PHYSICAL REVIEW C 82, 068201 (2010)

Radiative decays of radially excited mesons π 0′, ρ0′, and ω′ in the Nambu–Jona-Lasinio model

E. A. Kuraev and M. K. VolkovBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia

A. B. ArbuzovBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research and Department of Higher Mathematics,

Dubna University, 141980 Dubna, Russia(Received 9 September 2010; revised manuscript received 15 November 2010; published 15 December 2010)

Radiative decays π 0(π 0 ′) → γ + γ , π 0 ′ → ρ0(ω) + γ , ρ0 ′(ω′) → π 0 + γ , and ρ0 ′(ω′) → π 0 ′ + γ areconsidered in the framework of the SU(2) × SU(2) Nambu–Jona-Lasinio (NJL) model. Radially excited mesonsare described with the help of a simple polynomial form factor. In spite of mixing of the ground and excitedmeson states in this model, the decay widths of π 0 → γ + γ and ρ0(ω) → π 0 + γ are found to be in goodagreement with experimental data, as in the standard NJL model. Our predictions for decay widths of radiallyexcited mesons can be verified in future experiments.

DOI: 10.1103/PhysRevC.82.068201 PACS number(s): 12.39.Fe, 13.20.Jf, 13.40.Hq

Introduction. In Ref. [1], a version of the Nambu–Jona-Lasinio (NJL) model with a polynomial form factor wasdeveloped. This model allows one to describe both theground and the first radially excited meson states. All thelow-energy chiral theorems are valid in it. Further applicationsof the model gave a description of the mass spectrum ofthe scalar, pseudoscalar, and vector meson nonets and of thecorresponding first radially excited states that is in satisfactoryagreement with the experimental data. The main strong decaysof these mesons were also described [2–6]. In this Brief Report,we describe radiative decays with participation of radiallyexcited mesons in the framework of the same model. Theamplitudes of processes considered here can further be usedfor predictions of production rates of radially excited mesonsat electron-positron colliders in processes like l+l− → γ ∗ →PV . In particular, experimental studies of all these processescan be performed at high-luminosity modern electron-positronaccelerators with center-of-mass energy of about several GeV[e.g., BEPC-II (Beijing), VEPP-2000 (Novosibirsk), DA�NE(Frascati)].

The mass spectra and properties of light mesons, includingtheir radially excited states, were also considered in theliterature with other approaches, such as nonrelativistic [7,8]relativized quark models [9,10], the Dyson-Schwinger andBethe-Salpeter equations [11–14], the Tamm-Dancoff method[15,16], the QCD sum rules [17,18], lattice QCD calculations[19], anti–de Sitter-space (AdS)/QCD models [20,21], etc.Theoretical studies of radiative decays involve detailed treat-ment of meson structure. This potentially leads to differencesin predictions for decay rates, so that future experiments candiscriminate among specific models.

The Lagrangian. Recall the main elements of the versionof the NJL model given in Refs. [1,2]. The SU(2) × SU(2)NJL model Lagrangian with a form factor in four-fermioninteractions can be written in the form [3]

L(q̄, q) =∫

d4xq̄(x)(i∂µγ µ − m0 + eQAµγ µ)q(x)

+∫

d4x

N∑i=1

3∑a=1

[G1

2

(jσ,i(x)jσ,i(x) + ja

π,i(x)jaπ,i(x)

)

−G2

2

(jaρ,i(x)ja

ρ,i(x) + jaa1,i

(x)jaa1,i

(x))]

,

jσ,i(x) =∫

d4x1

∫d4x2q̄(x1)Fσ,i(x; x1, x2)q(x2), (1)

jaπ,i(x) =

∫d4x1

∫d4x2q̄(x1)Fa

π,i(x; x1, x2)q(x2),

ja,µ

ρ,i (x) =∫

d4x1

∫d4x2q̄(x1)Fa,µ

ρ,i (x; x1, x2)q(x2),

ja,µ

a1,i(x) =

∫d4x1

∫d4x2q̄(x1)Fa,µ

a1,i(x; x1, x2)q(x2),

where q(q̄) is the doublet of light quarks, m0 is the currentquark mass, e is the electron charge, Q = diag( 2

3 ,− 13 ) is

the quark charge matrix, Aµ is the electromagnetic field,G1 and G2 are the four-quark interaction constants in the(pseudo)scalar and (axial) vector cases, respectively.

