleptonic decays of D +, D O and F + mesons are described, in general, by four types of amplitudes, a, b, c and d, depicted in fig. 3 . We call (Es) (au) [(~u) (as)] a color favoured operator for n + [~0] production, because d and u [d and s] are in a color singlet combination and it is straightforward for them to create a rr+ IK°]. (Eu) (as) [(Es)(au)] on the other hand is called a color disfavoured operator for lr+ [~0] production. The amplitudes a and b [c and d] in fig. 3 correspond to the color favoured and disfavoured operators, respectively, when the decay proceeds through the disintegration process (fig. 1a) [annihilation process (fig. lb)]. We estimate the contribution from the color disfavoured operator in comparison to that of the favoured one, using the soft hadron interaction picture, where, of course, all soft gluon effects are considered to be taken into account. As depicted in fig. 4, the matrix element of the disfavoured operator can be expressed in terms of that of the favoured one (and vice versa) when the non-planar loop correction is taken into account. In other words, the favoured operator turns to the disfavoured one when operated on by the non-planar loop operator C, which is of a similar nature to that called "cylinder" in the dual topological unitarization scheme [10]. It is also known that the cylinder contribution in the pseudoscalar channel is always negative [ 11 ], which can be seen from the fact that both the isoscalar states 77 and
2 June 1980
(a)
Fig. 4. The disfavoured operator (b) is expressed in terms of the favoured operator (a) together with the non-planar operator ("cylinder"). r/' are heavier than the isovector state 7r * a. Using the cylinder operator C we can express the matrix elements of the disfavoured operator in terms of those of the favoured one as
(rr+l (Eu) (as) Ihadron)
=
(rr+l e (~s) (au) Ihadron)
= k(n+l(~s)(au)lhadron),
(8)
where k is a negative number. The same operator appears also in the annihilation type interaction of D O decay, where one has to express the matrix element of the disfavoured combination (Es) (au) in terms of the favoured combination (~u) (-ds). We note that the standard picture of color matching corresponds to k = ~, while the limit adopted by Fritzsch [5] corresponds to k = 1 and finally the "empirical" parametrization by Deshpande et al. [6] to k = - 1 / 6 . The value of k is estimated, for example, from the analysis of the two-body decays of K mesons which, in the pole model, leads to k = - 0 . 7 .
Numerical results. Given the value of the color matching factor k, we can estimate the numerical magnitude of D + and D O non-leptonic decay widths from eq. (6). For p = 1 as an extreme case of the maximal interference, r = 0.55 and k = - 0 . 7 as estimated ab ove, FNL(D0)/E(D0 -~ e vX) = 20.0,
(a)
{b)
PNL(D+)/F(D + --* evX) = 1.85. The corresponding semileptonic branching ratios are given by BR(D 0 ~ e v x ) = 4 . 5 %
(c )
( < 4% or 6 k 5%) ,
(all
Fig. 3. Two-body decay diagrams of D and F meson decays. See the text for details.
,3 Cf. the case of 2+ mesons where mf < mA2 and the cylinder contribution is positive. 127
Volume 93B, number 1, 2 BR(D + --> e vX) = 26%
PHYSICS LETTERS (23 -+ 6% or 17 + 6%),
which is in good agreement with the data [2] given in the parentheses. If we disregard for the moment the contribution from the annihilation type interaction in the exclusive D-decays, we find
F(D°~" K-n+) = 2 [(1 + k)f+ p~X
R01r0 )
+ (1 - k ) f _ q 2
L~+~-_ ~Z k~-_j~ 2 . 5 , (9)
for k ~ - 0 . 7 which is consistent with the experimental result of eq. (2) *4 . The numerical value for this ratio practically does not change, when we take the contribution from the annihilation type interaction into account, assuming the ratio to the disintegration type interaction to be the same as in the total decay rates. In this way we see that general considerations based on the soft hadron dynamics give us a satisfactory picture of the charmed meson decays. Finally, several comments are in order. (a) We are predicting r -= annihilation/disintegration cx 1/M, which should be compared, for example, with r ~x (FD/mc)2 of the standard model, where M and m e are a parent meson mass and a h e a w quark mass constituting the parent. I f F D ~ X ~ c , as empirically seems to be the case, both predictions give the same mass dependence. This behaviour of r also implies that for the decays of sufficiently h e a w quarks, like the b- or t-quark, the simple q u a r k - p a r t o n picture of the quark disintegration dominance describes the process correctly. A detailed analysis of D and F meson decays, as well as B meson (b-quark) decays will help to clarify the complexity of the heavy meson decays. (b) The same duality diagrams that appeared in the D O decays contribute to the F + decays. We expect, however, somewhat smaller Ptot(F+), Ptot(F+)/Ptot(D 0) 0.7 due to the low intercept of the ~ trajectory exchanged in the annihilation diagram in F + decays. (c) We note that in the exclusive and also in the annihilation processes, the short distance coefficients f_. appear always in the combination of (1 + k)f+_, as is seen, .4 Strictly speaking we have to replace (1 + k) by (1 + k)/(1 - k =) which comes about when we sum the higher order contributions from the non-planar diagrams. But this is irrelevant as far as we are concerned with ratios.
