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Nucleon-nucleon charge symmetry breaking and the dd ->alpha pi0 reaction

We show that using parameters consistent with the charge symmetry violating difference between the strong nn and pp scattering lengths provides significant constraints on the amplitude for the dd ->alpha pi0 reaction.

The concepts of charge independence and charge symmetry provide powerful tools in organizing the multiplet structure of hadrons and nuclei. These symmetries are not perfect; diverse small but interesting violations have been discovered [1,2]. Our concern here is with the breaking of charge symmetry (CS). This symmetry is defined as invariance under a rotation by 180 • around the 2-axis in isospin space. In quantum chromodynamics (QCD), CS implies that dynamics are invariant under the exchange of the up and down quarks [1]. However, since the up and down quarks do have different masses (m u = m d ) [3], the QCD Lagrangian is not charge symmetric. This symmetry violation is called charge symmetry breaking (CSB). The different electromagnetic interactions of the up and down quarks also cause CSB as well as the breaking of charge independence. Observing the effects of CSB interactions therefore provides a probe of m u and m d , once the electromagnetic interactions are treated.
It has long been known that CSB is violated in the 1 S 0 state of nucleon-nucleon scattering, with a pp − a nn ≡ ∆a = 1.5 ± 0.5 fm [1], where a denotes the scattering length. The nn interaction is more attractive than the pp. There are a variety of explanations for this using meson exchange mechanisms [4,5]. Each of these mechanisms involving the strong interaction can be traced to the mass difference between the up and down quarks. Nucleon-nucleon potentials that are consistent with this scattering length difference are successful in reproducing (along with electromagnetic effects) the measured binding energy differences between mirror nuclei [5,6]. It is interesting to search for further manifestations of the up-down quark mass difference.
Two exciting observations of CSB in experiments involving the production of neutral pions have stirred interest in this subject. CSB was observed in the reaction np → dπ 0 at TRIUMF by measuring the CSB forward-backward asymmetry of the differential cross section as A fb = [17.2 ± 8(stat) ± 5.5(sys)] × 10 −4 [7]. Furthermore, the final experiment at the IUCF Cooler ring reported a relatively large dd → απ 0 cross section (σ = 12.7 ± 2.2 pb at T d = 228.5 MeV and 15.1 ± 3.1 pb at 231.8 MeV) [8]. The dd → απ 0 reaction violates CS since the deuterons and the α-particle are self-conjugate under the CS operator, with a positive eigenvalue, while the neutral pion wave function changes sign.
The study of CSB π 0 production reactions presents an exciting new opportunity to determine the influence of quark masses in nuclear physics, and to use effective field theory (EFT) to improve understanding of how QCD works [2]. This is because chiral symmetry of QCD determines the form of pionic interactions. Electromagnetic CSB is typically of the same order of magnitude as the strong one, and also can be handled using EFT. The EFT for the Standard Model at momenta comparable to the pion mass, Q ∼ m π , is chiral perturbation theory (χPT) [9]. This EFT has been extended to pion production [10,11,12,13,14], where typical momenta are Q ∼ √ m π M , with M the nucleon mass. EFT using the operators of [15] was used to correctly predict the sign of the forward-backward asymmetry in np → dπ 0 [16].
The purpose of the present note is to use information regarding CSB in the nucleon-nucleon system to constrain or inform the calculations of the dd → απ 0 reaction. We begin by describing the pion production calculations and then show how the nucleon-nucleon CSB is relevant for this calculation. We constrain the parameters by the requirement that NN CSB is consistent with observation. It turns out that this constraint reduces considerably the uncertainty in the predictions for the dd → απ 0 reaction.
