The decay ϕ-> f_0(980) γand the process e^+e^- -> ϕf_0(980)
Yu. M. Bystritskiy, M. K. Volkov, E. A. Kuraev, E. Bartos, M. Secansky
aa r X i v : . [ h e p - ph ] D ec The decay φ → f (980) γ and the process e + e − → φf (980) Yu. M. Bystritskiy, ∗ M. K. Volkov, † and E. A. Kuraev ‡ Joint Institute for Nuclear Research, Dubna, Russia
E. Bartoˇs § Institute of Physics, Slovak Academy of Sciences, Bratislava
M. Seˇcansk´y ¶ Joint Institute for Nuclear Research, Dubna, Russia andInstitute of Physics, Slovak Academy of Sciences, Bratislava (Dated: October 24, 2018)
Abstract
The decay φ → f (980) γ and process e + e − → φf (980) are considered within the local Nambu-Jona-Lasinio model. In the amplitudes of these processes contributions of s -quark and kaon loopsare taken into account. The kaon loop gives a dominant contribution. Our estimation for thedecay width of φ → f γ is in satisfactory agreement with recent experimental data. This allowsus to make some predictions for cross sections of the process e + e − → γ ∗ → φf which can betested in the C- τ factory. The total and differential cross sections of this process are calculatedand presented in the figures. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] . INTRODUCTION In the last years a lot of experimental [1, 2] and theoretical [3, 4, 5, 6, 7, 8, 9, 10]papers have been devoted to the description of φ -meson decays with the production ofscalar isoscalar f mesons.There are different theoretical interpretations of the f (980) meson structure. In papers[3, 11, 12, 13], for example, this meson is considered as a kaon molecule. In other papers,this meson is described as a four quark state [3, 4] or as an admixture of quark-antiquarkand diquark-antidiquark states [5, 6, 7]. Recently, the decays of φ -mesons were consideredwithin the ChPT [8].In this paper, the local Nambu-Jona-Lasinio (NJL) model will be used. All mesons aretreated as quark-antiquark states in this model. In particular, the f (980) meson is theadmixture of light u ¯ u and d ¯ d and strange s ¯ s quarks [9]. In the framework of this model wedescribe the decay φ → f γ .This decay channel was successfully described in terms of one-loop Feynman amplitudeswith intermediate state of K + K − mesons [3].The amplitude of this process in our approach we express in terms of s -quark and kaonloops. The obtained result is in satisfactory agreement with recent experimental data [14].Using the same approximations we calculate the total and differential cross sections forthe e + e − → γ ∗ → φf process. A comparison with the results obtained in ChPT approach[15] and the recent experimental data [16] are discussed in Conclusion. II. PROCESS φ → f (980) γ The inner parameters of the NJL model are the constituent quark masses m u = m d =263 MeV, m s = 407 MeV and the ultraviolet cut-off parameter Λ = 1 .
27 GeV [9, 17]. Theseparameters are fixed by a value of the weak pion decay π → µν constant f π = 92 . ρ → ππ , g ρ = 5 .
94 (that correspond to the width Γ ρ → ππ = 149 . [18]. Let us note that in [17] some other values of these parameters were used which corresponded to f π =93 MeV, g ρ = 6 .
