ЯДЕРНАЯ ФИЗИКА, 2004, том 67, № 8, с. 1593-1601
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
D+ — п+п+п- DECAY: THE 13P0ss COMPONENT IN SCALAR-ISOSCALAR MESONS
© 2004 V. V. Anisovich, L. G. Dakhno, V. A. Nikonov
Received May 5, 2003; in final form, December 2, 2003
We calculate the processes D+ — п+ss and D+ — resonance, correspondingly, in the spectator and W-annihilation mechanisms. The data on the reaction D+ — n+po, which owes to the W-annihilation mechanism only, point to a negligibly small contribution of the W annihilation to the production of scalar— isoscalar resonances D+ — f0. As to spectator mechanism, we evaluate the 13Poss component in the resonances fo(980), fo(1300), fo(1500) and broad state fo(1200-1600) on the basis of data on the decay ratios (D+ — fo)/(D+ — ф). The data point to a large ss component in the fo(980): 40 < ss < 70%. Nearly 30% of the 13Poss component flows to the mass region 1300—1500 MeV being shared by fo(1300), fo(1500), and broad state fo(1200-1600): the interference of these states results in a peak near 1400 MeV with the width around 200 MeV. Our calculations show that the yield of radial-excitation state 23Poss is relatively suppressed, r(D+ — (23Poss))/r(D+ — n+(13Poss)) < 0.05.
1. INTRODUCTION
The meson yields in the decay D+ — n+n+n- [1] evoked immediately a great interest and now they are actively discussed (see [2—5] and references therein). The matter is that in this decay the production of strange quarkonium is dominant, D+ — n+ss, with a subsequent transition s s — fJ — n+n-. Therefore, the reaction D+ — n+n+n- may serve us as a tool for the estimation of ss components in the fJ mesons. This possibility is particularly important in context of the determination of quark content of the f0 mesons, for the classification of qq scalar—isoscalar states is a key problem in the search for exotics.
In the D+ decay the production of f0 mesons proceeds mainly via spectator mechanism, Fig. 1a: this very mechanism implements the transition ss — — f0. Besides, the spectator mechanism provides a strong production of the 0(1020) meson. To evaluate the ss components in f0 mesons, we use the process D+ — n+0(1O2O) as a standard: we consider the ratio (D+ — n+fo)/(D+ — n+0(1020)), where the uncertainty related to the coupling c — n+s is absent.
Calculation of the transition of Fig. 1a is performed in the spectral integration technique, this technique was developed in the study of deuteron form factors [6] and deuteron photodisintegration [7], it was used in the analysis of radiative decays of light mesons [8, 9] and weak decays of D and B mesons [10]. The cuttings of the triangle diagram related to the double spectral integrals are shown in Fig. 1b.
In addition, the f0 production can originate from the W-annihilation process of Figs. 1c, Id. It is a relatively weak transition, nevertheless we take it into account, and the reaction D+ ^ n+p° serves us as a scale to determine the W-annihilation coupling cs ud. Let us emphasize that the processes shown in Fig. 1 are of the leading order in terms of the 1/Nc expansion rule [11].
In Section 2 we present the data set used for the analysis and write down the amplitudes for the spectator and W-annihilation processes. The results of calculations are presented and discussed in Section 3. In this section we demonstrate that (i) the W annihilation contributes weakly to the fj-meson production, and (ii) the 13P0 state dominates the transition ss — fo, while the production of the 23P0ss component is relatively suppressed, so the transition D+ — n+ss — n+f0 is in fact a measure for the 13P0ss component in scalar—isoscalar mesons.
In Conclusion we sum up what the data on the decay D+ — n+n- tell us, in particular, with respect to the identification of the lightest scalar qq nonet.
2. DATA SET AND THE AMPLITUDES FOR THE SPECTATOR AND W-ANNIHILATION MECHANISMS
In this section we present the data used in the analysis and write formulae for the spectator and W-annihilation mechanisms, Fig. 1a and Figs. 1c, 1d, respectively.
1593
5 fo, f2 œ
c
d
Fig. 1. Diagrams determining the decay D+ ^ ^ n+n+n-. (a) Diagram for the spectator mechanism; (b) energy-off-shell triangle diagram for the integrand of the double spectral representation; (c, d) diagrams for the W-annihilation mechanism D+ ud with subsequent production of uU and dd pairs.
D+
2.1. The Data Set In the recently measured spectra from the reaction [1], the relative weight of channels
n+n+n
In addition, the production of /2(1270) is seen in [1]: BR (n+/2(1270)) = (20 ± 4)% that makes it necessary to include tensor mesons into the calculation machinery.
2.2. Decay Amplitudes and Partial Widths
The spin structure of the amplitude depends on the type of the produced meson — it is different for scalar (/0), vector (0,u,p), or tensor (/2) mesons. Let us denote the momenta of the produced scalar (S), vector (V), and tensor (T) mesons by pM, where M = S, V, T; the D+-meson momentum is referred as p.