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for example, in eq. (9). This means that the negative k gives effectively a further enhancement (suppression) of the short distance coefficient f _ (f+), suggesting a possibility o f explaining the A1 = 1/2 rule in strange particle decays without penguin diagram. Especially, k = - 1 leads to perfect octet enhancement. In this limiting case, which corresponds to maximal r/-r~' mixing through the cylinder interaction in theO - + channel, the ratio F(D + -+ K-Tr+)/P(D ° ~ K0n0) is 2. We note, however, that our solution is not identical to the results of Fritzsch [5], who also gets 2 for this ratio. His solution corresponds to maximal mixing in the opposite direction, i.e., k = 1, which corresponds to the simple parton picture. S.K. wants to thank Prof. H. Satz for the kind hospitality extended to him at the University of Bielefeld. M.K. thanks Prof. R.J.N. Phillips for his warm hospitality at Rutherford Laboratory, where this work has been stimulated.
References [1] N. Cabibbo and L. Maiani, Phys. Lett. 73B (1978) 418; D. Fakirov and B. Stech, Nucl. Phys. B133 (1978) 315. [2] Talks by J. Kirkby and V. Luth at the Intern. Symp. on Lepton and photon interactions at high energies (FNAL, Batavia, IL, 1979). [3] M.K. Gaillard and B.W. Lee, Phys. Rev. Lett. 33 (1974) 108; G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 35; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B100 (1975) 313. [4] H. Fritzsch and P. Minkowski, Bern preprint (1979); M. Bander, D. Silverman and A. Soul, UCI technical report 79-73. [5] H. Fritzsch, Phys. Lett. 86B (1979) 343. [6] N. Deshpande, M. Gronau and D. Sutherland, Fermilab preprint, FERMILAB-PUB-79/70 (1979). [7] B. Guberina et al., Max Planck Institut preprint MPIPAE/PTh 45/79 (1979). [8] See, for example, Chart H.-M., Nucl. Phys. B86 (1975) 479;B92 (1975) 13; V. Barger and R.J.N. Phillips, Nucl. Phys. B32 (1971) 93. [9] G. Altarelli, N. Cabibbo and L. Maiani, Phys. Rev. Lett. 35 (1975) 635. [10] G.G. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) 265. [ 11] T. Inami, K. Kawarabayashi, and S. Kitakado, Phys. Lett. 61B (1976) 60.
PHYSICS LETTERS
2 June 1980
SOFT HADRON INTERACTION EFFECTS IN CHARMED MESON DECAYS Y. IGARASHI 1 , S. KITAKADO 1,2 and M. KURODA Department of Theoretical Physics, University o f Bielefeld, Fed. Rep. Germany Received 21 January 1980
The effects of soft hadron interactions are considered in estimating the non-leptonic weak decays of charmed mesons. We find (i) the contribution from the usually disregarded annihilation-type interactions to be large and (ii) the two-body decay rates to become quite different from that of the standard model and to be in good agreement with the recent experimental results.
The conventional understanding [ 1 ] of the nonleptonic decays of hadrons is based on the simple-minded assumption that hadrons are made of an almost free quark-antiquark pair or of three quarks, and that soft gluons, which are as well contained in hadrons, can be neglected. Therefore, it is not so surprising to find that the recently measured D meson decay widths are in complete contradiction to the standard weak interaction predictions. In particular, the following two experimental results [2]: Ptot(DO)/Ptot(D +) ~ 5 - 1 0 ,
(1)
r ( D 0 ~ K - ~ r + ) / r ( D 0 ~ R%0) ~ 1 - 2 ,
(2)
cannot be explained by the conventional model, which predicts ~1 and ~ 4 0 for the ratios (1) and (2), respectively. The standard model [ 1 ] starts from the effective weak hamiltonian, Heft, where the strong interaction effects at short distances due to hard gluon exchange are already taken into account, H e f f = (G/N/r~) coS20c [1 (f+ + f _ ) (cs) (du) 1
+ ~ ([+ - f _ ) (~u) ( a s ) ] ,
(Es) = EaTu(1 - 3,5)Sa
(a = color index).
