###### Abstract

We consider the weak decays of a B meson to final states that are mixtures of S-wave radially excited components. We consider non leptonic decays of the type , and where , and are higher , and resonances. We find such decays to have larger or similar branching ratios compared to decays where the final state , and are in the ground state. We also study the effect of radial mixing in the vector and the pseudoscalar systems generated from hyperfine interaction and the annihilation term. We find the effects of radial mixing to be small and generally negligible for all practical purposes in the vector system. However, in the system the effects of radial mixing are appreciable and seriously affect decay branching ratios for . In particular we find that nonstandard mixing can resolve the puzzles in decays.

UdeM-GPP-TH-01-81

February 2001

Weak decays to final states with Radial Excitation Admixtures

Alakabha Datta^{1}^{1}1email:
, Harry J. Lipkin ^{2}^{2}2email:
and
Patrick. J. O’Donnell ^{3}^{3}3email:

Laboratoire René J.-A. Lévesque, Université de Montréal,

C.P. 6128, succ. centre-ville, Montréal, QC, Canada H3C 3J7

Department of Particle Physics,

Weizmann Institute,

Rehovot 76100, Israel

and

School of Physics and Astronomy,

Tel-Aviv University,

Tel-Aviv 69978, Israel

Department of Physics and Astronomy,

University of Toronto, Toronto, Canada.

## 1 Introduction

Nonleptonic decays play a very important role in the study of CP violation. It is expected that these studies will test the standard model(SM) picture of CP violation or provide hints for new physics. Most studies of two body nonleptonic decays have concentrated on processes of the type where both and are mesons in the ground state configuration. Here we want to look at nonleptonic B decays to final states where one of the final state meson contain admixtures of radially excited components. We expect such decays to have larger or similar branching ratios compared to decays where the final state contains the same meson in the ground state. There is an easy explanation for such a statement. For simplicity let us consider a simple model in which and M is a simple flavor eigenstate with no flavor mixing beyond isospin; e.g. , or and we are interested in comparing the branching ratios for the final states containing the ground state configuration of meson and its radial excitation. Some possible examples are:

(1) |

(2) |

(3) |

where , and are radially excited states.

We assume an extreme factorization approximation in which the quark decays into a pion and a -quark and we neglect the relative Fermi momentum of the initial and the spectator . The quark transition for the processes (1) and (2) is then

(4) |

where denotes the final momentum of the . For the process (3) the quark transition is essentially similar to the one above

(5) |

where now denotes the final momentum of the .

Concentrating on the processes (1) and (2) the transition matrix for the full decay has the form

(6) |

where denotes the transition matrix for the hadronic decay which factors into a weak matrix element at the quark level denoted by and a fragmentation matrix element denoted by describing the transition of a quark with momentum and an antiquark with zero momentum to make a meson with momentum .

It is immediately clear that if the final momentum is large, the fragmentation matrix element will depend upon the high momentum tail of the meson wave function. This might tend to favor radial excitations over ground states, since the radial excitations are expected to have higher kinetic energies. We now note that the harmonic oscillator wave functions commonly used in hadron spectroscopy have a Gaussian tail for their high momentum components and this can suppress the fragmentation matrix element in comparison with wave functions from a different confining potential. Hence the branching ratio to a final state which is radially excited will be sensitive to the choice of the confining potential.

So far we have assumed the physical states to be pure radial excitations. However additional interactions can mix the various radial excited components. For instance hyperfine interactions can mix radial excitations with the same flavor structure and so in general in the , and system the various physical states will be admixtures of radial excitations [1, 2]. Flavor mixing in the vector system is known to be small but is important in the pseudoscalar sector. Here the mixing in the system receives an additional significant contribution from the annihilation diagram that leads to flavor mixing of the strange and non strange parts of the wavefunction. The mixing in the pseudoscalar sector, therefore, is different from the ideal mixing found in the system. It is also possible for the annihilation term to mix states that are radial excitations allowing the the wavefunction to contain radially excited components . Such non standard mixing can have important implication for the non leptonic decays .

