Flavor Symmetry and Charm Decays
aa r X i v : . [ h e p - ph ] O c t Proceedings of the CHARM 2007 Workshop, Ithaca, NY, August 5-8, 2007 Flavor Symmetry and Charm Decays
Bhubanjyoti Bhattacharya and Jonathan L. Rosner
Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL 60637
A wealth of new data in charmed particle decays allows the testing of flavor symmetry and the extraction ofkey amplitudes. Information on relative strong phases is obtained.
1. Introduction
The application of flavor symmetries, notablySU(3), to charmed particle decays can shed light onsome fundamental questions. Often it is useful toknow the strong phases of amplitudes in these decays.For example, the relative strong phase in D → K − π + and D → K − π + is important in interpreting decaysof B mesons to D X and D X [1, 2]. Such strongphases are non-negligible even in B decays to pairs ofpseudoscalar mesons ( P ) despite some perturbativeQCD expectations to the contrary, and can be evenmore important in D → P P decays. In the present re-port we shall illustrate the extraction of strong phasesfrom charmed particle decays using SU(3) flavor sym-metry, primarily the U-spin symmetry involving theinterchange of s and d quarks.We begin in Section 2 by discussing the overall di-agrammatic approach to flavor symmetry. In Section3 we treat Cabibbo-favored decays, turning to singly-Cabibbo-suppressed decays in Section 4 and doubly-Cabibbo-suppressed decays in Section 5. We note spe-cific applications to D and D decays to K − π + inSection 6, mention some other theoretical approachesin Section 7, and conclude in Section 8.
2. Diagrammatic amplitude expansion
We use a flavor-topology language for charmed par-ticle decays first introduced by Chau and Cheng [3, 4].These topologies, corresponding to linear combina-tions of SU(3)-invariant amplitudes, are illustrated inFig. 1. Cabibbo-favored (CF) amplitudes, propor-tional to the product V us V ∗ cs of Cabibbo-Kobayashi-Maskawa (CKM) factors, will be denoted by un-primed quantities; singly-Cabibbo-suppressed ampli-tudes proportional to V us V ∗ cs or V ud V ∗ cd will be de-noted by primed quantities; and doubly-Cabibbo-suppressed quantities proportional to V us V ∗ cd will bedenoted by amplitudes with a tilde. The relative hi-erarchy of these amplitudes is 1 : λ : − λ : − λ , where λ = tan θ C = 0 . ± .
002 [5]. Here θ C is the Cabibboangle.
3. Cabibbo-favored decays
A detailed discussion of amplitudes and their rel-ative phases for Cabibbo-favored charm decays wasgiven in Ref. [6]. The main conclusions of that analysiswere large relative phases of the C and E amplitudesrelative to the dominant T term, and an approximaterelation A ≃ − E . The present updated data confirmthese results.In Table I we show the results of extracting ampli-tudes A = M D [8 π B ¯ h/ ( p ∗ τ )] / from the branching ra-tios B and lifetimes τ , all from Ref. [5] unless otherwisenoted. Here M D is the mass of the decaying charmedparticle, and p ∗ is the final c.m. 3-momentum.The extracted amplitudes, with T defined to be real,are, in units of 10 − GeV: T = 2 .
71 + 0 i ; (1) C = − . − . i ; δ ( CT ) = − ◦ ; (2) E = − .
71 + 1 . i ; δ ( ET ) = 115 ◦ ; (3) A = 0 . − . i ; δ ( AT ) = − ◦ . (4)These values update (and are consistent with) thosequoted with less precision in Ref. [6]. New (mainlylower) preliminary branching ratios for many D s de-cays reported at this Conference [7] will change someof the results slightly once they are incorporated intoaverages.The Cabibbo-favored amplitudes are shown on anArgand diagram in Fig. 2. Here A was extracted from D s → π + η and D s → π + η ′ ; the amplitude A for D s → K K + is then predicted to be 2 . × − GeVvs. (2 . ± . × − GeV observed. Note the im-portance of the E and A ≃ − E amplitudes.
4. Singly-Cabibbo-suppressed decays4.1. SCS decays involving pions andkaons
We show in Table II the branching ratios, ampli-tudes, and representations in terms of reduced ampli-tudes for singly-Cabibbo-suppressed (SCS) charm de-cays involving pions and kaons. The ratio of primed(SCS) to unprimed (CF) amplitudes is expected to betan θ C ≃ . Proceedings of the CHARM 2007 Workshop, Ithaca, NY, August 5-8, 2007
Figure 1: Flavor topologies for describing charm decays. T : color-favored tree; C : color-suppressed tree; E exchange; A : annihilation.Table I Branching ratios, amplitudes, and graphical representations for Cabibbo-favored charmed particle decays.Meson Decay B p ∗ |A| Rep.mode (%) (MeV) (10 − GeV) D K − π + ± ± T + EK π ± ± C − E ) / √ K η ± ± C/ √ K η ′ ± ± − ( C + 3 E ) / √ D + K π + ± ± C + TD + s K K + ± ± C + Aπ + η ± ± T − A ) / √ π + η ′ ± ± T + A ) / √ The deviations from flavor SU(3) implicit in Ta-ble II are well known. We shall discuss amplitudesin units of 10 − GeV. If one rescales the CF ampli-tudes by the factor of tan θ C , one predicts |A ( D → π + π − ) | = |A ( D → K + K − ) | = 5 .
