a0(980)-f0(980) mixing and isospin violation in the reactions pN ->d a0, pd ->3He/3H a0 and dd ->4He a0

It is demonstrated that f_0-a_0 mixing can lead to a comparatively large isospin violation in the reactions pN ->da_0, pd ->^3He/^3H a_0 and dd ->^4He a_0 close to the corresponding production thresholds. The observation of such mixing effects is possible, e.g., by measuring the forward-backward asymmetry in the reaction pn ->d a_0^0 ->d eta pi^0.

provide evidence for a significant f 0 -a 0 mixing intensity as large as |ξ| 2 = 8 ± 3%. In this letter we discuss possible experimental tests of this mixing in the reactions pp → da + 0 (a), pn → da 0 0 (b), pd → 3 H a + 0 (c), pd → 3 He a 0 0 (d) and dd → 4 He a 0 0 (e) near the corresponding thresholds. We recall that the a 0 -meson can decay to πη or KK. In this paper we consider only the dominant πη decay mode.
Note that the isospin violating anisotropy in the reaction pn → da 0 0 due to the a 0 (980)-f 0 (980) mixing is very similar to that what may arise from π 0 -η mixing in the reaction pn → dπ 0 (see Ref. [9]). Recently charge-symmetry breaking was investigated in the reactions π + d → ppη and π − d → nnη near the η production threshold at BNL [9]. A similar experiment, comparing the reactions pd → 3 Heπ 0 and pd → 3 Hπ + near the η production threshold, is now in progress at COSY-Jülich (see e.g. Ref. [10]).

Phenomenology of isospin violation
In reactions (a) and (b) the final da 0 system has isospin I f = 1, for l f = 0 (S-wave production close to threshold) it has spin-parity J P f = 1 + . The initial NN system cannot be in the state I i = 1, J P i = 1 + due to the Pauli principle. Therefore, near threshold the da 0 system should be dominantly produced in P -wave with quantum numbers J P f = 0 − , 1 − or 2 − . The states with J P i = 0 − , 1 − or 2 − can be formed by an NN system with spin S i = 1 and l i = 1 and 3. Neglecting the contribution of the higher partial wave l i = 3 we can write the amplitude of reaction (a) in the following form T (pp → d a + 0 ) = = α + p · S k · ǫ * + β + p · k S · ǫ * + γ + S · k p · ǫ * , where S = φ T N σ 2 σφ N is the spin operator of the initial NN system; p and k are the initial and final c.m. momenta; ǫ is the deuteron polarization vector; α + , β + , γ + are three independent scalar amplitudes which can be considered as constants near threshold (for k → 0). Due to the mixing the a 0 0 may also be produced via the f 0 . In this case the da 0 0 system will be in S-wave and the amplitude of reaction (b) can be written as: where ξ is the mixing parameter and F is the f 0 -production amplitude. In the limit k → 0, F is again a constant. The scalar amplitudes α, β, γ for reactions (a) and (b) are related to each other by a factor √ 2, i.e., The differential cross sections for the reactions (a) and (b) have the form (up to terms linear in ξ) where Similarly, the differential cross section of the reaction pn → df 0 can be written as The mixing effect -described by the term C 1 cos Θ in Eq.(4) -then leads to an isospin violation in the ratio R ba of the differential cross sections for reactions (b) and (a), and in the forward-backward asymmetry for reaction (b): The latter effect was already discussed in Ref. [11] where it was argued that the asymmetry A a (Θ = 0) can reach 5 ÷ 10% at an energy excess of Q = (5 ÷ 10) MeV. However, if we adopt a mixing parameter |ξ| 2 = (8 ± 3)%, as indicated by the WA102 data, we can expect a much larger asymmetry. We note explicitly, that the coefficient C 1 in (5) depends not only on the magnitude of the mixing parameter ξ, but also on the relative phases with respect to the amplitudes of f 0 and a 0 production which are unknown so far. This uncertainty has to be kept in mind for the following discussion.
In case of very narrow a 0 and f 0 states, the differential cross section (3), dominated by P -wave near threshold, would be proportional to k 3 or Q 3/2 , where Q is the c.m. energy excess. Due to S-wave dominance in the reaction pn → df 0 one would expect that the cross section increases as σ ∼ k or ∼ √ Q.
In this limit the a 0 -f 0 mixing leads to an enhancement of the asymmetry A a (Θ) as ∼ 1/k near threshold. In reality, however, both a 0 and f 0 have a finite width of about 40-100 MeV. Therefore, at fixed initial momentum their production cross section should be averaged over the corresponding mass distributions, which will significantly change the threshold behavior of the cross sections. Another complication is that broad resonances are usually accompanied by background lying underneath the resonance signals. These problems will be discussed explicitly in Sects. 1.2 and 1.3.