In the transitions chosen in (1),(2) and (3) the radially excited meson must include the spectator quark and therefore must depend upon the fragmentation matrix element . For the case of decays, factorization results in the kaon leaving the weak vertex with its full momentum and the remaining quark carries the full momentum of the final meson. There is therefore a form factor in which there is a large internal momentum transfer needed to hadronize this quark with the spectator antiquark. This might favor the radial excitation if it has a much higher mean internal momentum. However things are more complicated here as the can also be produced by an pair in a penguin diagram without containing the spectator quark. One possibility is when the quark in the QCD penguin combines with the quark from the transition to form the . Another possibility is when a and pair (where ), appearing in the same current in the effective Hamiltonian, hadronizes to the . In the diagrammatic language this is often represented as a “gluon” splitting into a pair which then hadronizes into a . This term is usually called OZI suppressed [3, 4] as in most decays the contribution of this term is indeed suppressed with respect to other terms in the decay amplitude. This may not be the case in the decays where the OZI term can be of comparable size as other terms in the decay amplitude and in particular we show that can have significant contribution from the OZI suppressed term.

This paper is organized in the following manner: In the next section we study the mass mixing in the vector meson sector, involving the , and , and the pseudoscalar sector involving the . In section 3 we present a general treatment of nonleptonic decays using the effective Hamiltonian and the factorization assumption. We then show how this approach is related to the diagrammatic approach of studying nonleptonic decays. In section 4 we study nonleptonic decays of to final states involving higher resonant , and states. This is followed by section 5 where we make predictions for and and comment on the relevance of the OZI suppressed term in the calculation of these decays. Finally we present our conclusions.

## 2 Mass mixing in the vector meson and pseudoscalar sector

We start with the mixing for the system. To obtain the eigenstates and eigenvalues we diagonalize the mass matrix which has the form

(7) |

where and are the quark spin operators and masses. Here and the basis states are chosen as and . In the above equation is the binding energy of the radially excited state and is the strength of the hyperfine interaction. Note we are only presenting results for the neutral mesons. A similar treatment also can be used for the charged mesons. We will use a very simple model for confinement in our calculations as we do not intend to present a detailed study of light meson spectroscopy. Our aim, as already stated in the previous section, is to study non leptonic decays of the meson to radially excited light meson states as well as to study the effects of radial mixings in the non leptonic decays . We believe the conclusions reached on the basis of our calculations are likely to hold true in a more detailed model of confinement.

To begin with, we use the same harmonic confining potential as well as the other parameters used in Ref[2] to obtain the eigenstates and eigenvalues for the mass matrix in Eqn. 7. The various parameters used in the calculation are 0.350 GeV, 0.503 GeV, the angular frequency, 0.365 GeV and 0.09.

We obtain for the eigenvalues and eigenstates in the system

(8) |

To see how this result changes with a different confining potential we use a power law potential [5] . We will use a linear and a quartic confining potential and compare the spectrum with that obtained with a harmonic oscillator potential. To fix the coefficient we require that the energy eigenvalues of the Schrödinger equation are similar in the least square sense with the energy eigenvalues used in Ref[2]. So for example, for the linear potential, we demand that

is a minimum. This fixes the constant in and we obtain

(9) |

We follow the same procedure for the quartic potential and obtain

(10) |

We observe that the mass eigenstates and eigenvalues are not very sensitive to the confining potential and the radial mixing effects are small.

We next turn to mixing in the system. In the traditional picture the and mesons are mixtures of singlet and octet states and of .

(11) | |||||

(12) | |||||

(13) |

where the mixing angle lies between and [10].

To obtain the eigenstates and eigenvalues in the system we diagonalize the mass matrix

(14) | |||||

This has a similar structure as the system but now we have the additional annihilation contribution with strength that causes flavor mixing. In our calculations we use the phase convention in Ref[2] where the wavefunctions at the origin in configuration space, which enter in the hyperfine and annihilation terms in the mass matrix, are positive(negative) for the even(odd) radial excitations.

