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Semileptonic D decay into scalar mesons: a QCD sum rule approach H.G. Dosch 1,E.M.Ferreira 2, F.S. Navarra 3 and M. Nielsen 3 1 Institut für Theoretische Physik, Universität Heidelberg Philosophenweg 16,

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Semileptonic D decay into scalar mesons: a QCD sum rule approach H.G. Dosch 1,E.M.Ferreira 2, F.S. Navarra 3 and M. Nielsen 3 1 Institut für Theoretische Physik, Universität Heidelberg Philosophenweg 16, D-6900 Heidelberg, Germany 2 Instituto de Física, Universidade Federal do Rio de Janeiro C.P , Rio de Janeiro , RJ, Brazil 3 Instituto de Física, Universidade de São Paulo C.P , São Paulo, SP, Brazil Abstract Semileptonic decays of D-mesons into scalar hadronic states are investigated. Two extreme cases are considered: a) the meson decays directly into an uncorrelated scalar state of two two mesons and b) the decay proceeds via resonance formation. QCD sum rules including instanton contributions are used to calculate total and differential decay rates under the two assumptions. PACS Numbers : Lg, Ce, Dw 1 I. INTRODUCTION Low lying scalar mesons are an old problem in hadron physics, see the review by Spanier and Törnqvist on scalar mesons in [1] and the literature quoted there. In a recent analysis [2,3] of D- andd s -meson nonleptonic decays distinct signals for strong enhancements in S-wave ππ and Kπ channels have been observed, reviving the interest for these states. In the following we shall refer to the signal in the ππ channel as σ andinthekπ chanel as κ. The enhancements can be well described by a Breit-Wigner-type resonance form, with the correct threshold behaviour ρ X,BW (s) = Γ X (s)m X (s m 2 X) 2 + m 2 XΓ X (s) 2, (1) where the subscript X stands for σ or κ. The correct threshold behaviour is guaranteed through the s-dependent width Γ X (s) =Γ 0X λ 1/2 (s, m 2 a,m 2 b) λ 1/2 (m 2 X,m2 a,m2 b ) m 2 X s. (2) Here m a and m b are the masses of the mesons in the decay channel and λ(x, y, z) =x 2 + y 2 + z 2 2xy 2xz 2yz. In [3] the following parameters were found: 1) for the S-wave ππ channel, m a = m b = m π, m σ =0.478 ± ± GeV; Γ 0σ =0.324 ± ± GeV. (3) 2) for the S-wave Kπ channel, m a = m K,m b = m π, m κ =0.797 ± ± GeV; Γ 0κ =0.410 ± ± GeV. (4) The semileptonic D-decays offer, in principle, much cleaner samples than the nonleptonic decays since there occur no problems connected with the presence of a third strongly interacting particle. In the non-strange channel important experimental information comes indeed from the analysis of the final state interaction of the pions from leptonic K-decays. For the case of strange mesons, the analysis of D decays can play a similarly important role. In a recent paper by the FOCUS collaboration [4] clear evidence was found that in the semileptonic decay D + K π + µ + ν the D-meson does not decay exclusively into the hadronic vector channel, but that there is interference with a scalar contribution. In this paper we estimate the decay rates of semileptonic D-decays into the scalar Kπ and ππ channels. We use the method of QCD sum rules [5] which has been successfully applied to several semileptonic decay processes. For a recent review see [6] and the literature quoted there. Even if (broad) resonances in the hadronic decay channels exist, it is not clear whether they are due to interactions on the quark level, or if they are rather an effect of interactions in the purely hadronic channel, see for instance [7]. Here a theoretical analysis can be very helpful. The semileptonic decay is supposed to occur on the quark level and therefore the decay rate should depend crucially on the direct coupling of the resonances to the quark currents. We consider two limiting cases 2 FIGURES j µ D c q 2 q 1 X a b FIG. 1. Schematic picture of the semileptonic vertex of a D-meson decay. c, q 1, q 2 are quark lines, j µ denotes the weak current. X might be either a (broad) resonance or represent an uncorrelated two-meson state