On the mixing angle of the vector mesons ω(782) and ϕ(1020)
OOn the mixing angle of the vector mesons ω (782) and φ (1020) M. K. Volkov ∗ , A. A. Pivovarov † , K. Nurlan , , ‡ BLTP, Joint Institute for Nuclear Research, Dubna, 141980, Russia Institute of Nuclear Physics, Almaty, 050032, Kazakhstan Eurasian National University, Nur-Sultan, 01008, Kazakhstan
Abstract
In the present work, the mixing angle of the vector ω (782) and φ (1020) mesons is estimated in the frameworkof Nambu–Jona-Lasinio model. The decay φ → π γ is considered as a basic process to determine this angle.The obtained value is compared with the results of the other authors. Besides, the width of the decay φ → π and the cross-section of the process e + e − → π φ are calculated by using this angle. The mixing of the isoscalar vector mesons ω (782) and φ (1020) plays an important role inthe description of the different processes of meson interactions. In the pseudoscalar case, thesinglet-octet mixing leading to the physical mesons η and η (cid:48) (958) has been described by manyauthors. For example, in the works [1, 2, 3], in the framework of Nambu–Jona-Lasinio (NJL)model [2, 4, 5, 6, 7, 8, 9, 10, 11], it is shown that the ’t Hooft interaction allows one to describethe masses of pseudoscalar mesons and their singlet-octet mixing angle. In the vector case, as amechanism of mesons ω (782) and φ (1020) mixing one may consider their interaction via the kaonloops. This mechanism in the framework of the hidden local symmetry Lagrangian was describedin [12, 13]. In the first work, the value 4 . ◦ was obtained at the energies of the φ (1020) meson mass.In the second one, this value was changed to 3 . ◦ . The effect of the ω (782) and φ (1020) mesonsmixing has been considered in the numerous other theoretical works. In the work [14] this anglewas defined by using the process φ → π γ calculated with the chiral SU (3) symmetric Lagrangian.The value 3 . ◦ was obtained. In the work [15], the authors received the angle 3 . ± . ◦ in theframework of the Chiral perturbation theory. In the work [16], in the light-front quark model withthe QCD-motivated effective Hamiltonian including the hyperfine interaction, the value 5 . ◦ wasobtained. The collaboration KLOE experimentally received the value 3 . ± . ◦ [17].In the present work, we do not consider the nature of these mesons mixing in detail. Howeverwe make the estimation of this angle by using different decays calculated in the NJL model. As abasic process, like in the work [14], we use the decay φ → π γ , because it has been measured withenough precision. First of all we calculate the mesons ω (782) and φ (1020) mixing angle by usingthis decay in the framework of NJL model. Than we apply this angle to investigate the processes φ → π and e + e − → π φ . ∗ [email protected] † tex [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] M a y The interaction Lagrangian of the extended NJL model
In the NJL model, a fragment of u and d quark part of the quark-meson interaction La-grangian, containing the vertices we need, takes the form [7]:∆ L int = ¯ q g a γ µ γ (cid:88) j = ± , λ a j a j µ + g φ α ) γ µ λ φ φ µ + g ω α ) γ µ λ ω ω µ + ig π γ (cid:88) j = ± , λ πj π j q, (1)where q and ¯ q are the u and d quark fields with the constituent masses m u ≈ m d = m = 280 MeV,s quarks do not appear here since they do not participate in the studied processes. The sine inthe second term and the cosine in the third term take into account the mixing of the ω (782) and φ (1020) mesons.The coupling constants are: g a = g φ = g ω = (cid:18) I (cid:19) − / ≈ . , g π = (cid:18) Z π I (cid:19) − / ≈ . , (2)where Z π = (cid:32) − m M a (cid:33) − ≈ . . (3)Here Z π is the additional renormalization constant appearing in the π − a transition, M a = 1230 ±
40 MeV is the mass of the axial-vector a (1260) meson [18].The integral appearing in the quark loops as a result of renormalization of the Lagrangianhas the form I = − i N c (2 π ) (cid:90) Θ(Λ + k )( m − k ) d k, (4)where Λ = 1 .
