Radiative decays of scalar mesons \sigma(600), f_{0}(980) and a_{0}(980) in the local Nambu-Jona-Lasinio model

# Radiative decays of scalar mesons σ(600), f0(980) and a0(980) in the local Nambu-Jona-Lasinio model

M. K. Volkov Joint Institute for Nuclear Research, Dubna, Russia    E. A. Kuraev Joint Institute for Nuclear Research, Dubna, Russia    Yu. M. Bystritskiy Joint Institute for Nuclear Research, Dubna, Russia
###### Abstract

The two-photon decay widths of scalar mesons , and as well and are calculated in the framework of the local Nambu-Jona-Lasinio model. The contributions of the quark loops (Hartree-Fock approximation) and the meson loops (next -approximation where is the number of colors) are taken into account. These contributions, as we show, are the values of the same order of magnitude. For the decay the -loop contribution turns out to play the dominant role. The results for two-gamma decays are in satisfactory agreement with modern experimental data. The predictions for and widths are given.

PACS: 12.39.Fe, 13.20.Jf, 13.40.Hq

## I Introduction

In recent paper Bystritskiy:2007wq (), the radiative decay width within the local Nambu-Jona-Lasinio (NJL) model Volkov:1986zb (); Volkov:1982zx (); Ebert:1982pk (); Volkov:1984kq (); Ebert:1985kz (); Ebert:1994mf (); Volkov:2006vq (); Vogl:1991qt (); Klevansky:1992qe () has been calculated. In this work, same as in Bystritskiy:2007wq (), we took into account not only the quark loop contributions (Hartree-Fock approximation) but also the meson loop contributions (next -approximation where is the number of colors). We should note that meson loops give the contribution of the same order as the Hartree-Fock approximation due to fractional charge of quarks and integer charge of mesons. The gauge invariance of the amplitudes leads to absence of ultraviolet divergences in the relevant loop integrals. Thus, the explicit dependence of these amplitudes of the external momenta was obtained Ebert:1996pc ().

In this paper, we will consider the two-photon decays of the scalar mesons , and as well as radiative decays and .

In the case of the quark loop we consider only a real part of the relevant loop integral. This prescription permits us to take into account the condition of the ”naive” quark confinement. Some theoretical arguments supporting this procedure can be found in Pervushin:1985yi (). As for the meson loops, both the real and the imaginary parts were taken into account.

The structure of our paper is the following. In Section II, the NJL quark-meson Lagrangians, corresponding parameters and the coupling constants of our model are defined. In Section III, the methods of quark and meson loop calculation are described; the contributions of quark and meson loops to the amplitudes and the widths of two-photon decays of the scalar meson are presented.

In Section IV, we consider the and decay widths.

In Section V, we discuss the results obtained.

## Ii Lagrangian of the NJL model

The initial four-quark Lagrangian of the local Nambu-Jona-Lasinio model has the form Volkov:1986zb (); Ebert:1985kz (); Vogl:1991qt (); Klevansky:1992qe ():

 L = ¯q[i^∂+eQ^A−M0]q+ (1) + G2((¯qq)2+(¯q(iλaγ5)q)2)−GV2((¯qγμλaq)2+(¯q(γμγ5λa)q)2),

where , and , , are the quark fields, is the matrix of quarks current masses, is the quark electric charge matrix, is the elementary electric charge (), , are the well-known Gell-Mann matrices and , and are effective coupling constants of four-fermion interactions.

The procedure of bosonization and the renormalization of meson fields leads to the effective quark-meson Lagrangian. The part of this Lagrangian which we will use is the following Volkov:1986zb (); Ebert:1994mf (); Volkov:2006vq ():

 Leff = ¯q[i^∂+eQ^A−M]q+¯q[gσuλuσu+gσsλsσs+gσuλ3a0+iγ5gπ(λπ+π++λπ−π−)+ (2) +iγ5gK(λK+K++λK−K−)+gρ2(λ3^ρ0+λu^ω)]q,

where is the constituent quark mass matrix and , , , , , . Taking into account the six-quark interaction of t’Hooft leads to mixing of and states Volkov:2006vq (); Vogl:1991qt (); Klevansky:1992qe (); Ebert:1994mf (). Scalar isoscalar mesons , are the mixed states

 f0 = σusinα+σscosα, σ = σucosα−σssinα, (3)

with the mixing angle Volkov:2006vq (); Vogl:1991qt ().

