Penguins, Trees and Final State Interactions in B Decays in Broken SU3

\authorHarry J. Lipkin

Department of Particle Physics

Weizmann Institute of Science

Rehovot 76100, Israel

and

School of Physics and Astronomy

Raymond and Beverly Sackler Faculty of Exact Sciences

Tel Aviv University

Tel Aviv, Israel

and

High Energy Physics Division

Argonne National Laboratory

Argonne, IL 60439-4815, USA

June 29, 1995

\abstractThe availability of data on decays to strange quasi-two-body final states, either with or without charmonium opens new possibilities for understanding different contributions of weak diagrams and in particular the relative contributions of tree and penguin diagrams. Corresponding and decays to charge conjugate final states are equal in the SU(3) symmetry limit and the dominant breaking mechanism is given by ratios of CKM matrix elements. Final State Interactions effects should be small, because strong interactions conserve and should tend to cancel in ratios between charge conjugate states. Particularly interesting implications of decays into final states containing and are discussed.

A large number of relations between ratios of and amplitudes to charge conjugate final states are obtainable by extending the general SU(3) symmetry relations for decays found by Gronau et al\REF\ROSGROMichael Gronau, Jonathan L. Rosner and David London, Phys. Rev. Lett. 73 (1994) 21; Michael Gronau, Oscar F. Hernandez, David London and Jonathan L. Rosner, Phys. Rev. D52 (1995) 6356 and 6374 beyond the two-pseudoscalar case to all quasi-two-body charmless strange decays and charmonium strange decays. These relations are of particular interest because: (1) they relate a large number of decay ratios in the SU(3) symmetry limit, and (2) strong final state interactions should cancel out in ratios between decay amplitudes to charge conjugate final states which have the same strong final state rescattering. We extend the treatment of ref. by noting the following points:

Many relations are obtainable with the U spin SU(2) subgroup of SU(3) and in particular the discrete transformation (Weyl reflection) which simply interchanges the d and s flavors. \pointU spin relations can be valid also for contributions from the electroweak penguin diagrams which break SU(3) because the photon and the Z are both singlets under U spin (they couple equally to and quarks) while they contain octet components in SU(3) (their couplings to quarks differs from those to and ). \pointRelations obtained from the discrete transformation do not require that both final hadrons are in the same SU(3) octet. Thus they apply equally well to other channels than PP. \pointRatios of amplitudes that go into one another under the transformations and have final states which are charge conjugates of one another should be insensitive to strong final state interactions which are invariant under charge conjugation.

With this approach, we find the following relations

where denotes or any resonance, denotes or any resonance, denotes any charmonium state and denotes an SU(3)-breaking parameter which may be different for different final states. Similarly for the charge conjugate states,

Note that only tree and penguin diagrams contribute to these transitions and that the individual tree and penguin diagrams, including both gluonic and electroweak penguins, also go into one another under this transformation.

The final states in the numerator and denominator of each ratio go into one another under charge conjugation. Thus final state strong interactions which conserve should be the same and therefore not disturb the equalities. These ratios may then give information about the relative contributions of different weak diagrams without the usual caveats about unknown strong phases.

In the limit of exact SU(3) symmetry . Thus the relations (1) hold when SU(3) is broken by the same factor in all cases. This is not expected to be valid everywhere. Thus the relations (1) provide a means for selecting groups of related decay modes which all have the same SU(3) breaking factor.

Since experimental data generally quote branching ratios rather than partial widths or amplitudes, we note that the relations (1b) can be rearranged to give ratios of branching ratios from the same initial state; e.g.

These relations can be used to distinguish between decays having the same SU(3) breaking factor and those having different SU(3) breaking factors.

The dominant SU(3) breaking effect is in the difference between the weak strangeness-conserving and strangeness-changing vertices. For the charmless tree diagrams this breaking introduces a common factor into each ratio, thereby leaving all the ratios (1a) and (1b) equal to one another and only changing the value to instead of unity. For the charmonium and charmed pair tree diagrams the appropriate breaking factor which is nearly the same as that of the charmless tree diagrams.

If only tree diagrams contribute, relations for the charmonium branching ratios analogous to eqs. (2) can also be written.

Penguin contributions will have a different SU(3) breaking factor ; e.g or .

The penguin diagram is expected to dominate in the charmless decays, and perhaps also in the charmless decays, since the charmless tree diagram is Cabibbo suppressed. The tree diagram is expected to dominate in the charmonium and charmed pair decays, where the tree is Cabibbo favored while the penguin requires the creation of a heavy quark pair by gluons from the vacuum. These features can be checked out by experimental tests of the relations (1). The most interesting cases are those in which both penguin and tree contributions are appreciable and CP violation can be observed in the interference. These decay modes can be identified by violations of the relations (1). The most favorable candidates seem to be the decays where one of the two weak vertices is Cabibbo favored and will have a better chance to compete with the penguin.

We can correct the relations (1) for the difference between penguin and tree SU(3) breaking by writing for example:

where , , and denote respectively the contributions to the decay amplitude and to the charge conjugate decay amplitude from tree and penguin diagrams and , , and denote respectively the analogous contributions to the corresponding decay amplitudes. We can obtain similar relations for final states containing the instead of the by noting that the corresponding and decay modes are related if electroweak penguins are neglected, because the tree diagram produces both and via their common component and the penguin produces both via their common component\REF\PENGRHOHarry J. Lipkin, Physics Letters B353 (1995) 119.

We can also consider linear combinations that project out direct and interference terms:

where we have noted that and .

Any violation of the relations (1) could indicate existence of both tree and penguin contributions and also offer the possibility of measuring their relative phase. Since the penguin and tree can have different weak phases, violation can be observable as asymmetries in decays of charge-conjugate mesons into charge-conjugate final states and also in differences between the charge-conjugate ratios (4a) and (4b).