26 GeV is the four-dimensional cutoff parameter [7], N c = 3 is the number of colorsin QCD. The matrices λ are linear combinations of Pauli matrices: λ a = λ π = (cid:32) − (cid:33) , λ a − = λ π − = √ (cid:32) (cid:33) ,λ φ = λ ω = (cid:32) (cid:33) , λ a + = λ π + = √ (cid:32) (cid:33) . (5) φ → π γ The diagram of the process φ → π γ is presented in the Fig. 1.The amplitude of the considered process in the NJL model takes the form: M ( φ → π γ ) = 34 √ α em π / F π g φ sin( α ) e µνλδ e µ ( p φ ) e ∗ ν ( p γ ) p πλ p γδ , (6)where α em is the electromagnetic constant, F π = mg π ≈
93 MeV is the pion decay constant, e ∗ µ ( p φ )and e ∗ ν ( p γ ) are the polarisation vectors of the meson φ (1020) and the photon. The similar amplitudefor the process ω → π γ was obtained in the NJL model in [7].The experimental value of this decay width [18]:Γ( φ → π γ ) exp = 5 . ± . . (7)By using this known experimental value one can fix the mixing angle of the mesons ω (782)and φ (1020). The result is α = 3 . ◦ . φ → π γ . φ → π In the NJL model, the decay φ → π is described with two types of the anomalous quarkdiagrams: anomalous box quark diagram (Fig. 2) and diagram with the intermediate ρ mesonwhich connects the anomalous triangle diagram with the vertex ρ → ππ (Fig. 3). Figure 2: The box diagram of the decay φ → π . This process is similar to the well-known decay ω → π investigated in the NJL modelrecently in [19] and can be obtained from it due to ω − φ mixing: M ( φ → π ) = − g φ sin( α ) π F π (cid:20) b + g ρ F π (cid:88) i =+ , − , M ρ − q i − i (cid:113) q i Γ ρ ( q i ) (cid:21) e µνλδ e µ ( p φ ) p ν p + λ p − δ . (8)where p , p − , p + are the pions momenta, p φ is the momentum of the φ meson, q i = p φ − p i ( i = + , − ,
0) are the momenta of the intermediate ρ mesons, M ρ = 775 .
49 MeV [18] is the ρ mesonmass. The first term of the amplitude describes the contribution of the box diagram. The π − a transitions on the pion lines lead to the following form of the constant b : b = 1 − a + 32 a + 18 a , (9)where a = 1 .
84 [19]. The second, the third and the fourth terms describe one, two and three π − a φ → π with the intemediate ρ meson. transitions respectively. The decay width of the intermediate ρ meson depends on the momentum:Γ ρ ( q ) = g ρ ( q − M π ) / πq . (10)The result for the full width of the decay φ → π in the NJL model isΓ( φ → π ) = 0 .
67 MeV , (11)The experimental value [20]:Γ( φ → π ) exp = (0 . ± . , (12) e + e − → φπ The diagrams of the processes e + e − → φπ are shown in Figs. 4, 5. Figure 4: The contact diagram of the process e + e − → φπ . The amplitude of the process e + e − → φπ is calculated in the extended NJL model [21, 22,23] (see Appendix), because the meson ρ (1450) considered as the first radially excited state cancontribute to this process. As a result, the amplitude takes the form: e + e − → φπ with the intermediate ρ, ρ (cid:48) mesons. M ( e + e − → φπ ) = 8 πα em s m sin( α )( T W + T ρ + T ρ (cid:48) ) l µ (cid:15) µνλδ e ∗ ν ( p φ ) p φλ p πδ , (13)where s = ( p e + + p e − ) , l µ = ¯ eγ µ e is the lepton current.The terms corresponding to contributions from the contact diagram and the diagram withthe intermediate ρ meson are T W = g π I φ , (14) T ρ = C ρ g π I ρφ g ρ sM ρ − s − i √ s Γ ρ ( s ) . (15)The contribution of the amplitude with the intermediate meson ρ (1450): T ρ (cid:48) = C ρ (cid:48) g π I ρ (cid:48) φ g ρ sM ρ (cid:48) − s − i √ s Γ ρ (cid:48) ( s ) . (16)The constants C ρ , C ρ (cid:48) appear in the quark loops of photon transition into the intermediatemeson: C ρ = 1sin (cid:16) θ ρ (cid:17) (cid:104) sin (cid:16) θ ρ + θ ρ (cid:17) + R ρ sin (cid:16) θ ρ − θ ρ (cid:17)(cid:105) ,C (cid:48) ρ = − (cid:16) θ ρ (cid:17) (cid:104) cos (cid:16) θ ρ + θ ρ (cid:17) + R ρ cos (cid:16) θ ρ − θ ρ (cid:17)(cid:105) . (17)The integrals, which appear in the quark loops are I φ = − i N c (2 π ) (cid:90) A φ ( k )( m − k ) Θ(Λ − k )d k,I ρφ = − i N c (2 π ) (cid:90) A ρ ( k ) A φ ( k )( m − k ) Θ(Λ − k )d k,I ρ (cid:48) φ = − i N c (2 π ) (cid:90) B ρ ( k ) A φ ( k )( m − k ) Θ(Λ − k )d k, (18) here Λ is the three-dimensional cutoff parameter. The parameters used here are defined in theAppendix.A comparison of the cross-section of the process e + e − → φπ with the experimental datais presented in Fig. 6. The obtained results are in satisfactory agreement with the experimentaldata. Figure 6: The cross-section of the process e + e − → φπ . The experimental points are taken from the work of theBaBar collaboration [24]. σ [ n b ] E c.m. [MeV] NJL model BaBar