The coupling constants from the Lagrangian (2) are defined in the following way Volkov:1986zb ():

 gσu = (4IΛ(mu,mu))−1/2=2.43, gσs = (4IΛ(ms,ms))−1/2=2.99, gπ = muFπ=2.84, gK = mu+ms2FK=3.01, gρ = √6gσu=5.95,

where we use the Goldberger-Treiman relation for and constants, and , and is the logarithmically divergent integral which has the form:

 I(m,m) = Nc(2π)4∫d4kθ(Λ2−k2)(k2+m2)2= = Nc(4π)2(ln(Λ2m2+1)−Λ2Λ2+m2),Nc=3.

This integral is written in the Euclidean space. The cut-off parameter is taken from Volkov:1986zb (); Bystritskiy:2007wq ().

## Iii Two-gamma decays of σ(600), f0(980) and a0(980)

The amplitudes of the decay can be expressed in terms of the quark and meson loop integrals.

The quark loop contribution to the amplitude is given by two triangle type Feynman diagrams (see Fig. 1, a):

 Tqμν = −α4π∫d4kiπ2Sp[γν(^k+mq)γμ(^k+^q1+mq)(^k−^q2+mq)](k2−m2q)((k+q1)2−m2q)((k−q2)2−m2q)+ (4) + ((q1,μ)↔(q2,ν)).

Applying the Feynman procedure of joining of the denominators

 1(k2−m2q)((k+q1)2−m2q)((k−q2)2−m2q)= (5)

where , , we obtain for :

 Tqμν = απ(gμν(q1q2)−q1νq2μ)Tq, (6) Tq = 2mq1∫0dx1∫0ydy1−4y2x¯xm2q−M2Sy2x¯x. (7)

For meson loops an additional Feynman diagram with two photon-two meson vertex contributes as well (see Fig. 2, b, c). To restore the general gauge invariant form of the amplitude, we can nevertheless consider only two triangle type Feynman diagrams:

 ΔTMμν = (8) + ((q1,μ)↔(q2,ν)).

Extracting the term and adding the relevant term we obtain:

 TMμν = απ(gμν(q1q2)−q1νq2μ)TM, (9)

with

 TM = 21∫0dx1∫0ydyy2x¯xM2−M2Sy2x¯x. (10)

Standard evaluation of these integrals leads to

 Tq = 1mqF(zqS), (11) TM = zMS4M2Φ(zMS), (12)

where , ,

 F(z) = Re[1+(1−z)Φ(z)], Φ(z) = zϕ(z)−1, ϕ(z) = ⎧⎪ ⎪⎨⎪ ⎪⎩14[π2−ln21+√1−z1−√1−z]+iπ2ln1+√1−z1−√1−z,z<1,(arctan1√z−1)2,z>1. (13)

We remind that for the quark loop contribution the imaginary part of the function must be omitted (see Pervushin:1985yi ()), and for the meson loop contribution both the real and possible imaginary parts are relevant.

Similar expressions were obtained in Ebert:1996pc (), where the imaginary part of the quark loop contribution was taken into account.

The vertices of the quark-meson and quark-photon interactions were given in (2). The vertices of the meson-meson interaction in the framework of the NJL model have the form (for details see Volkov:1986zb (); Ebert:1994mf (); Bystritskiy:2007wq ()):

 Vσsπ+π− = Va0π+π−=0, VσuK+K− = Va0K+K−=−2(2mu−ms)g2Kgσu, (14) VσsK+K− = 2√2(2ms−mu)g2Kgσs, Vσuπ+π− = −4mug2πgσu.

