Radiative decays of pseudoscalar P and vector V mesons and the process e+e−→PV

Nuclear Physics B (Proc. Suppl.) 181–182 (2008) 210–214 www.elsevierphysics.com

Radiative decays of pseudoscalar P and vector V mesons and the process e+e− → P V Yu. M. Bystritskiya ∗ , E. A. Kuraeva† , M. K. Volkova‡ a

Joint Institute for Nuclear Research, Dubna, Russia

Radiative decays of pseudoscalar and vector mesons are calculated in frames of chiral Nambu-Jona-Lasinio (NJL) model. We use triangle quark loops of anomalous type. During the evaluation of this loop integrals we used two methods. In first one we neglect the dependence of external momenta. In that case we reproduce the Wess-Zumino-Witten terms of effective chiral meson lagrangian. In the second method we take into account the momenta dependence of loop integrals omitting their imaginary part. That lets us to take into account quark confinement. The application of both methods are in qualitative agreement with each other and with experimental data. Second method allows us to describe the electron-positron annihilation with production pseudoscalar and vector mesons in center of mass energy range from 1 to 5 GeV. The comparison with the recent experimental data are presented.

1. Introduction In local NJL model [1], [2], [3] mesons interactions are described in terms of quark loops. If one neglect in corresponding integrals the dependence of external momenta the result will not violate chiral symmetry. In such a way we can reproduce effective chiral lagrangian corresponding to U (3) × U (3) symmetry [4], [2], [5]. In this lagrangian strong interaction vertices are expressed in terms of logarithmically divergent part of corresponding loop amplitudes. Radiative interaction of mesons are described in terms of quark Feynman diagrams of anomalous type which does not contain ultraviolet divergencies. To describe the radiative decays we use this part which is connected with anomalous quark loops. A whole set of problems, however require the knowledge of external momenta dependence of the amplitudes. In this way we encounter with the serious problem of providing of quark confinement condition. To solve this problem one usually uses nonlocal version of NJL model. In this version to describe the interaction of mesons with quarks it is necessary to introduce relevant ∗ [email protected]

† [email protected] ‡ [email protected]

0920-5632/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2008.09.038

formfactors. The choice of formfactor leads to the functional unambiguity. Among the different ways to introduce this formfactors we should mention the QCD approach [6], another form of formfactor arises from instanton model [7]. It is necessary to mention the models suggested by G. Efimov [8] and Yu. Simonov [9], Roberts and so on. It is necessary to note that different form of formfactors leads to different behavior of amplitudes in physical region. It becomes essential at large values of external momenta. It is the reason why we apply rather simple and rough method which consists in exact calculation of amplitudes and neglecting its imaginary parts to avoid the production of free quarks. As a result we obtain rather satisfactory description of radiative decay widths of vector and pseudoscalar mesons. This fact allows us to hope that this approach can be applied to description of processes ee → P V which can be measured in series of existing and planned experiments with colliding electron-positron beams. For the problems of the last type for center of mass energy range 1 − 3 GeV we hope to obtain only qualitative results. In our approach we do not introduce any quark-meson formfactor. When describing the decays of light mesons we

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ignore the dependence of corresponding loop integrals of external momenta – Approximation I. The problem of confinement in that approximation is automatically solved. In the case of heavy mesons we use exact external momentum dependence of relevant loop amplitudes and neglecting possible imaginary part – Approximation II. We use only Approximation II for description of processes at electron-positron colliders.

γ(p2 ) k + p2 k V (p1 ) k − p3 P (p3 )

2. Radiative decays of vector and pseudoscalar mesons For the description of interaction of mesons with quarks we use the NJL model lagrangian [2,3]:  Lint = q¯ eQAˆ + (iγ5 ) (gu λu ηu + gs λs ηs ) +  gρ  λ3 ρˆ0 + λu ω + ˆ + λs φˆ q, (1) 2   ¯ s¯ with u, d, s are the where q¯ = u ¯, d, quark fields, Q = diag (2/3, −1/3, the √ −1/3) is √ 2λ0 + λ8 / 3, quark charge matrix, λu = √  √  λs = −λ0 + 2λ8 / 3, where λi are Gell Mann matrices and λ0 = 2/3 diag (1, 1, 1). gu = mu /fπ , gs = ms /fs are the mesonquark coupling constants which are evaluated via Goldberger-Treiman relation (mu = 263 MeV, ms = 407 MeV are quark masses [10], and fπ = 92.4 MeV is the pion decay constant and fs = 1.3fπ ). gρ = 5.94 is the ρ → 2π coupling constant. Physical states of η and η  mesons are obtained after taking into account of singlet-octet mixing of ηu and ηs with angle θ = 51.3o [11,2]: η η

= =

−ηu sin θ + ηs cos θ, ηu cos θ + ηs sin θ.