Model calculations
In order to estimate the isospin-violation effects in the ratio R ba of the differential cross-section and in the forward-backward asymmetry A a we use the two-step model (TSM), which has successfully been applied to the description of η-, η ′ -, ω-and φ-meson production in the reaction pN → dX in Refs. [12,13]. Recently, this model has been also used for an analysis of the reaction pp → da + 0 [14].
The diagrams in Fig. 1 describe the different mechanisms of a 0 -and f 0 -meson production in the reaction NN → da 0 /f 0 within the TSM. In the case of a 0 production the amplitude of the subprocess πN → a 0 N contains three different contributions: i) the f 1 (1285)-meson exchange ( Fig. 1 a); ii) the ηmeson exchange (Fig. 1b); iii) s-and u-channel nucleon exchanges ( Fig. 1c and d). As it was shown in Ref. [14] the main contribution to the cross section for the reaction pp → da + 0 stems from the u-channel nucleon exchange (i.e. from the diagram of Fig. 1d and all other contributions can be neglected in a leading order approximation. In order to preserve the correct structure of the amplitude under permutations of the initial nucleons (which is antisymmetric for the isovector state and symmetric for the isoscalar state) the amplitudes for a 0 and f 0 production can be written as the following combinations of the t-and u-channel contributions where s = (p 1 + p 2 ) 2 , t = (p 3 − p 1 ) 2 , u = (p 3 − p 2 ) 2 and p 1 , p 2 , p 3 , and p 4 are the 4-momenta of the initial protons, meson M and the deuteron, respectively. The structure of the amplitudes (9) guarantees that the S-wave part vanishes in the case of direct a 0 production since it is forbidden by angular momentum conservation and the Pauli principle. Also higher partial waves are included in the model calculations in contrast to the simplified discussion in Sect. 1.1.
In the case of f 0 production the amplitude of the subprocess πN → f 0 N contains two different contributions: i) the π-meson exchange ( Fig. 1 b); ii) s-and u-channel nucleon exchanges ( Fig. 1 c and d). Our analysis has shown that similarly to the case of a 0 production the main contribution to the cross section of the reaction pn → df 0 is due to the u-channel nucleon exchange ( Fig. 1 d); the contribution of the combined ππ exchange (Fig. 1 b) as well as the s-channel nucleon exchange can be neglected. In this case we obtain for the ratio of the squared amplitudes If we take g a 0 N N = 3.7 (see e.g. Ref. [15]) and g f 0 N N =8.5 [16] then we find for the ratio of the amplitudes R(f 0 /a 0 ) = g f 0 N N /g a 0 N N = 2.3. Note, however, that Mull and Holinde give a different value for the ratio of the coupling constants R(f 0 /a 0 ) = 1.46, which is about 37% lower. In the following we thus use R(f 0 /a 0 ) = 1.46 ÷ 2.3.
The forward differential cross section for reaction (a) as a function of the proton beam momentum is presented in Fig. 2. The bold dash-dotted and solid lines (taken from Ref. [14] and calculated for the zero width limit Γ a 0 = 0) describe the results of the TSM for different values of the nucleon cut-off parameter, Λ N = 1.2 and 1.3 GeV/c, respectively.
In order to take into account the finite a 0 width we use a Flatté mass distribution with the same parameters as in Ref. [18]: K-matrix pole at 999 MeV, Γ a 0 →πη = 70 MeV, Γ(KK)/Γ(πη) = 0.23 (see also [19] and references therein). The thin dash-dotted and solid lines in Fig. 2 are calculated within the TSM using this mass distribution with a cut M(π + η) ≥ 0.85 GeV and for Λ N = 1.2 and 1.3 GeV, respectively. The corresponding π 0 η invariant mass distribution for the reaction pn → da 0 0 → dπ 0 η at 3.4 GeV/c is shown in Fig. 3 by the dashed line.
In case of the f 0 , where the branching ratio BR (KK) is not yet known [19], we use a Breit-Wigner mass distribution with m R = 980 MeV and Γ R ≃ Γ f 0 →ππ = 70 MeV. The calculated total cross sections for the reactions pn → da 0 and pn → df 0 (as a function of the beam energy T lab for Λ N = 1.2 GeV ) are shown in Fig. 4. The solid and dashed lines describe the calculations with zero and finite widths, respectively. In case of f 0 production in the ππ decay mode we choose the same cut in the invariant mass of the ππ system, i.e. M ππ ≥ 0.85 GeV. The lines denoted by 1 and 2 are obtained for R(f 0 /a 0 ) = 1.46 and 2.3, respectively. Comparing the solid and dashed lines it is obvious that near threshold the finite width corrections to the cross sections are quite important in particular for the energy behavior of the a 0 -production cross section (see also bold and thin curves in Fig. 2).
In principle, a 0 -f 0 mixing can modify the mass spectrum of the a 0 and f 0 . However, in the a 0 -f 0 case the effect is expected to be less pronounced as for the ρ-ω case, where the widths of ρ and ω are very different (see e.g. the discussion in Ref. [9] and references therein). Nevertheless, the modification of the a 0 0 spectral function due to a 0 -f 0 mixing can be measured by comparing the invariant mass distributions of a 0 0 with that of a + 0 . According to our analysis, however, a much cleaner signal for isospin violation can be obtained from the measurement of the forward-backward asymmetry in the reaction pn → da 0 0 → dπ 0 η integrating over the full a 0 mass distribution. For the following calculations, the strengths of the a 0 and f 0 thus will be integrated over the mass interval 0.85-1.02 GeV.
The magnitude of the isospin violation effects is shown in Fig. 5, where we present the differential cross section of the reaction pn → da 0 0 at T p = 2.6 GeV as a function of Θ c.m. for different values of the mixing intensity |ξ| 2 from 0.05 to 0.11. For reference, the solid line shows the case of isospin conservation, i.e. |ξ| 2 = 0. The dashed-dotted curves include the mixing effect. Note that all curves in Fig. 5 were calculated assuming maximal interference of the amplitudes describing the direct a 0 production and its production through the f 0 . The maximal values of the differential cross section may also occur at Θ c.m. = 0 • depending on the sign of the coefficient C 1 in Eq.(4).
It follows from Fig. 5 in either case that the isospin-violation parameter A a (Θ) for Θ c.m. = 180 • may be quite large, i.e.
A a (180 • ) = 0.86 ÷ 0.96 or 0.9 ÷ 0.98 (11) for R(f 0 /a 0 ) = 1.46 or 2.3, respectively. Note that the asymmetry depends rather weakly on R(f 0 /a 0 ). It might be more sensitive to the relative phase of a 0 and f 0 contributions, which has to be settled experimentally.