Including the short distance renormalization effects, the coefficients f± have been calculated in ref. [3] : ( 33 2Nf m 2 ] 12/(33-2Nf) f_= I + 121r as In-~2.15 kt2 / (4) -
f+ = f _ - l / 2 ..~ 0 . 6 8 .
When evaluating the matrix elements of neff, one usually assumes a rather naive picture of the valence quark approximation, which consists of the following two assumtions: (i) Of the possible two types (figs. la, b) of hard interactions induced by neff, one assumes the dominance of the charged quark decay type interaction c -+ sdu (fig. 1a) over that of the annihilation type cfi -'- sd (fig. lb), which is based on the assumption of factorization and helicity arguments (c -~ sdu disintegration dominance). (ii) When taking the matrix elements of the product of the two quark bilinears whose color structure is mis-
(3) s
where 1 Alexander von Humboldt Foundation Fellow. 2 Present address: Faculty of Physics, University of Wuppertal, 5600 Wuppertal, Fed. Rep. Germany. Permanent address: Institute of Physics, Faculty of General Education, University of Tokyo, Japan.
c
o(32
(a,,
~-~d)
( bJ
Fig. 1. Two possible types of weak interaction induced by Heft: c ~ sdu disintegration (a) and c~ ~ s] annihilation (b). 125
Volume 93B, n u m b e r 1, 2
PHYSICS LETTERS
matching with that of the participating hadrons, one uses the color Fierz identity, e.g.,
(eu) Os) = ~-(~s) Ou) + ~(~Xas)(d~au),
(5)
and completelydisregards the secondpart of this equation (color matching). The first assumption of the c -+ sdu disintegration dominance predicts Ptot(D0)/Ftot(D+) = I, which contradicts the experimentalresult (I). This strongly suggests the inadequacy of the assumption of c -+ sdu dominance and we have to take the annihilation type cfi -+ sd interaction into account,which is present in DOdecaysbut is absent in D+ decays.The breakdown of assumption (i) is related to the existence of gluons inside hadrons [4] and to the consequentbreakdown of factorizationat the weak interactionvertex. The same fact also invalidates assumption (ii), which leads to F(D0 -+ K-n+)/F(D 0 -+I(07r0) ~ 40. The soft gluon can always change the color nonsing]et combinations to color singlet configurations [5,6]. Abandoning these two assumptions, we present, in the following, arguments for the possible description of the nonleptonic decays of charmed mesons on the basis of soft hadron dynamical considerations and give some quantitative estimates. We first discuss assumption (i) which is mainly related to the problem of total decay rates. We note that out of the two hard interaction types of fig. 1 we can construct the six duality diagrams depicted in fig. 2, the imaginary part of which gives the total decay rates. Only the first two diagrams (figs. 2a, b), corresponding to the charm quark disintegration with a passive spectator quark, have the parton limit. The third and the
(a)
(b)
(c )
(d)
(e )
(f )
Fig. 2. Duality diagrams for D m e s o n decays. The cross in the diagrams stands for the current creating D mesons.
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2 June 1980
fourth diagrams (figs. 2c, d) have a similar soft behaviour to the first two diagrams, although they do not have the parton limit, being of non-diagonal type. The last two diagrams come from the annihilation type interaction. The conventional argument that disfavours the annihilation contribution is valid only for the last diagram 2f, while the contribution from diagram 2e, in which the annihilation occurs only at the final state interaction is not, in general, small, because the c~ system (and consequently the d~ system) does not remember its original jPC. In figs. 2 a - d , the direct channel (decay channel) intermediate state is an exotic qglq?t state, which corresponds to the Pomeron exchange in the t-channel while in fig. 2e the direct channel intermediate state is a non-exotic ds state and correspondingly the t-channel is described by the u~ reggeon exchange. The nonleptonic decay widths are then written in general as [3,9] FNL(DO)/F(DO~ evX) = ( 2 f 2 +f2)_ + ~r ,
(6a)
FNL(D+)/F(D + ~ e vX) = (2f+2 + f2_) + (2f2 _ f2)p, (6b) with * 1 = 9 [(1+ k)f+ - (1 - k ) f _ ] 2 .