The relations (4) also provide additional input from decays that can be combined with isospin analyses of decays to separate penguin and tree contributions\REF\PBPENGYosef Nir and Helen Quinn Phys. Rev. Lett. 67 (1991) 541; Harry J. Lipkin, Yosef Nir, Helen R. Quinn and Arthur E. Snyder, Physical Review D44 (1991) 1454. A similar additional input is obtainable from combining decay modes with isospin analyses of other decay modes\REF\GRONAUM. Gronau and D. Wyler, Phys. Lett. B265 172 (1991); M. Gronau and D. London, Phys. Lett. B253 483 (1991); I. Dunietz, Phys. Lett. B270 74 (1991). .

Similar relations, with different values of T and P hold for the other ratios. Before extending this result to other cases, we note that additional SU(3) breaking can arise from differences in hadronic form factors. This can be seen at the quark level by noting the quark couplings in the color-favored and color-suppressed tree diagrams and penguin diagrams:

where and denote respectively color-favored and color suppressed form factors which are point-like and proportional to wave functions at the origin; e.g. to factors like or , while and denote respectively color-favored and color suppressed form factors which involve overlap integrals on a hadronic scale. The pairs of color favored transitions (6) are seen to involve different form factors. One has a hadronic nonstrange form factor and a pointlike strange form factor; the other has a hadronic strange form factor and a pointlike nonstrange form factor. This form factor difference has been recently pointed out\REF\CLOLIP12Frank E. Close and Harry J. Lipkin, Physics Letters B405 (1997) 157 as possibly responsible for a reversal of relative phase of the two contributions for exclusive decay modes where there are nodes in wave functions.

The color suppressed tree and the penguin transitions (7) are seen to involve identical form factors in both cases. The trees have the same or form factor and charge-conjugate hadronic and form factors. Pairs of penguin diagrams always have the same form factors, since the hadronization into the final state occurs from charge conjugate intermediate states of a single pair and a gluon or electroweak boson. The only possible difference arises from the slight difference in the hadronic scale of the and wave functions.

We therefore extend the result to all cases where form-factor corrections are expected to be small: those having no color-favored tree contribution and no component in the wave function as in and .

where can denote any neutral isovector or ideally mixed nonstrange isoscalar meson; e.g. , , or . Similarly will denote any charged meson pair; e.g. , or . Each ratio is equal to an expression analogous to the right hand side of (3) with appropriate different values for and .

The color-favored transitions to charged final states may have form factor corrections. Let denote this form factor correction, where A and B denote the two particles in the final state. Then

Here the approximate equalities are exact if the tree contribution is negligible, and will be violated where both contributions are appreciable. In the latter case, each ratio is again equal to an expression analogous to the right hand side of (3) with appropriate different values for and .

The SU(3)-breaking effect is different in decays to final states containing and because the discrete transformation (Weyl reflection) which simply interchanges the d and s flavors interchanges the and components in the and system, which we denote by and . These decays are of particular interest since recently reported high branching ratios\REF\CLEOB. Behrens, [CLEO] talk at Physics and CP Violation, Waikiki, HI (March 1997) for strange decays to final states has led to suggestions for new types of diagrams\REF\ATSOND. Atwood and A. Soni, Phys. Lett. B405 (1997) 150.

We first note that the Cabibbo-favored tree diagram is expected to be dominant in and decays with charmonium and that in these decay modes they are produced via in decay and via in decay. Thus these decays immediately provide a measure of the mixing. We first obtain the SU(3) symmetry result

This immediately gives the values of the strange and nonstrange components in the and and a condition which must be satisfied if the mixing is described by a matrix.

A failure of the relation (11b) would indicate a breakdown of the simple mixing model.

We now investigate the decays into strange final states with and . The standard penguin diagram predicts\REF\PKETAHarry J. Lipkin, Phys. Lett. B254 (1991) 247, and B283 (1992) 421

where denotes the theoretical partial width without phase space corrections. We have assumed SU(3) symmetry with one of the standard mixings:

and noted that the penguin diagram creates the two states and , with a relative phase depending upon the orbital angular momentum of the final state.

where is a parameter describing SU(3) symmetry breaking and can also denote any resonance. The sum rule inequality (12c) holds generally for all mixing angles and for all positive values of .

The dramatic reversal of the ratio in the final states with occurs naturally in this penguin interference model and does not occur in any other suggestion for enhancing the . Present data indicate suppression. Better data showing significant suppression will rule out most other enhancement mechanisms.

A violation of the inequality (12c) would require an additional contribution. The Cabibbo favored charmed tree diagram can contribute via hidden or intrinsic charm in the wave function and may contribute appreciably even though the charm in the is quite small.

We now estimate the effect of an additional contribution from the production of the and via an additional diagram which in the SU(3) symmetry limit produces the states , and with equal amplitudes.

where defines the extra contribution strength. Consider for example the case

The inequality (16c) holds for all mixing angles and all . Thus a comparatively small contribution interfering constructively with the dominant penguin can give an appreciable enhancement. With sufficiently large to give (25/3) for ( and a 50:1 ratio favoring over , the enhancement of over is only a factor of two for the final state. The drastic difference between the and branching ratios still persists if both the penguin and the extra contribution are present, in contrast to the case where the extra contribution is dominant. Thus the data are important for determining the exact mechanism for the enhancement.

Discussions with M. Gronau and J. L. Rosner are gratefully acknowledged.

This work was supported in part by grant No. I-0304-120-.07/93 from The German-Israeli Foundation for Scientific Research and Development and by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38.