Background
The dash-dotted line in Fig. 3 shows our estimate of the possible background from nonresonant π 0 η production in the reaction pn → dπ 0 η at T lab = 2.6 GeV (see also Ref. [20]). The background amplitude is described by the diagram shown in Fig. 1 e, where the η and π mesons are created through the intermediate production of a ∆(1232) (in the amplitude πN → πN) and a N(1535) (in the amplitude πN → ηN). The total cross section for the nonresonant πη production due to this mechanism was found to be σ BG ≃ 0.8 µb for a cut-off in the one-pion exchange of Λ π = 1 GeV.
We point out that the background is charge-symmetric and cancels in the difference of the cross sections σ(Θ)−σ(π −Θ). Therefore, a complete separation of the background is not crucial for a test of isospin violation due to the a 0 -f 0 mixing. There will also be some contribution from π-η mixing as discussed in Refs. [9,10]. According to the results of Ref. [9] this mechanism yields a charge-symmetry breaking in the ηNN system of about 6%: A similar isospin violation due to π-η mixing can also be expected in our case.
The best strategy to search for isospin violation due to a 0 -f 0 mixing is a measurement of the forward-backward asymmetry for different intervals of M ηπ 0 . It follows from Fig. 3 that σ a 0 (σ BG ) = 0.3(0.4), 0.27(0.29) and 0.19(0.15) µb for M ηπ 0 ≥ 0.85, 0.9 and 0.95 GeV, respectively. For M ηπ 0 ≤ 0.7 GeV the resonant contribution is rather small and the charge-symmetry breaking will dominantly be related to π-η mixing and, therefore, be small. On the other hand, for M ≥ 0.95 GeV the background does not exceed the resonance contribution and we expect a comparatively large isospin-breaking signal due to a 0 -f 0 mixing.