Here we consider only the ground (i = 1) and the first ex-cited meson states (i = 2). The form factors (in the momentumspace) are chosen as a simple polynomial type [1–3]:

Fσ,2 = fπ (k⊥2), F a

π,2 = iγ5τafπ (k⊥2

),

Fa,µ

ρ,2 = γ µτafρ(k⊥2), F

a,µ

a1,2= γ5γ

µτafρ(k⊥2),

(2)

fπ,ρ(k⊥2) = cπ,ρf (k⊥2

), k⊥ = k − kp

p2p,

f (k⊥2) = (1 − d|k⊥2|)(2 − |k⊥2|),

where k and p are the quark and meson four-momenta,respectively. In the rest frame of the external meson k⊥ ={0, �k} and k⊥2 = −�k2. The parameter cπ,ρ defines only themeson masses and can be omitted in the description ofmeson interactions. The cutoff parameter is taken to be = 1.03 GeV.

In our model, the slope parameter d is chosen from thecondition so that the excited scalar meson states do notinfluence the value of the quark condensate, i.e., do not changethe constituent quark mass (see Refs. [1,2,22]). This condition

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BRIEF REPORTS PHYSICAL REVIEW C 82, 068201 (2010)

can be written in the form

If

1 = −iNC

(2π )4

∫d4kf (k⊥2

)

m2 − k2= 0, (3)

where NC = 3 is the number of colors. This gives d ≈1.78 GeV−2. This means that the quark tadpole connectedwith excited scalar mesons vanishes. Therefore, the quarkcondensate acquires only a contribution from the quark tadpoleconnected with the ground scalar state [23].

After bosonization of the SU(2) × SU(2) chiral-symmetricfour-quark Lagrangian and renormalization of the mesonfields, we get the following form of the quark-meson inter-action (we show only the terms relevant to the present study):

Lint = q̄(k){eQγµAµ(p) + τ 3γ5[gπ1π1(p)

+ gπ2π2(p)f (k⊥2)] + 1

2γµ

[gρ1

(ρ

µ

1 (p)τ 3 + ωµ

1 (p))

+ gρ2f (k⊥2)(ρ

µ

2 (p)τ 3 + ωµ

2 (p))]}

q(k′) + · · · (4)

Here πi , ρi , and ωi are nonphysical pseudoscalar and vectormeson fields. The coupling constants have the form [1,2]

gπ1 =[

4I2

(1 − 6m2

u

M2a1

)]−1/2

, gπ2 = [4I

f 2

2

]−1/2,

(5)

gρ1 =[

2

3I2

]−1/2

, gρ2 =[

2

3I

f 2

2

]−1/2

,

where

I2 = −iNc

∫d4k

(2π )4

(2 − �k2)(m2

u − k2)2 ,

(6)

I f n

m = −iNc

∫d4k

(2π )4

(f (k⊥2))n(

m2u − k2

)m , n,m = 1, 2.

Note that in gπ1 we take into account π -a1 transitions [23],where Ma1 = 1.23 GeV is the a1 meson mass. In the constantgπ2 these transitions can be neglected; see Refs. [1,2].

The free part of the Lagrangian for the pion fields containsnondiagonal kinetic terms:

Lfreeπ = p2

2

(π2

1 + 2�ππ1π2 + π22

) − M2π1

2π2

1 − M2π2

2π2

2 .

The mass terms have a diagonal form because of condition (3),and

�π = If

2√I2I

f 2

2

, M2π1

= g2π1

[1

G1− 8I1

],

M2π2

= g2π2

[1

G1cπ

− 8If 2

1

],

where G1 = 3.47 GeV−2 is the interaction constant ofscalar and pseudoscalar quark currents in the initial NJLLagrangian (1).

The following transformation allows us to get a diagonalform of the free meson Lagrangian [2,3]:

π0 = π1 cos(α − α0) − π2 cos(α + α0),(7)

π0′ = π1 sin(α − α0) − π2 sin(α + α0),

where

sin α0 =√

1 + �π

2,

(8)

tan(2α − π ) =√

1

�2π

− 1

[M2

π1− M2

π2

M2π1

+ M2π2

].