78, to be com-pared with a smaller observed value for π + π − and alarger observed value (by a factor of √
2) for K + K − .One can account for some of this discrepancy via theratios of decay constants f K /f π = 1 .
22 and form fac-tors f + ( D → K ) /f + ( D → π ) >
1. Furthermore,one predicts |A ( D → π π ) | = 4 .
45 (larger than ob-served) and |A ( D + → π + π ) | = 2 .
25 (smaller thanobserved), which means that the ππ isospin triangle[associated with the fact that there are two indepen-dent amplitudes with I = (0 ,
2) for three decays] hasa different shape from that predicted by rescaling theCF amplitudes. One predicts |A ( D + → K + K ) | = |A ( D s → π + K ) | = 5 .
79; experimental values are(11%,1%) higher. The decay D → K K is for- bidden by SU(3); the branching ratio of 2 B ( D → K S K S ) = (2 . ± . ± . ± . × − reportedby CLEO [7] is more than a factor of two below thatquoted in Table II (based on the average in Ref. [5])and so does offer some evidence for the expected sup-pression. η, η ′ The amplitudes C and E extracted from Cabibbo-favored charm decays imply values of C ′ = λC and E ′ = λE which may be used in constructing am-plitudes for singly-Cabibbo-suppressed D decays in-volving η and η ′ . In Table III we write amplitudesmultiplied by factors so that they involve unit co-efficient of an amplitude SE ′ describing a discon-nected “singlet” exchange amplitude for D decays[8]. Similarly the decays D + → ( π + η, π + η ′ ) and D + s → ( K + η, K + η ′ ) may be described in terms of a2 roceedings of the CHARM 2007 Workshop, Ithaca, NY, August 5-8, 2007 Table II Branching ratios, amplitudes, and decomposition in terms of reduced amplitudes for singly-Cabibbo-suppressed(SCS) charm decays involving pions and kaons.Meson Decay B p ∗ |A| Rep.mode (10 − ) (MeV) (10 − GeV) D π + π − ± ± − ( T ′ + E ′ ) π π ± ± − ( C ′ − E ′ ) / √ K + K − ± ± T ′ + E ′ ) K K ± ± D + π + π ± ± − ( T ′ + C ′ ) / √ K + K ± ± T ′ − A ′ D + s π + K ± ± − ( T ′ − A ′ ) π K + ± ± − ( C ′ + A ′ ) / √ C + T , C + A ,and E + T correspond to measured processes; the magni-tudes of other amplitudes listed in Table I are also neededto specify the reduced amplitudes T , C , E , and A . disconnected singlet annihilation amplitude SA ′ , writ-ten with unit coefficient in Table III. For experimen-tal values we have used new CLEO measurements asreported in Ref. [9]. (See Table IV.)We show in Fig. 3 the construction proposed in Ref.[8] to obtain the amplitude SA ′ . Two solutions arefound. In one, | SA ′ | is uncomfortably large in com-parison with the “connected” amplitudes, while in theother | SA ′ | is smaller, but nonzero. Correspondingstudies of the D decays listed in Table III [10], whichawait further analysis by the CLEO Collaboration,will permit determination of the corresponding am-plitude SE ′ if one or more consistent solutions arefound. Figure 3: Graphical construction to obtain the discon-nected singlet annihilation amplitude SA ′ from magni-tudes of SCS D + and D + s decays involving η and η ′ . Black: D + → ηπ + . Green: D + → η ′ π + . Blue: D + s → ηK + .Red: D + s → η ′ K + . The small black circles show the solu-tion regions.
5. Doubly-Cabibbo-suppressed decays
In Table V we expand amplitudes for doubly-Cabibbo-suppressed decays in terms of the reducedamplitudes ˜ T ≡ − tan θ C T , ˜ C ≡ − tan θ C C , ˜ E ≡ Proceedings of the CHARM 2007 Workshop, Ithaca, NY, August 5-8, 2007
Table III Real and imaginary parts of amplitudes for SCS charm decays involving η and η ′ , in units of 10 − GeV aspredicted in Ref. [8]. Amplitude Expression Re Im A exp −√ A ( D → π η ) 2 E ′ − C ′ + SE ′ − √ A ( D → π η ′ ) ( C ′ + E ′ ) + SE ′ − .