(7)
Here k stands for a color matching factor which represents the relative importance of the color mismatching quark bilinears to that of the matching one. This point is discussed in detail in the subsequent paragraph in connection with the exclusive two-body decays of D mesons. The symbols r and p represent the relative importance from the final state interactions of the reggeon exchange type (fig. 2e) and of the non-diagonal Pomeron exchange type (figs. 2c, d) in comparison with that of figs. 2a and b, which can be regarded as the Pomeron exchange contribution. In the case of complete interference [7] we expect p = 1, although for the heavier meson decays p becomes smaller, leaving the spectator quark less active. The reggeon exchange contribution r of the annihilation diagram is estimated based on Regge pole phenomenology [8], which gives r = r 0 X (S/So)-l/2 with r 0 ~ 1.2. Setting r 0 = 1 as a typical
,1 Note that the annihilation contains an extra N c factor compared with the disintegration. This leads, besides to 7r° ~ % to another possibility of "directly measuring" the colour
factor in the weak interaction. .2 To be more explicit ro = (~3~rK + ~ K ) / ~ K
Volume 93B, number 1, 2
PHYSICS LETTERS
value, s o = 1 GeV2 as a hadronic energy scale and s = (D meson mass) 2, we have r = 0.55 for charm meson decays. Let us next turn to assumption (ii) which is mainly related to the exclusive two-body decay widths. The two-body nonleptonic decays of D +, D O and F + mesons are described, in general, by four types of amplitudes, a, b, c and d, depicted in fig. 3 . We call (Es) (au) [(~u) (as)] a color favoured operator for n + [~0] production, because d and u [d and s] are in a color singlet combination and it is straightforward for them to create a rr+ IK°]. (Eu) (as) [(Es)(au)] on the other hand is called a color disfavoured operator for lr+ [~0] production. The amplitudes a and b [c and d] in fig. 3 correspond to the color favoured and disfavoured operators, respectively, when the decay proceeds through the disintegration process (fig. 1a) [annihilation process (fig. lb)]. We estimate the contribution from the color disfavoured operator in comparison to that of the favoured one, using the soft hadron interaction picture, where, of course, all soft gluon effects are considered to be taken into account. As depicted in fig. 4, the matrix element of the disfavoured operator can be expressed in terms of that of the favoured one (and vice versa) when the non-planar loop correction is taken into account. In other words, the favoured operator turns to the disfavoured one when operated on by the non-planar loop operator C, which is of a similar nature to that called "cylinder" in the dual topological unitarization scheme [10]. It is also known that the cylinder contribution in the pseudoscalar channel is always negative [ 11 ], which can be seen from the fact that both the isoscalar states 77 and
2 June 1980
(a)
Fig. 4. The disfavoured operator (b) is expressed in terms of the favoured operator (a) together with the non-planar operator ("cylinder"). r/' are heavier than the isovector state 7r * a. Using the cylinder operator C we can express the matrix elements of the disfavoured operator in terms of those of the favoured one as
(rr+l (Eu) (as) Ihadron)
=
(rr+l e (~s) (au) Ihadron)
= k(n+l(~s)(au)lhadron),
(8)
where k is a negative number. The same operator appears also in the annihilation type interaction of D O decay, where one has to express the matrix element of the disfavoured combination (Es) (au) in terms of the favoured combination (~u) (-ds). We note that the standard picture of color matching corresponds to k = ~, while the limit adopted by Fritzsch [5] corresponds to k = 1 and finally the "empirical" parametrization by Deshpande et al. [6] to k = - 1 / 6 . The value of k is estimated, for example, from the analysis of the two-body decays of K mesons which, in the pole model, leads to k = - 0 . 7 .
Numerical results. Given the value of the color matching factor k, we can estimate the numerical magnitude of D + and D O non-leptonic decay widths from eq. (6). For p = 1 as an extreme case of the maximal interference, r = 0.55 and k = - 0 . 7 as estimated ab ove, FNL(D0)/E(D0 -~ e vX) = 20.0,
(a)
{b)
PNL(D+)/F(D + --* evX) = 1.85. The corresponding semileptonic branching ratios are given by BR(D 0 ~ e v x ) = 4 . 5 %
(c )
( < 4% or 6 k 5%) ,
(all
Fig. 3. Two-body decay diagrams of D and F meson decays. See the text for details.