14 (in that case the width Γ ρ → ππ = 155 MeV). Here we use modern experimental data[18] for fixing our model parameters. α that describes the deviation from the angle of ideal mixingof scalar mesons in the singlet-octet sector. In the case of ideal mixing we have two states:the state σ u consists of light u and d quarks, and the state σ s consists of s quarks only. Theangle α allows us to express physical states f (980) and σ through the states σ u and σ s : σ = σ u cos α − σ s sin α,f = σ u sin α + σ s cos α. The value of α = 11 . o was obtained by using t’Hooft interaction and mass difference of η and η ′ mesons [19, 20].Part of Lagrangian corresponding to a quark-meson interaction has the form∆ L int = ¯ q (cid:26) g σ u λ u σ u + g σ s λ s σ s + iγ g K (cid:16) λ K + K + + λ K − K − (cid:17) + g φ γ ν λ s φ ν (cid:27) q. (1)Using the parameters of the model it is possible to calculate all meson-quark coupling con-stants and the constant corresponding to additional renormalization of the pseudoscalarfields Z π and Z K which takes into account the transition of pseudoscalar mesons to axial-vector mesons [17] g σ u = (4 I ( m u , m u )) − / = g ρ √ . , g σ s = (4 I ( m s , m s )) − / = 2 . , (2) g K ∗ = (4 I ( m u , m s )) − / = 2 . , g φ = √ g σ s = 7 . ,Z K = − m u + m s ) M K ! − = 1 . , g K = g K ∗ Z / K = 3 . , where M K = 1403 MeV is the mass of the strange axial-vector meson K and I ( m, m ) = 3(2 π ) Z d k θ (Λ − k )( k + m ) = 3(4 π ) ln Λ m + 1 ! − Λ Λ + m ! ,I ( m , m ) = 3(2 π ) Z d k θ (Λ − k )( k + m ) ( k + m ) == 3(4 π ) ( m − m ) m ln Λ m + 1 ! − m ln Λ m + 1 !! . We used in (1) the following combinations of the Gell-Mann matrices: λ u = = ( λ + √ λ ) √ , λ s = −√ = ( − λ + √ λ ) √ , µ ( p ) f ( p ) γ ν ( p ) = φ µ ( p ) k + p γ ν ( p ) kf ( p ) k − p | {z } s-quark contibution + φ µ ( p ) k + p γ ν ( p ) kf ( p ) k − p + φ µ ( p ) γ ν ( p ) f ( p ) | {z } kaon contibution FIG. 1: Feynman diagrams describing the amplitude of the decay φ → f γ . λ K + = √ = λ − iλ √ , λ K − = √
20 0 00 0 0 = λ + iλ √ . The decay φ → f γ in the local NJL model is described by the diagrams shown in Fig.1, where the first diagram contains the s -quark loop and presents the contribution of thefirst order of 1 /N c expansion (where N c = 3 is the number of quark colors), and the lasttwo diagrams describes the contribution of kaon loops (next order of 1 /N c expansion). Allvertices of these diagrams were calculated in terms of the quark loops. Only the divergentpart of this quark loop integrals with the appropriate ultraviolet regularization with thecut-off parameter Λ was taken into account. As a result, all coupling constants in formula(2) was calculated. Let us consider, for example, the calculation of the vertex f K + K − .The integral corresponding to this vertex has the form g f K + K − = i π ) g K Z dk ×× n Sp [( λ s g σ s cos α + λ u g σ u sin α ) S ( k − p + ) λ K + γ S ( k ) λ K − γ S ( k + p − )] f ++ Sp [( λ s g σ s cos α + λ u g σ u sin α ) S ( k − p − ) λ K − γ S ( k ) λ K + γ S ( k + p + )] f , o (3)where S ( k ) is the matrix of quark propagators: S ( k ) = diag ˆ k + m u k − m u , ˆ k + m d k − m d , ˆ k + m s k − m s ! , (4)and Sp[ . . . ] f is the trace over flavour indices. Calculation of this trace leads to the followingexpression: g f K + K − = i π ) g K Z dk × (cid:16) − √ (cid:17) g σ s cos α Sp h(cid:16) ˆ k − ˆ p + + m s (cid:17) γ (cid:16) ˆ k + m u (cid:17) γ (cid:16) ˆ k + ˆ p − + m s (cid:17)i(cid:16) ( k − p + ) − m s (cid:17) ( k − m u ) (cid:16) ( k + p − ) − m s (cid:17) ++ 2 g σ u sin α Sp h(cid:16) ˆ k − ˆ p + + m u (cid:17) γ (cid:16) ˆ k + m s (cid:17) γ (cid:16) ˆ k + ˆ p − + m u (cid:17)i(cid:16) ( k − p + ) − m u (cid:17) ( k − m s ) (cid:16) ( k + p − ) − m u (cid:17) . After calculation of the trace over the Dirac matrices we separate out the divergent termsof the quark loop and calculate them in Euclidean metric: g f K + K − = i π ) g K Z dkiπ ((cid:16) − √ (cid:17) g σ s cos α k − m u ) ( m u − m s ) + (finite terms))( k − m u ) ( k − m s ) ++ 2 g σ u sin α k − m u ) ( m s − m u ) + (finite terms))( k − m s ) ( k − m u ) ) == g K ((cid:16) − √ (cid:17) g σ s cos α ( m u − m s ) π ) Z dk ( k + m s ) ( k + m s ) ! ++ 2 g σ u sin α ( m s − m u ) π ) Z dk ( k + m u ) ( k + m s ) ! ) . (5)Recalling (2) the expression in the round brackets can be rewritten as ( ... ) = 4 I ( m s , m s ) =( g σ s ) − and ( ... ) = 4 I ( m u , m s ) = (cid:16) g K ∗ (cid:17) − and the vertex obtains the form g f K + K − = 2 √ g σ s cos α (2 m s − m u ) g K g σ s ! − g σ u sin α (2 m u − m s ) g K g K ∗ ! == 5 .
