Implications of factorization for the determination of hadronic form factors in D + s → φ transition
چکیده
Using factorization we determine the allowed domains of the ratios of form factors, x = A2(0)/A1(0) and y = V (0)/A1(0), from the experimentally measured ratio Rh ≡ Γ(D + s → φρ )/Γ(D s → φπ ) assuming three different scenarios for the q-dependence of the form factors. We find that the allowed domains overlap with those obtained by using the experimentally measured ratio Rsl = Γ(D + s → φl νl)/Γ(D + s → φπ ) provided that the phenomenological parameter a1 is 1.23. Such a comparison presents a genuine test of factorization. We calculate the longitudinal polarization fraction, ΓL/Γ ≡ Γ(D + s → φLρ + L)/Γ(D + s → φρ ), in the three scenarios for the qdependence of the form factors and emphasize the importance of measuring ΓL/Γ. Finally we discuss the q -distribution of the semileptonic decay and find that it is rather insensitive to the scenarios for the q-dependence of the form factors, and unless very accurate data can be obtained it is unlikely to discriminate between the different scenarios. Useful information on the value of x might be obtained by the magnitude of the q-distribution near q = 0. However the most precise information on x and y would come from the knowledge of the longitudinal and left-right transverse polarizations of the final vector mesons in hadronic and/or semileptonic decays. Postal address: LPTHE, Tour 16, 1er Etage, Université Pierre & Marie Curie and Université Denis Diderot, 4 Place Jussieu, F-75252 Paris CEDEX 05, France. Permanent address : Department of Physics, University of Alberta, Edmonton, Alberta T6G2JI, Canada Unité associée au CNRS URA 280 Since the Cabibbo-Kobayashi-Maskawa(CKM)-favoured decays of the strange-charmed meson D s involve, at the quark level, ss pair, the vector meson φ emerges naturally as one of the decay products. Among its decay modes are the three most important ones, D s → φπ , D s → φρ + and D s → φl νl(l = e, μ) measured by several groups [1], and the topic of this paper. The analysis of the experimental data for these modes involves three axial form factors, A0, A1 and A2, and a vector form factor V , in the notation of Bauer, Stech and Wirbel (BSW) [2]. In spite of this fact which renders the analysis of experimental data involving the φ meson more complicated than the corresponding one with pseudoscalar mesons such as η or η, it turns out that the polarization of the φ meson may provide crucial information that can be used to test the factorization assumption which is one of the basic ingredients in the computation of hadronic decay rates. The fact that the π meson is light allows us to ignore the variation of the hadronic form factor As 0 between q 2 = mπ, where it enters the description of D + s → φπ , and q = 0. This then allows us to eliminate As 0 (0) in favour of A Dsφ 1 (0) and A Dsφ 2 (0) through [2], 2mφA Dsφ 0 (0) = (mDs +mφ)A Dsφ 1 (0)− (mDs −mφ)A Dsφ 2 (0) (1) Since the other decays, D s → φρ + and D s → φl νl, involve the form factors A Dsφ 1 , As 2 and V s, it proves convenient to factor out As 1 (0) which then cancels in the ratio of the rates, and define the following ratios of the form factors at q = 0, x = As 2 (0) As 1 (0) y = V s(0) As 1 (0) (2) In our analysis we do not adopt any of the several models of hadronic form factors for D s → φ transition available in the literature to compute decay rates as such a procedure would not tell whether the factorization assumption or the model-form factors were being tested in the comparison between theory and experiment. Instead, we simply consider different scenarios for the q-variation of the form factors ( and do not make any assumptions about their absolute normalizations ) and show that the measured ratio of the two hadronic rates, Rh ≡ Γ(D + s → φρ )/Γ(D s → φπ ) (3) constrains the parameters x and y to an allowed domain of the x, y plane ( limited by x and y non-negative a suggested by all phenomenological models ). Such a domain can then be compared with the values of x and y already measured in the semileptonic modes [1, 3, 4]. We remind the reader that semileptonic data are analyzed with monopole qdependence for all form factors which is just one of the scenarios we discuss. A procedure such as ours that eschews reliance on a particular set of model-form factors produces not only a more bias-free test of factorization but also one that is independent of the BSW phenomenological parameter a1 [2]. Further, since the φmeson is longitudinally polarized inD s → φπ , the ratio Γ(D s → φπ)/Γ(D s → φρ ) behaves physically like the longitudinal polarization fraction ΓL/Γ in D s → φρ , where ΓL Γ ≡ Γ(D s → φLρ + L ) Γ(D+ s → φρ +) (4)