We obtained the values α0 = 59.06◦ and α = 59.38◦ forthe angles. The free pion Lagrangian takes the standardform

Lfreeπ = p2

2

(π02 + π0′2) − M2

π

2π02 − M2

π ′

2π0′2

. (9)

For cπ = 1.36 the values Mπ ≈ 134.8 MeV and Mπ ′ ≈1308 MeV were obtained, in agreement with the experimen-tal values 134.9766 ± 0.0006 MeV and 1300 ± 100 MeV,respectively [24].

As a result, the interaction Lagrangian for physical pionfields with quarks takes the form

Lintπ = q̄(k)τ 3γ5

×{[

gπ1

sin(α + α0)

sin(2α0)+ gπ2f (k⊥2

)sin(α − α0)

sin(2α0)

]π0(p)

−[gπ1

cos(α +α0)

sin(2α0)+ gπ2f (k⊥2

)cos(α − α0)

sin(2α0)

]π0′

(p)

}× q(k′). (10)

An analogous procedure for the vector mesons leadsto [2,3]

Lintρ,ω = q̄(k)

γµ

2

{[gρ1

sin(β + β0)

sin(2β0)+ gρ2f (k⊥2

)sin(β − β0)

sin(2β0)

]

× (τ 3ρ0

µ(p) + ωµ(p))−[

gρ1

cos(β + β0)

sin(2β0)

+ gρ2f (k⊥2)cos(β − β0)

sin(2β0)

](τ 3ρ0′

µ (p) + ω′µ(p)

)}q(k′),

where the angles β0 = 61.53◦ and β = 76.78◦ are definedanalogously to Eq. (8), using

M2ρ1

= 3

8G2I2, M2

ρ2= 3

8cρG2If 2

2

. (11)

For cρ = 1.15 and G2 = 13.1 GeV−2 we get Mρ = Mω ≈783 MeV and Mρ ′ = Mω′ = 1450 MeV. The correspondingexperimental values are Mρ = 775.49 ± 0.34 MeV, Mω =782.65 ± 0.12 MeV, Mρ ′ = 1465 ± 25 MeV, and Mω′ =1400–1450 MeV [24].

Results for radiative decays. Let us consider now thestandard two-photon decay described by the triangle quarkdiagram of the anomaly type. The decay amplitude has the

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BRIEF REPORTS PHYSICAL REVIEW C 82, 068201 (2010)

TABLE I. Ground-state meson radiative decay widths.

Decay π 0 → γ γ ρ0 → π 0γ ω → π 0γ

Theory 7.7 eV 77 keV 710 keVExperiment 7.5 ± 1.1 eV 88 ± 12 keV 700 ± 30 keV

form

Aπ0→γ γµν = 8muεµνγσ q

γ

1 qσ2 e2

(Q2

u − Q2d

) (−i)NC

(2π )4

×∫

d4k

{gπ1

sin(α +α0)

sin(2α0)+ gπ2f (k⊥2

)sin(α−α0)

sin(2α0)

}

× 1(k2 − m2

u + i0)(

(k − q1)2 − m2u + i0

)× 1(

(k + q2)2 − m2u + i0

) ,

where Qu,d are the u and d quark charges and q1,2 are thephoton momenta. The amplitude contains two types of one-loop integrals: with and without form factor in the quark-pion vertex. The expression for the π0′ → γ γ amplitude onlydiffers from the above expression by the coupling constant ofpion and quarks [see Eq. (10)] and by the mass of the decayingparticle. In the calculation we take into account only the realpart of the loop integrals. This ansatz corresponds to the naı̈veconfinement definition [25], which was used in some our recentworks [26–28].

Consider now two-particle decay modes of pseudoscalarand vector mesons with a single photon. The amplitude ofρ0 → π0γ takes the form

Aρ0→π0γµν = 4muεµνγσ q

γ

1 qσ2 2e(Qu + Qd )

(−i)NC

(2π )4

×∫

d4k(k2 − m2

u + i0)(

(k − q1)2 − m2u + i0

)× 1(

(k + q2)2 − m2u + i0

)×

{gρ1

2

sin(β + β0)

sin(2β0)+ gρ2

2f (k⊥2

)sin(β − β0)

sin(2β0)

}

×(

gπ1

sin(α + α0)

sin(2α0)+ gπ2f (k⊥2

)sin(α − α0)

sin(2α0)

),

where q1 and q2 are the vector meson and photon momenta,respectively.