51 GeV . (6)Similar calculations give us the following expressions for other constants: g φ µ K + K − = g φ √ g K ∗ g σ s ! Z K (cid:16) p + + p − (cid:17) µ ,g A µ K + K − = e (cid:16) p + + p − (cid:17) µ , (7)where p ± are the K ± momenta and e is the electric charge ( e / π = 1 / g φ µ K + K − and g φ µ γνK + K − the factor Z K disappears after taking intoaccount K + → K +1 transitions on the kaon line. A similar situation takes place in thedecay of ρ → ππ [17]. The vertex g φ µ K + K − was derived in [21, 22] and leads to satisfactoryagreement with the experiment – we get the decay width Γ φ → KK = 1 .
88 MeV while theexperimental value is Γ expφ → KK = 2 . φ → f γ . The quark loop gives the amplitude M ( s ) φ → f γ = C ( s ) A ( s ) φ → f γ ( g µν ( p p ) − p ν p µ ) e µ ( p ) e ν ( p ) , (8)5 ( s ) = e (4 π ) g ρ g σ s cos α,A ( s ) φ → f γ = Z dx − x Z dy m s (4 xy − m s − y (1 − y ) M φ + xy (cid:16) M φ − M f (cid:17) + iǫ . Following the quark confinement condition we take into account only the real part of thisamplitude. Then the amplitude square is (cid:12)(cid:12)(cid:12) Re (cid:16) M ( s ) φ → f γ (cid:17)(cid:12)(cid:12)(cid:12) = 12 (cid:16) M φ − M f (cid:17) (cid:12)(cid:12)(cid:12) C ( s ) Re (cid:16) A ( s ) φ → f γ (cid:17)(cid:12)(cid:12)(cid:12) . (9)It gives the following contribution to the decay width:Γ ( s ) φ → f γ = 12 π (cid:16) M φ − M f (cid:17) M φ (cid:12)(cid:12)(cid:12) C ( s ) Re (cid:16) A ( s ) φ → f γ (cid:17)(cid:12)(cid:12)(cid:12) = 6 .
75 eV . (10)The kaon loop gives the amplitude M ( K ) φ → f γ = C ( K ) A ( K ) φ → f γ ( g µν ( p p ) − p ν p µ ) e µ ( p ) e ν ( p ) , (11) C ( K ) = e (4 π ) g ρ √ g f K + K − ,A ( K ) φ → f γ = Z dx − x Z dy xy ) M K − y (1 − y ) M φ + xy (cid:16) M φ − M f (cid:17) + iǫ . Its contribution to the decay width φ → f γ is dominantΓ ( K ) φ → f γ = 12 π (cid:16) M φ − M f (cid:17) M φ (cid:12)(cid:12)(cid:12) C ( K ) A ( K ) φ → f γ (cid:12)(cid:12)(cid:12) . (12)It is worth noticing that a theoretical prediction has a strong dependence on the mass of the f -meson value. Experimental value is M f = 980 ±