The general structure of the two-photon scalar meson decay amplitudes has the form

 TSγγ=−αgσuπmu(gμν(q1q2)−qν1qμ2)aSγγ. (15)

The expression for the width has the form:

 ΓSγγ=M3S64πα2g2σuπ2m2u|aSγγ|2. (16)

### iii.1 The decay a0→2γ

The amplitude of contains the contribution of , quarks and the -meson intermediate states. The color-charge factor associated with , quarks is . Thus,

 au,da0γγ = F(zua0). (17)

Taking the -meson loop contribution we obtain:

 aa0γγ = au,da0γγ+aKa0γγ= (18) = F(zua0)−mugσu2(2mu−ms)g2K4gσuM2KzKa0Φ(zKa0)= = 0.482−0.114=0.367.

The corresponding width is

 Γa0(980)→γγ = (2.25 keV)∣∣aa0γγ∣∣2=0.29 keV.

This value is in satisfactory agreement with the experimental values (see Table 1).

### iii.2 The decay f0(980)→γγ

In the case of the decay we also have the contribution of , and quarks and the -meson and pion intermediate states. We should recall that meson consists of two components: and (see (3)). The color-charge factor associated with , quarks is for -component of and for -component of . Taking the -meson and the -meson loop contribution we obtain

 af0→γγ = 53F(zuf0)sinα−√23F(zsf0)(gσsmugσums)cosα+ (19) + (−g2Kg2σumu4M2K2(2mu−ms)sinα+ +g2Kgσugσsmu4M2K2√2(2ms−mu)cosα)zKf0Φ(zKf0)− − sinαm2uM2πg2πg2σuzπf0Φ(zπf0)= = 0.157−0.417−0.022+0.589+0.082−0.038i= = 0.385−0.038i.

For the width we have

 Γf0(980)→γγ = (2.25 keV)∣∣af0γγ∣∣2=0.33 keV.

This value is also in satisfactory agreement with the experimental values (see Table 1).

### iii.3 The decay σ→γγ

In the case of the decay we have

 aσ→γγ = 53F(zuσ)cosα+√23F(zsσ)(gσsmugσums)sinα (20) − (g2Kg2σumu4M2K2(2mu−ms)cosα+ +g2Kgσugσsmu4M2K2√2(2ms−mu)sinα)zKσΦ(zKσ)− − cosαm2uM2πg2πg2σuzπσΦ(zπσ)= = 1.89+0.057−0.041−0.043+0.92−0.98i= = 2.78−0.98i.

Let us notice that unlike decay, where contributions of and quarks have opposite signs, here they have the same sign.

The corresponding width is

 Γσ(600)→γγ = (0.51 keV)∣∣aσγγ∣∣2=4.3 keV. (21)

The experimental value of the mass and the width of the meson is not well established. We present the width of for two other masses: and . They are

 Γσ(450)→γγ=2.18 keV, Γσ(550)→γγ=3.53 keV.

The comparison of our results with the experimental data and some other model predictions is given in Table 1.

## Iv The decays a0→ω(ρ)γ, f0→ω(ρ)γ

In this section, we will consider the following decays 111 Let us notice that in Radzhabov:2007wk () the decays were considered. The meson intermediate state was taken into account. :

 a0(p) → ω(q)+γ(k1), a0(p) → ρ(q)+γ(k1), f0(p) → ω(q)+γ(k1), f0(p) → ρ(q)+γ(k1),
 p2=M2S,q2=M2V,k21=0, (22)

where is the mass of decaying scalar meson Amsler:2008zzb () and is the mass of vector meson. Now the matrix element in the general case has the form:

 Mi = (24) eγ=e(k1),eV=e(q),

where , and contains all vertex constants and dynamic information of the processes. The radiative decay width then has the form:

 Γi=α(M2S−M2V)332M3Sg2ρ|Ai|2. (25)

To calculate quark and meson loop contributions to coefficients , we need quark-meson (see (2)) and meson-meson vertices (see (14)), and vertices of interaction of vector mesons with the pseudoscalar ones Volkov:1986zb (); Volkov:1993jw (); Ebert:1994mf ():

 gωμK+K− = gρμK+K−=gρ2(p+−p−)μ, gρμπ+π− = gρ(p+−p−)μ.