First, we recalculate the width of radiative decays of theground meson states; see Table I. The results are in satisfactoryagreement with the experimental data [24]. Note that thesimilar strong decays of radially excited mesons ρ ′ → ωπ

and ω′ → ρπ , considered within the same nonlocal model,were found previously in Ref. [3] to be also in reasonable

TABLE II. Radiative decay widths of π 0 ′ and ρ0 ′.

Decay π 0 ′ → γ γ π 0 ′ → ρ0γ ρ0 ′ → π 0γ ρ0 ′ → π 0 ′γ

Theory 3.2 keV 1.8 keV 450 keV 24 keV

TABLE III. Decay widths of radiative processes with ω and ω′

mesons.

Decay π 0 ′ → ωγ ω′ → π 0γ ω′ → π 0 ′γ

Theory 17 keV 3.7 MeV 93 keV

agreement with observations [24,29]:

�theor.ρ ′→ωπ ≈ 75 MeV, �

exper.ρ ′→ωπ = 65.1 ± 12.6 MeV,

(12)�theor.

ω′→ρπ ≈ 225 MeV, �exper.ω′→ρπ = 174 ± 60 MeV.

We recall that the decay widths of the ground states wasobtained within the local NJL model in 1986 [23], in goodagreement with the experimental data. In the considerednonlocal version of the NJL model the radially excited mesonstates are mixed with the ground states. However, this mixingdoes not lead to the distortion of the description of the groundmeson state interactions with each other, obtained in the localmodel.

Tables II and III contain our results of theoretical cal-culations of radiative decay widths of the radially excitedstates of π0, ρ0, and ω mesons calculated in the frameworkof the NJL model. For the calculations of the phase spacewe used the present experimental values for the excitedmeson masses. The widths of these decay channels are notyet measured experimentally. So we made predictions thatcould be relevant to future experiments. Note also that thedecay widths ω′(ρ0′

) → π0′γ are very sensitive to the meson

masses, which have, at the present time, large experimentaluncertainties [24].

Table IV shows a considerable discrepancy between ourresults and the ones obtained earlier within a nonrelativisticquark model [8]. However, we have several arguments infavor of our approach. First, within our model we obtaineda theoretical description of the ground meson state decays ingood agreement with experimental data. On the other hand, thestrong decay widths of ω′ → ρ0π0 and ρ0′ → ωπ0 computedwithin the same model in Ref. [3] also show a satisfactoryagreement with the experiment; see Eq. (12). It is worth notingthat all these decays in our model have the same mechanism:they are described by triangular quark loop diagrams of theanomaly type. So we hope that the predictions for radiallyexcited meson decay widths will also be in a reasonableagreement with the future experimental data. Let us emphasizethat in the present study to describe the radiative decays wedid not use any additional parameters with respect to earlierworks [2,3]. Moreover, the 3P0 model applied in Ref. [8] has

TABLE IV. Comparison with results of Ref. [8].

Decay ω→π 0γ ρ0 ′→π 0γ ρ0 ′→π 0 ′γ ω′→π 0γ ω′→π 0 ′

γ

This work 710 keV 450 MeV 24 keV 3.7 MeV 93 keVRef. [8] 520 keV 61 keV 5.9 keV 510 keV 29 keV

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BRIEF REPORTS PHYSICAL REVIEW C 82, 068201 (2010)

serious problems in the description of excited meson states, asdiscussed in that paper.

With our model, using the interaction amplitudes ob-tained here, one can compute the production probabilitiesof the ground and excited meson states at lepton collid-ers. There, processes like e+e→γ ∗ → PV can be studied,where P and V are the ground and excited pseudoscalarand vector meson states, respectively. In these processes,it is necessary to take into account the transitions γ ∗ →

V (V ′). The Primakoff effect and the two-photon productionmechanism of radially excited mesons can be considered aswell. We plan to perform similar calculations to describeradiative decay channels of radially excited states of η,η′, and φ mesons in the framework of the U(3) × U(3)NJL model.

Acknowledgments. The authors are grateful toA. Akhmedov and S. Gerasimov for helpful discussions. Thework was supported by RFBR Grant No. 10-02-01295-a.

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