87 0.56 √ A ( D → ηη ) C ′ + SE ′ − . − . − √ A ( D → ηη ′ ) ( C ′ + 6 E ′ ) + SE ′ − .
99 2.63 √ A ( D + → π + η ) T ′ + 2 C ′ + 2 A ′ + SA ′ . − .
68 8.29 ± − √ A ( D + → π + η ′ ) ( T ′ − C ′ + 2 A ′ ) + SA ′ − .
92 4.03 ± −√ A ( D + s → ηK + ) − ( T ′ + 2 C ′ ) + SA ′ ± √ A ( D + s → η ′ K + ) (2 T ′ + C ′ + 3 A ′ ) + SA ′ − .
84 3.88 ± D + and D + s SCS decays involving η and η ′ .Meson Decay B p ∗ A mode (10 − ) (MeV) (10 − GeV) D + π + η ± ± π + η ′ ± ± D + s K + η ± ± K + η ′ ± ± − tan θ C E , and ˜ A ≡ − tan θ C A .With tan θ C ≃ .
23 one predicts |A ( D → K + π − ) | = 1 . × − GeV and |A [ D + → K + ( π , η, η ′ )] = (0 . , . , . × − GeV, in qual-itative agreement with experiment. D → ( K π , K π ) interference The decays D → K π and D → K π are re-lated to one another by the U-spin interchange s ↔ d ,and SU(3) symmetry breaking is expected to be ex-tremely small in this relation [11]. Graphs contribut-ing to these processes are shown in Fig. 4.The CLEO Collaboration [12] has reported theasymmetry R ( D ) ≡ Γ( D → K S π ) − Γ( D → K L π )Γ( D → K S π ) + Γ( D → K L π ) (5)to have the value R ( D ) = 0 . ± . ± . R ( D ) =2 tan θ C ≃ . R ( D ) if π is replaced by η or η ′ [11]. Moreover, by similar argu-ments, one expects A [ D → K ( ρ , f , . . . )] /A [ D → K ( ρ , f , . . . )] = − tan θ C . D + → ( K π + , K π + ) interference In contrast to the case of D → ( K π , K π ), thedecays D + → ( K π + , K π + ) are not related to oneanother by a simple U-spin transformation. Ampli-tudes contributing to these processes are shown in Fig.5. Although both processes receive color-suppressed( C or ˜ C ) contributions, the Cabibbo-favored processreceives a color-favored tree ( T ) contribution, whilethe doubly-Cabibbo-suppressed process receives anannihilation ( ˜ A ) contribution. In order to calculatethe asymmetry between K S and K L production inthese decays due to interference between CF and DCSamplitudes, one can use the determination of the CFamplitudes discussed previously and the relation be-tween them and DCS amplitudes. Thus, we define R ( D + ) ≡ Γ( D + → K S π + ) − Γ( D + → K L π + )Γ( D + → K S π + ) + Γ( D + → K L π + ) (6)and predict R ( D + ) = − C + ˜ AT + C = 2 tan θ C Re C + AT + C = 0 . ± . . (7)This is consistent with (though slightly larger in cen-tral value than) the observed value R ( D + ) = 0 . ± . ± .
018 [14]. The relative phase of C + A and T + C is about 70 ◦ , as can be seen from Fig. 2. Thereal part of their ratio hence is small. A similar exer-cise can be applied to the decays D + s → K + K and D + s → K + K , which are related by U-spin to the D + decays discussed here. The corresponding ratio R ( D + s ) ≡ Γ( D + s → K S K + ) − Γ( D + s → K L K + )Γ( D + s → K S K + ) + Γ( D + s → K L K + ) (8)is predicted to be R ( D + s ) = − C + ˜ TA + C roceedings of the CHARM 2007 Workshop, Ithaca, NY, August 5-8, 2007 Table V Branching ratios, amplitudes, and representations in terms of reduced amplitudes for doubly-Cabibbo-suppressed decays. Amplitudes denoted by (a) involve interference between the doubly-Cabibbo-suppressed processshown and the corresponding Cabibbo-favored decay to K + X .Meson Decay B p ∗ |A| Rep.mode (10 − ) (MeV) (10 − GeV) D K + π − ± ± T + ˜ EK π (a) 860 (a) ( ˜ C − ˜ E ) / √ K η (a) 772 (a) ˜ C/ √ K η ′ (a) 565 (a) − ( ˜ C + 3 ˜ E ) / √ D + K π + (a) 863 (a) ˜ C + ˜ AK + π ± ± T − ˜ A ) / √ K + η ± ± − ˜ T / √ K + η ′ < . < .