of mesons a and b. the observed two-meson final state couples to the corresponding quark-antiquark state only through the σ or κ resonance. the quark antiquark state couples to uncorrelated final mesons in an S-wave. Since the signals we investigate in this paper are very broad the finite width has to be taken into account. Otherwise our analysis is based on the same principles and assumptions as the used in the sum-rule analysis of other semileptonic D-decays [8]. We therefore refer to this paper for details and clarifications. We investigate the semileptonic decays II. KINEMATICS D Xl ν l, (5) where X might be the σ, κ or an uncorrelated ππ or Kπ pair in the S-state, as depicted in Fig. 1. The semileptonic decay of a D-meson with momentum p D into a scalar state with total momentum p X and invariant mass s = p 2 X is described by the two form factors of the matrix element of the weak current X j µ D =(p D + p X ) µ f +X (t)+(p D p X ) µ f X (t), (6) where t =(p D p X ) 2. In the decay rate the form factor f X is multiplied by the difference of the lepton masses and hence is negligible for both e and µ decays. The differential semileptonic decay rate is given by d 2 Γ(s, t) ds dt = G2 F V cq 2 2 λ 3/2 (m 2 192π 3 m 3 D,s,t)f +X 2 (t)ρ X(s) D π, (7) where G F is the Fermi coupling constant and V cq2 the CKM transition element from the charmed quark to the quark q 2. The total width is 3 m 2 D (md s) 2 Γ= ds dt d2 Γ(s, t). (8) (m a+m b ) 2 0 ds dt The spectral distribution in the invariant mass s of the hadronic final state is given by dγ d s =2 s (md s) 2 0 dt d2 Γ(s, t). (9) ds dt III. SUM RULES The D meson in the initial state is interpolated by the pseudoscalar current j D (x) = c(x)iγ 5 q 1 (x), (10) where c isthefieldofthecharmedquarkandq 1 that of an up or down quark, summation over spinor and colour indices being understood but not indicated explicitly. The final hadronic state X is interpolated by the scalar current j X (x) = q 1 (x)q 2 (x), (11) where q 2 represents a light quark field for X = σ, and a strange quark field for X = κ. The semileptonic decay rate is obtained from the time ordered product of the two interpolating fields in Eqs. (10) and (11) and the weak current jµ W = q 2γ µ (1 γ 5 )c T µx (p 2 D,p2 X,t)=i2 d 4 xd 4 y 0 T[j D (x)jµ W (0)j X(y)] 0 e i(p D.x p X.y). (12) In order to select the semileptonic decay rates into the lowest lying hadrons we insert intermediate states and obtain the following double dispersion relation in the phenomenological side T phen µx (p 2 D,p 2 X,t)= 1 ds π 2 D ds X 0 j D (0) D D jµ W (0) X X j X (0) contributions of higher resonances. (13) s D p 2 D s 2 X m 2 X Introducing 0 j D (0) D = A D ρ D (s D ); p 2 D = s D, (14) X j X (0) 0 = A X ρ X (s X ); p 2 X = s X, (15) and using Eq. (6) we obtain T phen µx (p 2 D,p 2 X,t)= 1 ρ D (s D ) ρ X (s X ) ds π 2 D ds X A D A X s D p 2 D s X p 2 X ( ) f +X (t, s D,s X )(p D + p X ) µ + f X (t, s D,s X )(p D p X ) µ + contributions of higher resonances. (16) 4 In the following we concentrate on the relevant form factor f + and introduce the factor T + which multiplies the vector (p D + p X ) µ T phen +X (p 2 D,p 2 X,t)= 1 ρ D (s D ) ρ X (s X ) ds π 2 D ds X A D A X f s D p 2 D s X p 2 +X (t, s D,s X ) X + contributions of higher resonances. (17) For the D-meson the density ρ D (s D ) introduced in Eq. (14) is given by and we obtain ρ D (s D )=πδ(s D m 2 D ), (18) A D = f D m 2 D/m c, (19) f D being the D-meson decay constant in the conventional notation. For the density ρ X we use the two extreme ansätze mentioned above: The quark-antiquark current j X in the J =0 + state couples to the σ or κ-resonance described by the Breit-Wigner distribution (1) ρ X (s) =ρ X,BW (s). (20) The quark-antiquark current j X couples to an uncorrelated meson pair, the density being described by the density of two-particle phase space ρ X (s) = π λ(s, m 2 a,m 2 b) 1/2, (21) 16π 2 s where m a, m b are the masses of the mesons in the final state. The three-point function can be evaluated