(2)

Figure 1. The radiative decay amplitude (3).

×

(e1 e2 p1 p2 ),

(3)

where (abcd) ≡ εαβγδ aα bβ cγ dδ , gV = gρ /2, gP = gu in case of light quarks in the loop and gP = gs if strange quark is in the loop. CP V is flavour-color multiplier corresponding to quarkmeson interaction, Cηω = 2 sin θ, Cηρ = 6 sin θ, Cη ω = 2 cos θ, Cη ρ = 6 cos θ, Cηφ = 4 cos θ, Cη φ = 4 sin θ. Mq is the loop quark mass and 

1 dk 2 2 2 = J(p1 , p2 , p3 ) = Re iπ 2 (0) (1) (2)  1 1−x 1 , (4) = Re dx dy D 0 0 where z = 1 − x − y and (0) =

Mq2 − k 2 − i0,

(1) =

Mq2 − (k + p2 )2 − i0,

(2) =

Mq2 − (k − p3 )2 − i0,

D

=

Mq2 − xyp21 − yzp22 − xzp23 − i0.

We will consider the following processes below: ρ(ω) → ηγ, η  → ρ(ω)γ, φ → η(η  )γ. Vector meson decay

Neglecting the external momentum dependence we obtain Approximation I:

V (p1 ) → γ(p2 ) + P (p3 ),

J(p21 , p22 , p23 ) =

is described by amplitude of one loop with quark (see Fig. 1): MV →P γ

=

iegP gV CP V (2π)

2

Mq J(p21 , 0, p23 ) ×

1 . 2Mq2

(5)

In this approximation a wide set of decays of light mesons was described in [2] and the results were found to be in good agreement with the experimental data. Matrix element square can be writ-

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ten in the form: 2

|MV →P γ | =

2 2 2 e2 gP gV CP V (2π)4

×       2 1 2 2 2 . (6) M − M × Mq J(MV2 , 0, MP2 ) V P 2

V (p1 )

e− (p− )

Then if the phase volume of final state is: dΦP γ =

d3 p2 d3 p3 1 MV2 − MP2 = . 2E2 2E3 8π MV2

then the decay width obtains the form:  2 3 2 MV −MP ΓV →P γ = 13 27απ4 × MV

2 × gP gV CP V Mq J(MV2 , 0, MP2 ) .

e+ (p+ )

In Table 1 we present the theoretical result for both methods – Approximation I (5) and Approximation II (4) – and compare it with the relevant experimental data. In particular we note that the ratio Rth. = Γ(φ → η  γ)/Γ(φ → ηγ) = 2.49 × 10−3 is in a qualitative agreement with the result of the KLOE collaboration Rexp. = (4.70 ± 0.47(stat.) ± 0.31(syst.)) × 10−3 [13]. 3. Associative production of pseudoscalar and vector mesons in electron-positron annihilation Matrix element of the processes of associative production of pseudoscalar and vector mesons (10)

where s = (p+ + p− )2 , p2± = m2 , p21 = MV2 , p23 = MP2 , in the lowest order of QED coupling constant α have a form (see Fig. 2): 4πα em Aμ J J , (11) s μ with QED lepton current Jμem = v¯(p+ )γμ u(p− ) and the anomalous current MPV = i

gP gV CP V × (2π)2 2 × (e1 μp1 p2 ) Mq J(p1 , s, p23 ),

P (p3 )

(8)

Relevant expression for radiative pseudoscalar meson decays P → V γ have a form:  2 3 MP −MV2 × ΓP →V γ = 27απ4 MP

2 (9) × gP gV CP V Mq J(MV2 , 0, MP2 ) .

e+ (p+ ) + e− (p− ) → V (p1 ) + P (p3 ),

q

(7)

J Aμ =

(12)

Figure 2. The processes at electron-positron colliders.

where q = p+ + p− = p2 = p1 + p3 and e1 is the polarization vector of the vector meson (i.e. (e1 p1 ) = 0). The cross section build by general rules 1  dσ = |M P V |2 dΦP V , (13) 8s with the phase volume of the final state having form: dΦP V =

d3 p1 d3 p3 1 × 2E1 2E3 (2π)2

×δ 4 (p+ + p− − p1 − p3 ).