The production amplitude is written as
A(D+ — n+M) = Om(P,PM)Am(q2), (4)
where the spin operators OM(p,pM) for scalar, vector, and tensor mesons read as follows:
Os (p,ps) = 1, Ov (p,pv )= py±^, (5) ot{P,Pt) = —ri--^v
P2T±
n+/o(980) and n+p0(770) is evaluated,
BR(n+/o(980))=(57 ± 9)%, (1)
BR (n+po(770)) = (6 ± 6)%, and the ratio of yields, r(D+ — n+n+n-)/r(D+ — n+0(1020)) = (2) = 0.245 ± 0.028-0:019,
is measured. These values are the basis to determine relative weight of the ss component in the /0(980).
Besides, in [1] the bump in the wave (IJPC = = 00++) is seen at 1434 ± 18 MeV with the width 173 ± 32 MeV; this should be a contribution from the nearly located resonances /0(1300), /0(1500) and the broad state /0(1200-1600). Relative weight of this bump is equal to
BR(n+ (/0(1300) + /0(1500) + (3)
+ /0(1200-1600))) = (26 ± 11)%.
This magnitude allows us to determine the total weight of the 13P0ss component in the states /0(1300), /0(1500), /0(1200-1600).
Now the data of FOCUS Collaboration [12] on the decay D+ — n+/0(980) are available. These data are compatible with those of [1], so we do not use them in our estimates, and we base on the ratios n+/0/n+0 measured in [1].
The momenta pv± and pT± are orthogonal to the D+-meson momentum p:
pv ±u = 9ßß< pvß', pT ±u = 5V pTß'
(6)
5 un' =
pßpß'
p2
In the spectral integration technique, the invariant production amplitude AM(q2) is calculated as a function of q2 = (p—pM )2 = mn.
In terms of the spin-dependent operators OM(p, pM), partial width for the transition D+ ^ n+M reads:
mDsr(D+ ^ n+M) = \A
m (
= ml)\2 x (7)
x iom (p,pm)
pM±
8nmDs
where
\ 2
Os (p,ps )) = 1,
2
Ov(p,pv)J = -pV±, (8)
Ot (p,pt ) =
2
In (7), the value \J—p2M1_ is equal to the center-
of-mass relative momentum of mesons in the final state; it is determined by the magnitudes of the meson masses as follows:
2
2
D+ — n+n- DECAY
1595
\J-Pm± = \J[mDs ~ (mM + mn)2][m2Ds - (mM - m7r)2]/(2mDJ.
2.3. Amplitudes for the Spectator and W -Annihilation Processes in the Light-Cone Variables
In the leading order of the 1/Nc expansion, there exist two types of processes which govern the decays D+ — n+/0, n+/2, n+0/u, n+p0. Theyare shown in Figs. 1a, 1c, 1d. We refer to the process of Fig. 1a as a spectator one, while that shown in Figs. 1c, 1d is called the W-annihilation process. The transition ss — meson is a characteristic feature of the spectator mechanism, it contributes to the production of isoscalar mesons: D+ — n+/0, n+/2, n+0, n+u, whereas p0 cannot be produced within spectator mechanism. The W annihilation contributes to the production of mesons with both I = 1 and I = 0, D+ — n+/o, /2, n+0, n+u and D+ — n+p0. Therefore, the latter reaction, D+ — n+p0, allows us to evaluate relative weight of the effective coupling constant for W annihilation, thus giving a possibility to estimate the W-annihilation contribution to the channels of interest: D+ — n+/0, n+/2, n+0. This estimate tells us that the W annihilation is relatively weak that agrees with conventional evaluations, see, for example, [10].
The amplitudes for the spectator production of mesons (Fig. 1a) and for W annihilation (Figs. 1c, 1d) can be calculated in terms of double spectral integral representation developed for the quark three-point diagrams in [10, 8]. The calculation scheme for the diagram of Fig. 1a in the spectral integration technique is as follows. We consider the relevant energy-off-shell diagram shown in Fig. 1b for which the momentum of the cs system, P = k1 + k2, obeys the requirement P2 = s > (mc + ms)2, while the ss system, with the momentum P' = k'1 + k2, satisfies the constraint P'2 = s' > 4m2s; here, ms,c are the masses of the constituent s,c quarks which are taken to be ms = 500 MeV and mc = 1500 MeV. The next step consists in the calculation of double discontinuity of the triangle diagram (cuttings I and II in Fig. 1b) which corresponds to real processes, and the double discontinuity is the integrand of double dispersion representation.
The double dispersion integrals may be rewritten in terms of the light-cone variables, by introducing the light-cone wave functions for the D+ meson and produced mesons /0, /2, 0, u, p0: the calculations performed here are done by using these variables.
Our calculations have been carried out in the limit of negligibly small pion mass, mn — 0, that
is a reasonable approach, for in the ratio (D+ — — n+/J)/(D+ — n+0) the uncertainties related to this limit are mainly cancelled.
2.3.1. Spectator-production form factor. The
form factor for the spectator process given by the triangle diagram of Fig. 1a reads:
p(sp)/ _ G'sp f
FM {q ] J
dx
(9)
16n3 7 x(1 — x)2 0
x J d2k±^Ds (s)^M(s')Sns^nM(s,s',q2).
Here, Gsp is the vertex for the decay transition c — — n+s; the light-cone variables x and k^ refer to the momenta of quarks in the intermediate states. The energies squared for initial and final quark states (cs and ss) are written in t
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