Theoretical study on \eta'-->4 pions

The $\eta'$ meson is associated with the U(1) anomaly. In this paper, a successful effective chiral theory of mesons has been applied to study the anomalous decay of $\eta'\rightarrow \pi^+\pi^-\pi^{+(0)}\pi^{-(0)}$. In this study, the contributions of triangle and box anomalies are calculated. It is shown that the contribution of box diagrams is important in this process. We predict branching ratios of $Br(\eta' \rightarrow \pi^+ \pi^- \pi^+ \pi^-)={1 \over 2}Br(\eta' \rightarrow \pi^+ \pi^- \pi^0 \pi^0) = (8.3 \pm 1.2) \times 10^{-5}$, which is in good agreement with BESIII measurement.

This work is organized as follows: In section 2, we briefly review the effective chiral theory of mesons which has been applied in this paper. In section 3, we calculate η → π + π − π + π − branching ratio by evaluating triangle and box anomalies, and with the use of isospin relation we connect this branching ratio to η → π + π − π 0 π 0 . We summarize our results in section 4.

Review of the Effective Chiral Theory
It is well known that the current algebra is very successful in the study of hadron physics. Based on current algebra and large N C expansion of QCD, the Lagrangian of U (3) L × U (3) R chiral field theory of quarks and mesons (0 −+ , 1 −− and 1 ++ ) has been constructed [18,19] where i = 1, 2, 3 and a = 4, 5, 6, 7. The ψ in Eq. (1) is u, d, s quark fields. The scheme of nonlinear σ−model is used to introduce pseudoscalar mesons into Eq. (1) and parameter m is originated in quark condensation and it leads to the dynamical chiral symmetry breaking. In this Lagrangian (Eq. (1)) meson fields are coupled to the corresponding quark field bilinears. The η 8 and η 0 are octet and singlet, respectively. By assuming the mixing angle θ = −20, η and η fields are defined as: Mesons are bound state solutions of QCD and are not independent degrees of freedom. Thus, in Eq. (1) there are no kinetic terms for meson fields. The kinetic terms of the meson fields are generated from quark loops. This theory is an effective theory, therefore, a cut-off is necessary to be introduced [31,32]. In the chiral limit m q → 0, the cut-off Λ is defined [18] By normalizing the kinetic terms of pion, η and ρ fields, physical meson fields are defined [18,19] 2 f π is the pion decay constant and g is a universal coupling constant which are defined as f π and g are two inputs and f π = 186M eV and g = 0.395 are taken. Thus, the cut-off is determined to be Λ ∼ 1.8 GeV. All the masses of mesons are below the cut-off and the theory is self-consistent. The input values of f π and g are chosen such that the theory fits the experimental data for different meson processes [20][21][22][23][24][25][26][27][28][29][30].
As shown in Refs. [18,19] the VMD is a natural result of this meson theory instead of an input. According to Sakurai [33], the VMD is revealed from a Lagrangian in which photon and vector mesons are coupled to quarks symmetrically. These symmetries are shown in the Lagrangian (1), in which the photon field is added. At the fourth order in covariant derivative, the ρ − γ vertex is derived from the quark vertex of photon and ρ meson [18] where A is the photon field. By using Eq. (7) the Γ th (ρ → e + e − ) = 6.75 KeV is predicted. The experimental value, as quoted by particle data group (PDG), is Γ exp (ρ → e + e − ) = 7.04(6) KeV [34]. There are similar terms for ω − γ and φ − γ vertex [18,19]. By employing the VMD of ρ − γ, ω − γ, and φ − γ, the pion form factor, the form factors of the charged and the neutral kaons are obtained [27,30].
In this theory the quark loop is always of order N C . The meson loops are at higher order in N c expansion and f π and g are both of order √ N C [18,19]. The meson physics studied are at the leading order of N C expansion. In the chiral limit m q → 0, the theory is explicitly chiral symmetric.
In this limit, f π and g are two parameters.