08 ( ˜ T + 3 ˜ A ) / √ D + s K K + (a) 850 (a) ˜ T + ˜ C Figure 4: Graphs contributing to D → ( K π , K π ). = 2 tan θ C Re C + TA + C = 0 . ± . . (9)
6. Strong phases in ( D , D ) → K − π + The relative strong phase in the CF decay D → K − π + and the DCS decay D → K − π + is of inter-est in studying B decays involving neutral D mesons,where these two processes often can interfere. It wasshown in Ref. [1] that one could measure this phase byproducing a CP eigenstate D CP , for example by tag-ging on a state of opposite CP at the ψ (3770). Definedecay amplitudes as h K − π + | D i ≡ Ae iδ R , h K − π + | D i ≡ ¯ Ae iδ W . (10)The difference δ = δ R − δ W of strong phases wouldvanish in the SU(3) limit. At ψ (3770) with K − π + produced opposite a state S ζ with CP eigenvalue ζ ,one would haveΓ( K − π + , S ζ ) ≈ A A S ζ (1 + 2 ζr cos δ ) , (11)so by choosing states with ζ = ± r cos δ ) / (1 − r cos δ ), where r = | ¯ A/A | = 0 . ≃ tan θ C . In an analysis of 281 pb − of CLEO data [15], theerror on cos δ is not yet conclusively determined, as aresult of uncertainty in fits to D – D mixing. For aneventual integrated luminosity at CLEO of 750 pb − and a cross section of σ ( e + e − → ψ (3770) → D ¯ D ) = 6nb one can estimate by rescaling the calculation inRef. [1] an eventual error of ∆ cos δ < .
7. Other theoretical approaches
One can invoke effects of final state interactions toexplain arbitrarily large SU(3) violations (if, for exam-ple, a resonance with SU(3)-violating couplings domi-nates a decay such as D → π + π − or D → K + K − ).As one example of this approach [16], both resonantand nonresonant scattering can account for the ob-served ratio Γ( D → K + K − ) / Γ( D → π + π − ) =2 . ± .
1. This same approach predicted B ( D → K K ) = 9 . × − , a level of SU(3) violation con-sistent with the world average of Ref. [5] but far inexcess of the recent CLEO value [7]. The paper ofRef. [16] may be consulted for many predictions for P V and
P S final states in charm decays, where V de-notes a vector meson and S denotes a scalar meson.Results for P V decays also may be found in Refs.5
Proceedings of the CHARM 2007 Workshop, Ithaca, NY, August 5-8, 2007
Figure 5: Amplitudes T and C contributing to D + → K π + ; amplitudes ˜ C and ˜ A contributing to D + → K π + . [6, 8, 17, 18].The recent discussion of Ref. [19] entails a predic-tion A ≃ − . E (recall we were finding A ≃ − E ),essentially as a consequence of a Fierz identity andQCD corrections. Tree amplitudes are obtained fromfactorization and semileptonic D → π and D → K form factors. The main source of SU(3) breaking in / T is assumed to come from f K /f π = 1 .
22. Predictionsinclude asymmetries R ( D , + = (2 tan θ C , . ± . D → K ∓ π ± and D + → K + π – and expectation of | δ | ≃ ◦ (to becompared with 0 in exact SU(3) symmetry).
8. Summary
We have shown that the relative magnitudes andphases of amplitudes contributing to charm decaysinto two pseudoscalar mesons are describable by fla-vor symmetry. We have verified that there are largerelative phases between the color-favored tree ampli-tude T and the color-suppressed amplitude C , as wellas between T and E ≃ − A .The largest symmetry-breaking effects are visible insingly-Cabibbo-suppressed (SCS) decays, particularlyin the D → ( π + π − /K + K − ) ratio which are at leastin part understandable through form factor and decayconstant effects. Decays involving η , η ′ are mostlydescribable with small “disconnected” amplitudes, apossible exception being in SCS D + and D + s decays.One sees evidence for the expected interferencebetween Cabibbo-favored (CF) and doubly-Cabibbo-suppressed decays in D , + → K S,L π , + decays. As aresult of CLEO’s present data on ( D , D ) → K − π + ,limits are being placed on the relative strong phase δ between these amplitudes, and the full CLEO datasample is expected to result in an error equal to orbetter than ∆(cos δ ) = 0 . Acknowledgments
J.L.R. wishes to thank C.-W. Chiang, M. Gronau,Y. Grossman, and Z. Luo for enjoyable collaborationson some of the topics described here. We are grateful to S. Blusk, H. Mahlke, A. Ryd, and E. Thorndike forhelpful discussions. This work was supported in partby the United States Department of Energy throughGrant No. DE FG02 90ER40560.
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