The consideration of the mesons ω (782) and φ (1020) mixing angle is necessary to describedifferent processes. This angle is determined by various authors in different ways. In the presentwork, by using the decays φ → π γ , φ → π and e + e − → π φ calculated in the NJL model theresult 3 . ◦ have been obtained. The error of this model is determined by the chiral symmetrybreaking effects and by the model parameters uncertainty. The error caused by the first reasonis M π M p ≈ . ± . ◦ . Theangle 3 ◦ has been used in our earlier works, particularly, in the process e + e − → π γ [25], and hasled to a good agreement with the experiment. This result is close to the results obtained in [14, 15],where the chiral symmetry has been used. Further theoretical and experimental researches aimedat an investigation of the nature of this mixing are of interest. ppendix. The Lagrangian of the extended NJL model In the extended NJL model, the part of the quark–meson interaction Lagrangian referringto the mesons involved in the process under consideration has the form [21, 22, 23]:∆ L int = ¯ q γ µ γ (cid:88) j = ± , λ a j (cid:16) A a a j µ + B a a (cid:48) j µ (cid:17) + 12 sin αγ µ λ φ (cid:16) A φ φ µ + B φ φ (cid:48) µ (cid:17) + 12 γ µ (cid:88) j = ± , λ ρj (cid:16) A ρ ρ jµ + B ρ ρ (cid:48) jµ (cid:17) + iγ (cid:88) j = ± , λ πj (cid:16) A π π j + B π π (cid:48) j (cid:17) q, (19)where the excited meson states are marked with prime, A M = 1sin(2 θ M ) (cid:104) g M sin( θ M + θ M ) + g (cid:48) M f M ( k ⊥ ) sin( θ M − θ M ) (cid:105) ,B M = − θ M ) (cid:104) g M cos( θ M + θ M ) + g (cid:48) M f M ( k ⊥ ) cos( θ M − θ M ) (cid:105) . (20)The subscript M specifies the corresponding meson.The first radially excited states are introduced using the form factor f ( k ⊥ ) = (1 + dk ⊥ ).The slope parameter d was obtained from the requirement of invariability of the quark condensateafter including the radially excited meson states and it depends only on the quark content of themeson: d uu = − . × − MeV − . (21)The transverse relative momentum of the inner quark-antiquark system can be representedas k ⊥ = k − ( kp ) pp , (22)where p is the meson momentum. In the rest system of a meson k ⊥ = (0 , k ) . (23)Therefore, this momentum may be used as a three-dimensional one.The parameters θ M are the mixing angles determined after diagonalization of the free La-grangian for the ground and first radially excited states [22, 23]: θ a = θ ρ = θ φ = 81 . ◦ , θ π = 59 . ◦ . (24)In addition, θ M are auxiliary parameters introduced for convenience assin ( θ M ) = (cid:113) R M ,R a = R ρ = R φ = I fuu (cid:113) I I f ,R π = I f (cid:113) Z π I I f , (25)The integrals appearing in the quark loops as a result of renormalization of the Lagrangian are I f m = − i N c (2 π ) (cid:90) f m ( k )( m − k ) Θ(Λ − k )d k, (26) here Λ = 1 .
03 GeV is the three-dimensional cutoff parameter.Then θ a = θ ρ = θ φ = 81 . ◦ , θ π = 59 . ◦ . (27)The constants g a , g φ and g π were defined in (2). The constant g ρ is the same as g φ . The constantsappearing due to introducing of excited states have the following form: g (cid:48) a = g (cid:48) ρ = g (cid:48) φ = (cid:18) I f (cid:19) − / , g (cid:48) π = (cid:16) I f (cid:17) − / . (28) Acknowlegments
The authors are grateful to A. A. Osipov and A. B. Arbuzov for useful discussions.
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