In this effective chiral theory [18,19] meson resonances are involved. As pointed out in Refs. [11,35,36], the coupling constants of effective chiral Lagrangian for strong interactions are essentially saturated by meson resonance exchange. In this regard, the anomalous decay ω → 3π is very interesting. In this decay ω → ρ+π, ρ → ππ and direct ω → 3π are involved. The amplitude derived in Ref. [18] is the same as the one derived byÖ. Kaymakcalan, S. Rajeev, and J. Schechter [4]. The vertex ω → ρ + π is from the triangle diagram of quarks and the direct vertex ω → 3π is from the box diagram. Both are in low energies. The vector meson resonance is not involved in the box diagram and as it is shown, the contribution of the box diagram is small. To be more precise, the contribution of box anomaly to the branching ratio Br(ω → 3π) is only 5% [18].
In this process there are two kinds of anomaly: 1) triangle anomaly [1,2] of two body decays η → ρρ, ρ → ππ; 2) box anomaly [3,4] of three body decays η → ρππ, ρ → ππ. The vertex η → ρρ comes from the triangle diagram of quarks and the vertex η → ρππ comes from the box diagram, shown in Fig.(1). The box anomaly proposed by Chanowitz has been applied to study the three body decay of η → π + π − γ in Ref. [37]. : Feynman diagrams for triangle and box anomalies contributing to η → π + π − π + π − . Feynman diagrams were generated using TikZ-Feynman [38] In the theory [18,19], the pion and η fields have two sources: one from the term u of the Lagrangian (Eq. (1)), the other from the shift caused by the mixing between the axial-vector and pseudoscalar fields Eqs. (8) are the result of mixing a µ and ∂ µ π i , and f µ and ∂ µ η , which are generated by corresponding quark loop diagrams. The constant 1 g (1 − 1 2π 2 g 2 ) − 1 2 is the renormalization constant of the a i µ and f µ fields.
In Ref. [19] the triangle anomaly of the η is expressed as where the θ is the mixing angle. The η → γγ; ργ; ωγ anomalous decay modes have been studied by using this Lagrangian (9). The theory agrees with experimental data without a new parameter [19].
From Eqs. (1,8) we can see that π and η are coupled to pseudoscalar and axial-vector currents and ρ 0 is coupled to vector current. Thus, by permuting the final states in the box diagram η → ρ 0 π + π − , we can see that there are 48 box diagrams. The effective Lagrangian of all box diagrams is obtained as The amplitude of the box diagrams is obtained with ρ 0 → π + π − T (2) = 1 g where Adding both the triangle and the box diagrams (Eqs. (10,13)) the total amplitude of the decay η → π + π − π + π − can be obtained.
To obtain the branching ratios, we insert the numerical values: 1) Pion decay constant: 1 √ 2 f π = 131.5(1.0) M eV is taken, where the uncertainty comes from the difference between the input value of f π and the PDG experimental value [34].
3) Weighted average pion mass M π = (3M π + + M π 0 )/4 = 138.4(1.2) M eV . To account for isospin breaking effects due to the phase space corrections, we consider the difference between the charged pion mass and weighted average value as uncertainty. 4) Total width of the η : Γ η = 0.196 MeV [34]. To obtain branching ratios we normalize partial widths by this value.
Of course, there is pentagon diagram for the decay η → π + π − π + π − . It has been shown that the coupling constants of effective chiral Lagrangian for strong interactions are saturated by meson resonance exchange [11,35,36]. There is no ρ-resonance in the pentagon diagram and it is believed that its contribution to this decay mode is small. Thus, calculation of pentagon diagrams is not presented in this work. We are certain that the error made thereby is well below our uncertainty estimate.

Summary
In summary, an effective chiral theory has been applied to study the two decay modes of η → π + π − π + π − and η → π + π − π 0 π 0 . In this work we have evaluated the triangle and box anomalous diagrams. We have also shown that the contribution of box diagrams is important in these processes. Theoretical predictions are: Br(η → π + π − π + π − ) = 1