We try to fit the values of and to the measured masses. The mass matrix is matrix which we diagonalize to make predictions for 6 masses and mixings. However, for the sake of brevity we will only give the predictions for and masses and wavefunctions. Several solutions that give acceptable values of the masses can be obtained. We choose solutions for the linear, quadratic and quartic confining potentials that make similar predictions for the masses:

For the linear potential we obtain with and

(15) | |||||

For the harmonic potential we obtain with and

(16) | |||||

Our results for the harmonic potential is similar to the results obtained in Ref[2] where a slightly different mass mixing matrix than the one used here has been used to obtain the mixing.

Finally, for the quartic potential we obtain with and

(17) | |||||

It is clear that the eigenstates of system are sensitive to the confining potential and there can be substantial radial mixing which can then affect the predictions for the decays . We note that as we move from the linear to the quartic potential the mixing deviates more significantly from the ideal mixing case. This may be understood from the fact that the fit to annihilation term, , is the largest for the quartic potential which leads to the largest deviation from the ideal mixing case. A standard mixing often used in the literature is given by [6, 7]

(18) |

We can then write the states obtained with the various confining potential and keeping only the ground states, in terms of the the states defined above. For the linear potential we find

(19) |

For the harmonic potential

(20) |

and finally for the quartic potential one finds

(21) |

This shows that all three confining potentials give mixings for the that have substantial overlap with the standard mixing but the mixing with the quartic potential is closest to the standard mixing in the sense that here one has the smallest component of the in .

We now turn to the system. As in the the system we diagonalize the mass matrix in Eqn. 14. We use the same value for the hyperfine interaction as used for the system. Again, for the sake of brevity, we only give the wavefunctions for the ground and the first excited states.

For the linear potential we obtain with and

(22) | |||||

For the harmonic potential we obtain with and

(23) | |||||

For the quartic potential we obtain with and

(24) | |||||

As in the system we find the mixing to be insensitive to the confining potential and we also find the effects of radial mixing to be small. We also find, as expected, a smaller value for the annihilation term in the fits to the masses as compared to the pseudoscalar system.

## 3 Effective Hamiltonian, Factorization and the Diagrammatic approach

In the Standard Model (SM) the amplitudes for hadronic decays are generated by the following effective Hamiltonian [8]:

(25) |

where the superscript indicates the internal quark, can be or quark and can be either a or a quark depending on whether the decay is a or process. The operators are defined as

(26) | |||||

where , and is summed over u, d, and s. and are the tree level and QCD corrected operators. are the strong gluon induced penguin operators, and operators are due to and Z exchange (electroweak penguins), and “box” diagrams at loop level. The Wilson coefficients are defined at the scale and have been evaluated to next-to-leading order in QCD. The are the regularization scheme independent values obtained in Ref. [9]. We give the non-zero below for GeV, , and GeV,

(27) |

where is the number of colors. The leading contributions to are given by: is given by . The function and

(28) |

All the above coefficients are obtained up to one loop order in electroweak interactions. The momentum is the momentum carried by the virtual gluon in the penguin diagram. When , develops an imaginary part. In our calculation, we use, for the current quark masses at the scale , MeV, MeV, MeV, GeV [10, 11] and use .

The structure of the effective Hamiltonian allows us to write the amplitude for as

(29) |

Now we can write the tree amplitude as

(30) |

where from Eqn. 27 . In the factorization assumption there are two contributions to the tree matrix element, . To be specific consider the decay . In this case there can be two contributions to , given in the factorization assumption by

(31) |

In the diagrammatic language the first term, , corresponds to a quark transition to a quark and a which turns into a . The quark then combines with the spectator quark to form the particle. In the term , the quark from the transition combines with the quark from the to form the particle while the quark from the combines with the spectator quark to form the . This is the color suppressed diagram and from the expression above we see that there is an additional suppression coming from the Wilson’s coefficients and so, effectively is suppressed by a factor of relative to .