(14)

As we are concerned of total cross section only we can use the property of anomalous current gauge invariance and thus rewrite the final state phase integral as 

JμA (JνA )∗ dΦP V = 

 A 2 1 qμ qν J  dΦP V . gμν − 2 = (15) μ 3 q Second term in the braces does not give the contribution due to gauge invariance of lepton current Jμem . First term contribution is proportional to  (e1 μp1 p2 ) (e∗1 μp1 p2 ) =  1  (16) = − λ s, MP2 , MV2 , 2

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Table 1 The table of radiative decays. The values of widths are in KeV. Approximation I – neglect external momenta dependence. Approximation II – real part of exact loop integrals. Decay Experiment Approximation I (5) Approximation II (4) ρ → ηγ 39.47 65 33.72 ω → ηγ 4.07 7.83 4.16 η  → ργ 59.68 76.18 41.09 η  → ωγ 6.15 7.59 4.04 φ → ηγ 55.59 71.01 117.9 φ → η γ 0.265 0.497 0.294 The experimental values are taken from ref. [12]. where λ(a, b, c) = a2 + b2 + c2 − 2ab − 2ac − 2bc is triangle function. Thus the value  Awell-known  Jμ 2 in (15) does not depend on vectors p1 and p2 itself but only of their squares p21 = MV2 , p23 = MP2 . This allows us to calculate the integral over final state phase volume which neglecting the masses of leptons can be written as:

3 d p1 d3 p3 4 δ (q − p1 − p3 ) = 2E1 2E3  1  π λ 2 s, MP2 , MV2 . (17) = 2 s Then the total cross section obtains the form:  α2 3  σ(s) = λ 2 s, MP2 , MV2 × 3 3 96π s   2 × gV gP CP V Mq J MP2 , MV2 , s  . (18) In Fig. 3 we put the total cross section (18) as a function of s for different annihilation channels. The differential cross section can be written in that way:   3 1 + cos2 θ dσ , (19) = σ(s) dΩ2 16π where θ is the center of mass angle between the 3-momenta of the initial electron p− and final vector particle with momentum p1 . 4. Conclusion In this paper we had investigated the radiative decays of vector and pseudoscalar decays which goes through the quark loops of the anomalous type.

Figure 3. Total cross sections (18) as a function of s for different annihilation channels.

Let us note that in paper [14] we considered process φ → f0 (980)γ within the same framework of NJL model. But there quark loop contribution was small enough and the main contribution arose from terms of next order of 1/Nc expansion (where Nc is the number of colors) – meson loops. In this paper we have another situation: meson loops are absent totaly and only quark loops of anomalous type give a contribution to the process amplitude. Both approaches (Approximation I and Approximation II) were considered. We show that the application of NJL approach leads to rather

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satisfactory agreement with the modern experimental data for decays. That allows us to use the Approximation II to calculate the cross sections of associative vector and pseudoscalar mesons production in electron-positron annihilation channel at lowest order by electromagnetic constant. In the last case the heavy virtual photon converts to the set of pseudoscalar and vector mesons. Concerning the energy range of modern C-τ factories up to 5 GeV we give some predictions for the set of experiments at e+ e− -colliders. The cross section are large enough to be measured at experiments with the sufficient precision at modern experimental facilities. For comparison with the experiment the precision of our results are worth to be mentioned. We should notice that for NJL model results precision is of order of 20-30 %. One of our important theoretical assumption is the absence of imaginary part of relevant amplitudes. The mechanism of elimination of imaginary part is tightly connected with the confinement nature and is not considered here. We do the elimination ”by hands”. We note however that if the imaginary part is taken into account the considerable disagreement with the experimental data will occur in decay case. For instance if imaginary part of amplitude is taken into account the decay width of φ → η  γ is Γth. φ→η  γ = 0.824 KeV while the experiment gives Γexp. φ→η  γ = 0.265 KeV (see Table 1). In conclusion we note that the analysis of same decays, the η−η  mixing angles and possible gluon admixture in η, η  states (neglecting on momentum dependence) was performed also in a number of papers [15], [16], [17] [18], [19]. We should note that our results for decays in Approximation I are in agreement with the ones obtained in [15] (compare Table 1 and Table 2 in [15]). We do not consider the possible gluon content of η and η  mesons, which is of serious interest now [16], [17]. We also use one-angle η − η  mixing scheme (see (2)). For analysis of two-angle mixing scheme or possible momentum dependence of this important model parameter see [18], [19]. Here we did not consider the production of strange and charmed particles which can be the

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