Attempts to understand the dd → απ 0 reaction began with a survey of various mechanism using initial state plane wave functions and simplified final state wave functions [11]. Next, recent significant advances in four-body theory [17,18] were used to in- clude the effects of deuteron-deuteron interactions in the initial state, and to use bound-state wave functions with realistic two-and three-nucleon interactions [19]. The resulting calculations are hybrid: the CSB pion-production operators are constructed using EFT, but the strong interactions are not. The result was that a cross section of the experimentally measured size could be obtained using leading order (LO) and next-to-next-to LO (NNLO) pion production operators. However, not all NNLO diagrams were included. A complete analysis would require a careful treatment of loop diagrams. The calculation made use of a variety of CSB mechanism which we will briefly review now. At formally leading order, there is only one contribution, represented by Fig. 1a: pion rescattering in which the CSB occurs through the seagull pion-nucleon terms linked to the nucleon-mass splitting.
There is no next-to-leading order (NLO) contribution. At NNLO, suppressed by O(m π /M ), there exists a recoil correction of the LO term, labelled by M rec . This is determined by the same parameters as the LO term. At the same order there are other operators. A term arises in which a one-body CSB operator (∝ β 1 +β 3 ) is sandwiched between initial and final state wave functions, as illustrated in, e.g., Fig. 1b. We refer to this as the one-body term (M 1b ). The terms β 1 = O(ǫm 2 π /M 2 ) andβ 3 = O(α/π) arise from, respectively, the quark-mass-difference and electromagnetic contributions to the isospin-violating pionnucleon coupling. Neither β 1 norβ 3 can be extracted from experiment yet. These terms were estimated by modeling [20] β 1 by π-η mixing, see Fig. 1b, where π 0 |H|η = −4200 MeV 2 is the π-η-mixing matrix element [21], andḡ η = g ηN N f π /M the η-nucleon coupling constant. Nucleon-nucleon elastic scattering data show little sensitivity to η exchange and highaccuracy fits can be achieved [22] using g 2 ηN N /4π = 0. Indeed, the possibility of a vanishing coupling constant had been raised earlier by the detailed analysis of N N total cross sections and pp data using dispersion relations [23] resulting in g 2 ηN N /4π ≈ 0. Also, in the Bonn full model [24] it is found that g 2 ηN N /4π ≈ 0 is consistent with the NN scattering data. A value of g 2 ηN N /4π = 0.51 was used in Ref. [19]. The effects of electromagnetic interactions as well as strong CSB were included in computing the α particle wave functions. These interactions generate a small isospin T = 1 component of the wave function that enables a non-zero contribution of charge-symmetry conserving (CSC) production operators. The one-body operator was used to generate π 0 production [19]. A number of other CSB mechanisms enter at N 3 LO or higher, including loop diagrams, and short-range interactions. The lowest order, where four-nucleon contact interactions start to contribute, is N 4 LO. To estimate their strength, Ref. [11] evaluated certain treelevel contributions as indicated by Figs. 1c. This figure represents the exchange of heavy mesons (σ, ω, ρ) via a Z-graph mechanism, with π-η mixing to generate CSB at pion emission (M σ , M ω , and M ρ ). Another Z-graph (labeled as M ρω ) arises in which the CSB occurs in the heavy-meson exchange via ρ-ω mixing along with strong pion emission at the vertex. The Z-graphs are believed to be important, because their inclusion lead to a quantitative description of the total cross section for the reaction pp → ppπ 0 near threshold [25]. The results [19] use the coupling constants and parameters of Ref. [11], see their Table I. It was found that the Z-graphs give unexpectedly large contributions, especially the ρ-ω exchange operator that add constructively and overwhelm the one-body term. This model of resonance saturation, gives results in vast disagreement with the power counting and therefore needs reassessment. Here we re-asses the coupling constants used by [19]. This is only a first step, because it is also necessary to justify the use of Z-graphs in resonance saturation, which would require further calculations.