We now turn to the penguin contribution, and for simplicity, we just concentrate on the the penguin, . We can write

(32) |

Again from the values of the Wilson’s coefficients given in Eqn. 27 we can write

(33) | |||||

where from Eqn. 27 and .

In the diagrammatic approach the quarks appearing in the operator appears from a “gluon” splitting into a pair while in the case the operators it is a “”, “Z” splitting into a pair.

Concentrating on only the term proportional to , we can write in the factorization assumption,

(34) |

We see that the second term, has a suppression factor of . This term is called OZI suppressed and in the diagrammatic language this is shown as a “gluon” splitting into a quark-antiquark pair which then hadronizes to a hadron. Of course for a real gluon this process is forbidden as the color octet gluon cannot form a color singlet hadron. Note that there can be other OZI violating diagrams that have been considered to explain the large branching ratios in the decay and the semi-inclusive decay . In these diagrams the enhanced branching ratios are due to the anomaly, gluon couplings to the flavour singlet component of the or the intrinsic charm content of the [12, 13]. We will not consider such diagrams in our analysis.

The terms represented by , in the diagrammatic approach, has a “gluon” splitting into a quark- antiquark pair but now the antiquark combines with the quark, coming from the transition, to form a meson while the other quark combines with the spectator quark to form the second meson in the final state. The two terms represented in represent the cases where the quark-antiquark pair from the “gluon” is a and a pair.

One can do the same exercise with the term proportional to in Eqn. 33 and in this case the OZI violating term is suppressed by . Note the term from the electroweak penguin term does not have any suppression. This is expected as a pair from a or boson is in a color singlet state and therefore can form a hadron.

For the case of the term and are similar to the one above.

(35) |

Note we again find the OZI term to be suppressed by a factor . However if terms in interfere destructively then the OZI terms may become important. Note the suppression in the OZI term can also be diluted if the contributions from the , and term interfere constructively in .

In the decay the OZI suppressed term does not play an important role, as these decays are not penguin dominated because the CKM factors in the tree and penguin terms are of the same order and the penguins are loop suppressed. This fact is also supported by recent experimental measurement of the branching ratios and [14]. Note that in Eqn. 34, from the flavor structure of the and wavefunction, it is obvious that the two terms in in interfere destructively for the but constructively for the . Furthermore the OZI suppressed term, for the vanishes, neglecting the electroweak contribution, but not for the . This means that the penguin term for is smaller than in . Hence if the penguin terms were dominant then there would be a significant difference between the branching ratios and . Hence the small measured difference in the branching ratios for and implies relatively small penguin effects in these decays.

However decays of the type are dominated by penguin terms because of large CKM factors in the penguin terms compared to the tree term. Here, the OZI suppressed terms may have significant effects on the predictions of these decays. Note that we expect the OZI suppressed terms to be more important in than in decays where is a pseudoscalar and is a vector state. This follows from the fact that in and decays we know that the OZI-forbidden process requires three gluons for coupling to a vector meson and two gluons for coupling to a pseudoscalar. Thus one would expect that the contribution of the OZI suppressed term should be much smaller in the and decays than in and decays [15, 16]. One of the authors of this work has shown that one can make definite predictions about the branching ratios and [17, 18] if one assumes that the OZI terms are forbidden. We will first derive these predictions in the language of effective Hamiltonian and using the factorization assumption. We then study how the predictions change if we use non standard mixing in the sector and if we include the OZI terms.

## 4 transitions.

In this section we study decays of the type where . As we found in section 2 the wavefunction of has the general form,

(36) |

We can then write,

(37) |

We now consider the ratios of the following decays

(38) |

(39) |

(40) |

(41) |

As discussed in the previous section we can neglect the penguin contribution to these decays. Note that for the decay there is no contribution from the QCD penguin and the dominant electroweak penguin term has the same structure as the tree amplitude and hence the ratio remains essentially the same even in the presence of penguin terms. We then obtain