,3 Cf. the case of 2+ mesons where mf < mA2 and the cylinder contribution is positive. 127
Volume 93B, number 1, 2 BR(D + --> e vX) = 26%
PHYSICS LETTERS (23 -+ 6% or 17 + 6%),
which is in good agreement with the data [2] given in the parentheses. If we disregard for the moment the contribution from the annihilation type interaction in the exclusive D-decays, we find
F(D°~" K-n+) = 2 [(1 + k)f+ p~X
R01r0 )
+ (1 - k ) f _ q 2
L~+~-_ ~Z k~-_j~ 2 . 5 , (9)
for k ~ - 0 . 7 which is consistent with the experimental result of eq. (2) *4 . The numerical value for this ratio practically does not change, when we take the contribution from the annihilation type interaction into account, assuming the ratio to the disintegration type interaction to be the same as in the total decay rates. In this way we see that general considerations based on the soft hadron dynamics give us a satisfactory picture of the charmed meson decays. Finally, several comments are in order. (a) We are predicting r -= annihilation/disintegration cx 1/M, which should be compared, for example, with r ~x (FD/mc)2 of the standard model, where M and m e are a parent meson mass and a h e a w quark mass constituting the parent. I f F D ~ X ~ c , as empirically seems to be the case, both predictions give the same mass dependence. This behaviour of r also implies that for the decays of sufficiently h e a w quarks, like the b- or t-quark, the simple q u a r k - p a r t o n picture of the quark disintegration dominance describes the process correctly. A detailed analysis of D and F meson decays, as well as B meson (b-quark) decays will help to clarify the complexity of the heavy meson decays. (b) The same duality diagrams that appeared in the D O decays contribute to the F + decays. We expect, however, somewhat smaller Ptot(F+), Ptot(F+)/Ptot(D 0) 0.7 due to the low intercept of the ~ trajectory exchanged in the annihilation diagram in F + decays. (c) We note that in the exclusive and also in the annihilation processes, the short distance coefficients f_. appear always in the combination of (1 + k)f+_, as is seen, .4 Strictly speaking we have to replace (1 + k) by (1 + k)/(1 - k =) which comes about when we sum the higher order contributions from the non-planar diagrams. But this is irrelevant as far as we are concerned with ratios.
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2 June 1980
for example, in eq. (9). This means that the negative k gives effectively a further enhancement (suppression) of the short distance coefficient f _ (f+), suggesting a possibility o f explaining the A1 = 1/2 rule in strange particle decays without penguin diagram. Especially, k = - 1 leads to perfect octet enhancement. In this limiting case, which corresponds to maximal r/-r~' mixing through the cylinder interaction in theO - + channel, the ratio F(D + -+ K-Tr+)/P(D ° ~ K0n0) is 2. We note, however, that our solution is not identical to the results of Fritzsch [5], who also gets 2 for this ratio. His solution corresponds to maximal mixing in the opposite direction, i.e., k = 1, which corresponds to the simple parton picture. S.K. wants to thank Prof. H. Satz for the kind hospitality extended to him at the University of Bielefeld. M.K. thanks Prof. R.J.N. Phillips for his warm hospitality at Rutherford Laboratory, where this work has been stimulated.
References [1] N. Cabibbo and L. Maiani, Phys. Lett. 73B (1978) 418; D. Fakirov and B. Stech, Nucl. Phys. B133 (1978) 315. [2] Talks by J. Kirkby and V. Luth at the Intern. Symp. on Lepton and photon interactions at high energies (FNAL, Batavia, IL, 1979). [3] M.K. Gaillard and B.W. Lee, Phys. Rev. Lett. 33 (1974) 108; G. Altarelli and L. Maiani, Phys. Lett. 52B (1974) 35; J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B100 (1975) 313. [4] H. Fritzsch and P. Minkowski, Bern preprint (1979); M. Bander, D. Silverman and A. Soul, UCI technical report 79-73. [5] H. Fritzsch, Phys. Lett. 86B (1979) 343. [6] N. Deshpande, M. Gronau and D. Sutherland, Fermilab preprint, FERMILAB-PUB-79/70 (1979). [7] B. Guberina et al., Max Planck Institut preprint MPIPAE/PTh 45/79 (1979). [8] See, for example, Chart H.-M., Nucl. Phys. B86 (1975) 479;B92 (1975) 13; V. Barger and R.J.N. Phillips, Nucl. Phys. B32 (1971) 93. [9] G. Altarelli, N. Cabibbo and L. Maiani, Phys. Rev. Lett. 35 (1975) 635. [10] G.G. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) 265. [ 11] T. Inami, K. Kawarabayashi, and S. Kitakado, Phys. Lett. 61B (1976) 60.