by perturbative QCD if the external momenta are in the deep Euclidean region p 2 D (m c + m 1 ) 2, p 2 X (m 1 + m 2 ) 2, t (m c + m 2 ) 2. (22) In order to approach the not-so-deep-euclidean region and to get more information on the nearest physical singularities, nonperturbative power corrections are added to the perturbative contribution T+X theor (p2 D,p2 X,t)= 1 ds π s 2 D ds X D,th s X,th σ +X (s D,s X,t) (s D p 2 D)(s X p 2 X) + ij C ij (p 2 D) i (p 2 X) j O ij. (23) The perturbative contribution is contained in the double spectral function σ +X. The Wilson coefficients C ij multiplying the power corrections can be evaluated in perturbative QCD. The operators O ij occur in the operator expansion of the time ordered product Eq. (12); their vacuum expectation values, the condensates O ij, are introduced as phenomenological parameters. 5 In order to suppress the condensates of higher dimension and at the same time reduce the influence of higher resonances, the series in Eq. (23) is Borel improved, leading to the mapping f(p 2 ) ˆf(M 2 ), 1 (p 2 m 2 ) ( 1)n e m2 /M 2 n (n 1)! (M 2 ). (24) n Furthermore, we make the usual assumption that the contributions of higher resonances are well approximated by the perturbative expression 1 ds π 2 D ds X s 0D s 0X σ +X (s D,s X,t) (s D p 2 D)(s X p 2 X), (25) with appropriate continuum thresholds s 0D and s 0X. By equating the Borel transforms of the phenomenological expression in Eq.(17) and that of the theoretical expression, Eq. (23), we obtain the sum rule f +X (t, m 2 m 2 D D, s X )f D e m2 D /M 2 A X D m c π = 1 π 2 s0d s D,th ds D s0x s0x (m a+m b ) 2 ds X ρ X (s X )e s X/M 2 X s X,th ds X σ +X (s D,s X,t)e s2 D /M 2 D e s X /M 2 X + ˆK +X (MD 2,M2 X,t) + ˆK +I (MD 2,M2 X,t), (26) where s X is some value below s 0X. In the zero width approximation we have of course s X = m 2 X. ˆK+X are the Borel transforms of the nonperturbative expressions due to the condensates and ˆK I is an approximation to the instanton contributions, which might be important in the scalar channel [9]. The decay constant f D and the coupling A X defined in Eqs. (14), (15), and(19) can also be determined by sum rules obtained from the appropriate two-point functions. Using the same procedure as described above we arrive at m 4 ( D f theor m 2 D (M 2 ) ) 2 s0d e m 2 D /M 2 = ds σ D (s)e s/m 2 + ˆK D (M 2 ), (27) c m 2 c and ( A theor X (M 2 ) ) 2 ( 1 π s0x s0x (m a+m b ) 2 dsρ X (s)e s/m 2) = ds σ X (s)e s/m 2 + ˆK X (M 2 )+ ˆK I (M 2 ). (28) m 2 q 2 The analysis of the two-point function for the scalar mesons, and the explicit expressions for the functions occuring in Eqs. (26), (27) and (28) are given in Appendices A and B respectively. The final sum rule for the form factor is obtained from Eq. (26) by inserting for f D and A X the expressions fd theor and A theor X of Eqs. (27) and (28) 6 f +X (t, m 2 D, s X)= ( m e m2 D /M 2 D 2 D f D (M 2 D )A X (M 2 X ) 1 ) s0x 1 ds X ρ X (s X )e s 2 X/M X m c π (m a+m b ) ( 2 s0d s0x ds D ds X σ +X (s D,s X,t)e s2 D /M 2 D e s X /M 2 X s D,th s X,th + ˆK +X (MD 2,M2 X,t) + ˆK ) +I (MD 2,M2 X,t). (29) The radiative corrections for the scalar and pseudoscalar channels are known to be large [10]. They are expected to be large in the three-point function too. By inserting the sum rule expressions for the two-point functions, Eqs. (27) and (28), in the denominator of the sum rule for the three-point function, Eq. (29), we expect, at least, a partial cancellation of these corrections [11,12]. IV. EVALUATION OF THE SUM RULES AND RESULTS The sum rule Eq. (29) is evaluated in the same way as described in [8], and we only sketch the main steps of this evaluation. In the complete theory, the right hand side of Eq. (29) should not depend on the Borel variables M 2. However, in a truncated treatment there will always be some dependence left. Therefore, one has to work in a region where the approximations made are supposedly acceptable and where the result depends only moderately on the Borel variables. To decrease the dependence of the results on the Borel variables M 2, we take them in the two-point functions at half the value of the corresponding variables in the three-point sum rules, i.e., in Eq.