10 MeV and changing M f in the interval970 MeV ≤ M f ≤
990 MeV we obtain the following interval for a theoretical predictionof decay width 2 .
39 KeV ≥ Γ ( K ) φ → f γ ≥ .
66 KeV. With the contribution of the quark looptaken into account these values slightly changeΓ ( K + s ) φ → f γ ≈ .
65 KeV . (13)The experimental value is Γ ( exp ) φ → f γ = 0 . ± .
03 KeV [14]. So we can see that our predictionis in qualitative agreement with experiment at M f = 990 MeV.6 II. SUBPROCESS γ ∗ → φf (980) Let us now consider the cross process for the φ → f γ decay, namely, the γ ∗ → φf . Dueto the off-mass-shell photon here the additional gauge invariant Lorentz structure appearsand the amplitude can be written in the form M ( γ ∗ ( p , ν ) → φ ( p , µ ) f ( p )) == X i = s,K C ( i ) π e µ ( p ) e ν ( p ) (cid:16) A ( i ) R µν (1) + B ( i ) R µν (2) (cid:17) , (14)where i denotes the type of a contribution ( i = s corresponds to the s -quark loop contributionand i = K to the kaon loop contribution). Two gauge invariant structures R µν (1 , are R µν (1) = g µν ( p p ) − p ν p µ ,R µν (2) = p − p p ( p p ) ! µ p − p p ( p p ) ! ν ,p µ R ( i ) µν = p ν R ( i ) µν = 0 , i = 1 , . (15)The quantities A ( i ) , B ( i ) in (14) depend only on momentum squares ( p , p , p ) and C ( i ) arethe product of the coupling constants.Quark-loop contribution takes the form (which differs from the case of φ → f γ bynonzero virtuality of photon) C ( q ) = e g ρ g σ s cos α,A ( q ) = − α ( q ) + β ( q ) p p ( p p ) ,B ( q ) = β ( q ) ,α ( q ) = Z dx − x Z dy m s (4 xy − m s − x z p − y z p − x y p − iǫ ,β ( q ) = Z dx − x Z dy m s (2( x + y ) − xy − m s − x z p − y z p − x y p − iǫ , where z = 1 − x − y . The kaon-loop contribution reads as: C ( K ) = e g φK + K − g f K + K − ,A ( K ) = − α ( K ) + β ( K ) p p ( p p ) ,B ( K ) = β ( K ) , ν ( p ) e + ( p + ) e − ( p − ) f ( p ) φ µ ( p ) FIG. 2: Feynman diagram of the process e + e − → γ ∗ → φf (980). α ( K ) = Z dx − x Z dy xy ) M K − x z p − y z p − x y p − iǫ ,β ( K ) = Z dx − x Z dy x + y ) − xy − M K − x z p − y z p − x y p − iǫ . IV. PROCESS e + e − → γ ∗ → φf (980) Using the amplitude (14) we can write the amplitude for the process e + e − → γ ∗ → φf (980) (see Fig. 2) M (cid:16) e + ( p + ) e − ( p − ) → γ ∗ ( p ) → φ ( p ) f ( p ) (cid:17) == 4 παs J QEDµ e ν ( p ) X i = s,K C ( i ) π (cid:16) A ( i ) R µν (1) + B ( i ) R µν (2) (cid:17) , (16)where J QEDµ = ¯ v ( p + ) γ µ u ( p − ) is the electromagnetic current of electron and positron annihi-lation ( J µ p µ = 0), and e ν ( p ) is the polarization 4-vector of the φ − meson ( p ν e ν ( p ) = 0).The square modulus of the amplitude (16) after summation over polarization states hasthe form X pol | M | = 8 παs ( s | A | − | A − ˜ B | s − | ˜ B | s M φ ! (cid:16) E φ (cid:16) − β φ c (cid:17) − M φ (cid:17)) , (17)where s = 2( p p ) = s + M φ − M f , ˜ B = B (4 sM φ /s ), E φ = (cid:16) s + M φ − M f (cid:17) / (2 √ s ) isthe φ -meson energy in the center-of-mass system c = cos θ = cos( ~p − , ~p ) is the cosine of theemission angle of the φ -meson, and β φ = r λ (cid:16) s, M φ , M f (cid:17) / (cid:16) s + M φ − M f (cid:17) is the velocity8f the φ -meson ( λ ( x, y, z ) = x + y + z − xy − xz − yz is the well-known trianglefunction). The quantities A and B in (17) are the sums of quark and kaon contributions A = C ( q ) A ( q ) + C ( K ) A ( K ) ,B = C ( q ) B ( q ) + C ( K ) B ( K ) . The phase volume is d Γ = d p E φ d p E f (2 π ) (2 π ) δ ( p + + p − − p − p ) = r λ (cid:16) s, M φ , M f (cid:17) πs dc. (18)The differential cross section can be written in the form dσ e + e − → φf d cos θ = πα s (cid:16) D ( s ) + E ( s ) cos θ (cid:17) , (19)where D ( s ) = 4 π r λ (cid:16) s, M φ , M f (cid:17) πs α ( s | A | − | A − ˜ B | s − | ˜ B | s M φ ! (cid:16) E φ − M φ (cid:17)) , (20) E ( s ) = 4 π r λ (cid:16) s, M φ , M f (cid:17) πs α β φ E φ | A − ˜ B | s − | ˜ B | s M φ ! . (21)The total cross section then reads as: σ ( s ) = 2 α s (cid:18) D ( s ) + 13 E ( s ) (cid:19) . (22)Unlike the φ → f γ decay, where the contribution of the quark loop was negligible, in theprocess e + e − → γ ∗ → φf (980) both contributions are of the same order. In Figs. 3 and 4we show the contributions of quarks and kaons separately for the values of D ( s ) and E ( s ).In Fig. 5, the same contributions are shown for the total cross section. V. CONCLUSION
The decay φ → f (980) γ was calculated in the framework of local NJL model. Wesuppose that all mesons are the quark-antiquark states. It turns out that the lowest orderof 1 /N c expansion (where N c = 3 is the number of colors, Hartree-Fock approximation)where only quark loops are taken into account does not give satisfactory agreement withthe experimental data (see (10)). In the next order of 1 /N c expansion we have to consider9 ,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,00,040,060,080,100,120,140,160,180,200,22 D ( s ) s , GeV FIG. 3: Coefficient function D ( s ) from differential cross section (see (20)). The dotted line is thequark loop contribution, dashed line is the kaon loop contribution and the solid line is the totalvalue of D ( s ). the meson loops and they give the dominant contribution to the amplitude of the process φ → f γ , which leads to satisfactory agreement with experimental data. By the way, asimilar approximation was also used in other models for description of this process (see[3, 8, 13]).In the same approximation of the local NJL model the total probability and the differ-ential cross section of the process e + e − → φf (980) were calculated.The recent experiment [16] of production K + K − π + π − in annihilation channel at highenergy of e + e − collision show some structure of 0 . √ s ≈ .
175 GeV,which was treated as a resonance state. The value of the cross section exceeds the theoretical(non-resonant) cross section calculated in frames of ChPT which is equal 0 . nb . Our resultexceeds the experimental value by a factor 1.2 in this energy range. The difference can beassociated with the background from the channel e + e − → φπ + π − with the effective mass of π + π − outside the f meson width.The investigation of this process and the set of similar ones with production of heavy10 ,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,00,000,020,040,060,080,100,12 E ( s ) s , GeV FIG. 4: Coefficient function E ( s ) from differential cross section (see (21)). The dotted line is thequark loop contribution, dashed line is the kaon loop contribution and the solid line is the totalvalue of E ( s ). and radially excited mesons could be part of the physical program of the BABAR and theBES-III experiment. 11 ,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,00,00,20,40,60,81,01,21,41,6 ( s ) , nb s , GeV FIG. 5: Total cross section of the e + e − → γ ∗ → φf (980) process (see (22)). The dotted line is thequark loop contribution, dashed line is the kaon loop contribution and the solid line is the totalvalue of cross section. Acknowledgments
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