The reaction pn → df 0 → dππ
The isospin-violation effects can also be measured in the reaction where, due to mixing, the f 0 may also be produced via the a 0 . The corresponding differential cross section is shown in Fig. 6. The differential cross section for f 0 production is expected to be substantiatially larger than for a 0 production, but the isospin violation effect turns out to be smaller than in the πη-production channel. Nevertheless, the isospin violation parameter A is expected to be about 10÷30% and can be detected experimentally.

Reactions (c) and (d)
We continue with pd reactions and compare the final states 3 H a + 0 (c) and 3 He a 0 0 (d). Near threshold the amplitudes of these reactions can be written as with Here D a and D f are the scalar S-wave amplitudes describing the a 0 and f 0 production in case of ξ=0. The ratio of the differential cross sections for reactions (d) and (c) is then given by The magnitude of the ratio R dc now depends on the relative value of the amplitudes D a and D f . If they are comparable |D a | ∼ |D f | or |D f | 2 ≫ |D a | 2 the deviation of R dc from 0.5 (which corresponds to isospin conservation) might be 100% or more. Only in the case |D f | 2 ≪ |D a | 2 the difference of R dc from 0.5 will be small. However, this seems to be very unlikely.
Recently the cross section of the reaction pd → 3 He K + K − has been measured by the MOMO collaboration at COSY-Jülich. It was found that σ = 9.6 ± 1.0 and 17.5 ± 1.8 nb for Q = 40 and 56 MeV, respectively [24]. The authors note that the invariant K + K − mass distributions in those data show broad peaks which follow phase space. However, as it was shown in Ref. [18], the shape of an invariant mass spectrum following phase space cannot be distinguished from an a 0 -resonance contribution at small values of Q. Therefore, the events from Ref. [24] might also be attributed to a 0 and/or f 0 production. Moreover, due to the phase space behavior near threshold one expects a dominance of two-body reactions. Thus the cross section of the reaction pd → 3 He a 0 0 → 3 He π 0 η is expected to be not significantly smaller than the upper limit of about 80÷150 nb at Q = 40 − 60 MeV which follows from the MOMO data (using Γ(KK)/Γ(πη) = 0.23 from [19]).

Reaction (e)
Any direct production of the a 0 in the reaction dd → 4 He a 0 0 is forbidden. It thus can only be observed due to f 0 -a 0 mixing: Therefore, it will be very interesting to study the reaction dd → 4 He (π 0 η) near the f 0 -production threshold. Any signal of the reaction (18) then will be related to isospin breaking. It is expected to be much more pronounced near the f 0 threshold as compared to the region below this threshold.

Summary
In summary, we have discussed the effects of isospin violation in the reactions pN → da 0 , pd → 3 He/ 3 H a 0 and dd → 4 He a 0 0 which can be generated by f 0 -a 0 mixing. It has been demonstrated that for a mixing intensity of about (8±3)%, the isospin violation in the ratio of the differential cross sections of the reactions pp → da + 0 → dπ + η and pn → da 0 0 → dπ 0 η as well as in the forwardbackward asymmetry in the reaction pn → da 0 0 → dπ 0 η not far from threshold may be about 50-100%. Such large effects originate from the interference of direct a 0 production and its production via the f 0 . The former amplitude is suppressed close to threshold due to the P -wave amplitude whereas the latter is large due to S-wave production. A similar isospin violation is expected in the ratio of the differential cross sections of the reactions pd → 3 H a + 0 (π + η) and pd → 3 He a 0 0 (π 0 η).
Finally, we have also discussed the isospin-violation effects in the reactions pn → df 0 (π + π − ) and dd → 4 He a 0 . All reactions together -once studied experimentally -are expected to provide detailed information on the strength of the f 0 -a 0 mixing.
Corresponding measurements are now in preparation for the ANKE spectrometer at COSY-Jülich [25].