(29) we put We furthermore choose M D2 = MD 2 /2 and M X 2 = MX 2 /2. (30) M 2 X M 2 D = m 2 X m 2 D m 2 c. (31) We have checked that the results do not depend crucially on this particular choice. If the momentum transfer t to the lepton pair is larger than a critical value t cr, non-landau singularities have to be taken into account [8]. Since anyhow we have to stay away from the physical region, i.e. we must have t (m c + m 1 ) 2, we limit our calculation to the region 0 t t cr. In this range the t-dependence can be obtained from the sum rule (29) directly. It can be fitted by a monopole, and extrapolated to the full kinematical region. Since we do not take into account radiative corrections we choose the QCD parameters at a fixed renormalisaton scale of about 1 GeV 2 : the strange and charm mass m s =0.16 GeV, m c =1.3 GeV, the up and down quark masses are put to zero. We take for the non-strange quark condensate qq = (0.24) 3 GeV 3, for the strange quark condensate ss =0.8 qq, and for the mixed quark-gluon condensate qσgq qq = m 2 0 qq with m2 0 =0.8 GeV2. For the continuum threshold in the D-channel we take from [8] s 0D =6 GeV 2. The standard value in the X channel would be s 0X (m X +0.5 GeV) 2, yielding s 0σ 1GeV 2 7 and s 0κ 1.6GeV 2. As an additional condition we use the mass constraint from the twopoint function, as described in Appendix A. We start with the decay D κl ν l and first consider the case where the scalar quark current (11) couples directly to the κ-signal through a Breit-Wigner distribution (20). In Fig. 2 we show the different contributions to the form factor f +κ (0) in Eq. (29), as a function of the Borel variable MD 2, using the continuum threshold s 0κ =1.6GeV 2. As the lower limit for the fiducial region in MD 2 we take that value of M D 2 where the perturbative contribution is one half of the total contribution. As upper limit we take the value MD 2 =15GeV2,which is motivated in Appendix A. For such a high value of the Borel parameter the result is very stable but it is largely determined by the choice of the continuum model. The instanton contribution in the fiducial region is completely negligible; the five dimensional mixed condensate is strongly suppressed compared to the three dimensional quark condensate. In the Borel variable of the X-channel the fiducial region corresponds approximately to the range 1.7GeV 2 MX 2 5GeV f +K (0) M 2 D (GeV 2 ) FIG. 2. Dependence of the form factor f +κ at t = 0 on the Borel variable MD 2. Here the decay is assumed to proceed through resonance formation and the state density is given by Eq.(20). Solid curve: total contribution; long-dashed: perturbative; dashed: quark condensate; dot-dashed mixed condensate; dotted : instanton contribution. In the range 0 t 0.5 GeV 2 no non-landau singularities occur for our choices of the continuum thresholds. The momentum dependence of f +κ can, in this t-range, be very well approximated by a monopole expression f +κ (t) = f +κ(0) 1 t M 2 P, (32) and extrapolated to the full physical region. 8 For 1.4 GeV 2 s 0κ 1.8 GeV 2 and for values of M 2 D discussed above, we find for the form factor at t =0 and for the pole mass, 0.48 f +κ (0) 0.55, (33) 1.9 GeV M P 2.2 GeV. (34) The pole mass is considerably smaller than the mass of the charmed pseudovector-meson D s1 (2536) which would fit into the t-channel. In the D Klν decay the pole mass also came out to be smaller [8] than the mass of the strange vector meson D s (2114). In the limits of the Borel variables and the continuum thresholds discussed above we obtain for the total semileptonic decay width Γ(D κlν) =(5.5 ± 1.0) GeV, (35) wherewehaveusedv cs =0.97. The same calculation done in an unjustified zero width approximation would yield a total semileptonic decay width which is about 20 % larger dγ/ds 1/2 (X10 14 ) s 