The various mechanisms generate pion-production kernels that are sandwiched between final and initial state wave functions to provide a transition matrix element M for T d =228.5 MeV. These matrix elements are given in Table 1 of [19]. The transition amplitude can be written as where the pion exchange term M PE , its recoil correction M rec , and the effects of CSB in the α wave function M WF are independent of g η and β 1 . The one-body term and sigma and rho exchange terms involving the Z-graphs, M 1b + M σ + M ρ , are proportional to β 1 . The terms M ω + M ρω , which arise from omega and rho-omega exchange, are proportional to β 1 and g ω , the strong ω-nucleon coupling constant. Given the contributions to M expressed in Table 1 of [19] in units of 10 −4 fm −2 , the cross section can be written as Results were obtained [19] using either the Argonne V 18 (AV18) [26] or CD-Bonn (CDB) [22] two-nucleon potentials combined with a properly adjusted Tucson-Melbourne (TM99) [27] three-nucleon force. The combination guarantees that the α particle binding energy is reproduced with high accuracy. We now turn to CSB in the nucleon-nucleon (NN) system due to meson-mixing. Our CSB NN calculation is described in Ref. [5]. Since we have to use mesonnucleon form factors in our NN scattering calculations, while in Refs. [11,19] no such form factors are applied, we explain the definition of meson-nucleon coupling constants in conjunction with form factors. For this, we define a coupling constant as a function of the momentum-transfer t by where α stands for any meson, m α denotes the meson mass and Λ α the so-called cutoff mass. We use Λ ρ,ω = 1400 MeV. In the relativistic three-dimensional theory used for CD-Bonn, the momentum transferred between the two nucleons is t = −(q ′ −q) 2 with q and q ′ the center-of-mass nucleon three-momenta before and after scattering. Coupling constants used in theories without form factors should be compared to coupling constants at t = 0, i.e., g α (0), of a theory with form factors. Thus, in our CSB NN calculations, we use the coupling constants given in Table I of Ref. [11] and identify them with g α (0), except for the omeganucleon coupling where we use instead of the 10.6 applied in Ref. [11]. Our lower value for the omega coupling is more consistent with a fixed-s dispersion relations analysis by Hamilton and Oades [28] in which a value of 5.7 ± 2.0 was obtained. Moreover, SU (3) flavor implies g ω = 3g ρ which, for the rho-coupling used, yields g 2 ω /4π = 3.9. Applying the omega coupling constant stated in Eq. (6) and the other meson parameters as given in Table I of Ref. [11] and using ρ 0 |H|ω = −4300 MeV 2 for the ρ-ω-mixing matrix element [21], we find a CSB contribution to the 1 S 0 scattering length difference from ρ − ω mixing of ∆a ρω = 1.45 fm .
Obviously, this term entirely accounts for the observed CSB scattering length difference. We have also calculated η − π 0 mixing to find ∆a πη = 0.33 fm (8) using the same strong coupling constants and mixing matrix element as in [19]. Since the ρω mixing explains all CSB, this might be considered as yet another argument that g η = 0.
To explore a variety of possibilities, we re-write the results of [19] as depending on the potential used. Ref. [19] observed that this model dependence visibly influences the cross section result, requiring a more consistent treatment of the NN interaction and production operator in the future. As a first step, we note that using the parameters of [19] leads to the results with cross sections Next we replace g 2 ω /4π by the value stated in Eq. (6), which yields M CDB = (−6.80 + 7.03i)10 −4 fm −2 , (15) M AV18 = (−4.42 + 4.54i)10 −4 fm −2 .
This would lead to about a 20% reduction of the cross section. However, using πη mixing along with ρω mixing overestimates ∆a. Therefore, it is reasonable to explore the consequences of using g η = 0, which results in with cross sections σ CDB = 111.5 pb, (19) σ AV18 = 34.5 pb.
The cross section for AV18 is now only a factor of 2 or so bigger than the data, while CD-Bonn is off by a factor of about 7.
In summary, past theoretical work [19] on the cross section of the dd → απ 0 reaction at 228.5 MeV was plagued by the problem that the predictions were off by factors between 15 and 30. In this note, we have shown that constraining the coupling constants involved by the requirement that the CSB in the 1 S 0 NN scattering length is correctly reproduced reduces the over-prediction to just a factor of about 2 (using the AV18 potential). In relative terms, this is substantial progress in understanding the dd → απ 0 reaction. However, significant differences between the use of the AV18 and CD-Bonn potentials remains, signaling that a deeper understanding is needed.