1/2 (GeV) FIG. 3. Invariant mass distribution dγ( s)/d s of the hadronic final state in the decay D (Kπ) S lν. Solid: resonance formation; dots: uncorrelated Kπ. The spectral distribution dγ/d s, Eq. (9), where s is the invariant mass of the πk state, is given in Fig. 3, solid line. Next we investigate the same decay under the assumption that the scalar current in Eq. (11) does not couple to a resonance, but to an uncorrelated πk pair in an S-state, i.e., we use the ansatz (21) for ρ κ (s) entering the sum rules. In Fig. 4 we show the dependence on the Borel variable M 2 D of the decay form factor f +κ(0) for s 0κ =1.6GeV 2.Notethatnow the density (21) describes a two-particle phase space and therefore the dimension of f +κ is different from the previous case. 9 15 12 f +K (0) (GeV 1 ) M 2 D (GeV 2 ) FIG. 4. Dependence of the form factor f +κ at t = 0 on the Borel variable MD 2. Here the decay is assumed to proceed through an uncorrelated πk-meson pair in an S-state, the density is given by Eq.(21). Solid curve total contribution; long-dashed perturbative; dashed quark condensate; dot-dashed mixed condensate; dotted instanton contribution. The masses of the pole fit to the t-dependence are practically the same as for the resonance case; for the form factor at t =0weobtain 5.0GeV 1 f +κ (0) 7.1GeV 1. (36) The total width comes out for the case of an uncorrelated πk pair in an S state as Γ(D (Kπ) S l ν l )=(3.7 ± 1.1) GeV. (37) The spectral distribution for this case is also shown in Fig. 3, with a dotted line. Although there is no resonance formation the resulting distribution shows a maximum at approximately the mass of the κ, which is an effect of the decrease of the total phase space near the kinematical limits. The evaluation of the decay D σlν follows exactly the same lines. Here the fiducial range in M 2 D is chosen according to the same criteria as before and goes approximately from M 2 D =8GeV2 to 18 GeV 2 corresponding approximately to a range 1 GeV 2 M 2 σ 2.3GeV2. The continuum limit s 0σ was chosen between 1 and 1.6 GeV 2. The resulting form factors and total decay width are, in the case of a resonance formation with a Breit-Wigner width 0.42 f +σ (0) 0.57, (38) Γ(D σl ν l )=(8.0 ± 2.5) GeV, (39) and for the case of two uncorrelated π-mesons in an S-state 10 5GeV 1 f +σ (0) 6GeV 1 (40) and Γ(D (ππ) S l ν l )=(4.5 ± 1.0) GeV. (41) dγ/ds 1/2 (X10 15 ) s 1/2 (GeV) FIG. 5. The invariant mass distribution of the hadronic final state of the rate in the decay D (ππ) S lν. Solid: resonance formation; dots: uncorrelated Kπ. The spectral distributions for both an satze are shown in Fig. 5. V. SUMMARY AND CONCLUSIONS The semileptonic decays of D-mesons into scalar hadrons are very similar to those into pseudoscalars. The same role played by the vector part in the pseudoscalar case is played by the weak pseudovector for the decay into scalars. Therefore, the theoretical expressions are very similar, only terms proportional to the mass of the strange quark change sign. This leads to a small reduction of the form factors compared to the decay into a pseudoscalar state. The main difference of the semileptonic decay rates into K and κ is due to the different phase space. The decay into σ mesons is suppressed by the small value of the weak (c, d) matrix element V cd The spectral distributions of the invariant masses of the Kπ and ππ sytems are given in Figs. 3 and 5. Their maxima are well below the masses of the κ and the σ, and also the forms are quite different from Breit-Wigner distributions. The increase of the spectral ditributions is steeper and the fall-off substantially faster than for the corresponding Breit-Wigner forms. This is an effect of the total phase space in the semileptonic decay. If the scalar current does not couple directly to the resonance but only to an uncorrelated meson pair, the decay rate is reduced by a factor two compared with the decay into resonances, but nevertheless an enhancem

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