### iv.1 The decays a0→ωγ, a0→ργ

Let us consider the decay of the isoscalar meson with -meson production. The amplitude of this decay will contain contributions from the quark loop and -meson loop. The -meson loop is forbidden since the vertex is absent.

Quark contribution consists of and quark loops (see Fig. 2, a) (since does not contain quarks):

 M(u,d)a0→ωγ = e gρ2C(u,d)a0→ωγ∫d4kiπ2Sp[(^q+^k+mu)(^k−^k1+mu)^eγ(^k+mu)^eω]((q+k)2−m2u)(k2−m2u)((k−k1)2−m2u), (26)

where , where factor is the color factor.

Standard Feynman procedure of denominators joining and loop momenta integration leads to:

 M(u,d)a0→ωγ = e gρ2C(u,d)a0→ωγRe(Iu)(qνk1μ−gμν(qk1))eνγeμω, (27)

where

 Iu = 4mu1∫0dx1∫0dyy(1−4y2x(1−x))m2u−y(1−y)(1−x)q2−x(1−x)y2p2+iϵ. (28)

Let us consider now kaon loop contributions. The kaon contribution consists of diagrams of two types (see Fig. 2 b, c). This contribution can be written in a form similar to (24):

 M(K)a0→ωγ = e gρ2C(K)a0→ωγIK(qνk1μ−gμν(qk1))eνγeμω, (29)

where and

 IK = 1∫0dx1∫0dy4y2x(1−x)yM2K−y(1−y)(1−x)q2−x(1−x)y2p2+iϵ. (30)

More details of this type of diagram calculation can be found in Bystritskiy:2007wq ().

The amplitude of the process then has the form

 Ma0→ωγ=e Aa0→ωγ(qνk1μ−gμν(qk1))eνγeμω, (31)

where

 Aa0→ωγ=3gσuRe(Iu)+ga0K+K−IK=−1.78374+0.159415=−1.62433. (32)

The decay width is:

 Γa0→ωγ = 115 keV.

The decay can be considered in complete analogy with the decay . Let us note that the quark contribution to is three times smaller than the quark contribution to the decay. However, the kaon loop contributions are the same. As a result, the amplitude of the decay has the form:

 Ma0→ργ = e gρ2Aa0→ργ(qνk1μ−gμν(qk1))eνγeμρ, (33) Aa0→ργ = gσuRe(Iu)+ga0K+K−IK=−0.598209+0.156921.

The decay width is

 Γa0→ργ = 8.5 keV.

Let us note that in both these processes the main contribution comes from quark loops.

### iv.2 The decays f0→ωγ and f0→ργ

The total amplitude for the decay has the form:

 Af0→ωγ = sinα Au,dσu→ωγ+sinα AKσu→ωγ+cosα AKσs→ωγ=−0.115+0.0195+(−1.068)=−1.1635. (34)

The decay width is then:

 Γf0→ωγ = 60 keV.

Let us consider the decay . The quark loop contribution to component here exceeds the relevant component of the decay:

 Af0→ργ = sinα A(u,d)σu→ργ+sinα Aπσu→ργ+sinα AKσu→ργ+cosα AKσs→ργ= (35) = −0.353+(0.0436−0.00542i)+0.027+(−1.05)=−1.3324−0.00542i.

The decay width is then:

 Γf0→ργ = 82 keV.

Note that in both the cases (the decays of to or ) the main contribution comes from the kaon loops related with the component.

Unfortunately, at present we do not have any experimental data for this decays.

## V Conclusion

The calculations of the radiative decays in the NJL model show an important role of both the quark and meson loops. Moreover, for the meson decay the kaon loop turns out to provide the dominant contribution. This can be understood if one takes into account that in the decays the fractional quark charge gives effectively factor into quark loop contribution. It is worth noticing that the situation here is similar to the one that takes place in the case of decay Bystritskiy:2007wq () and and . This fact permits one to understand the success of such models as the model of a kaon molecule Weinstein:1990gu (); Branz:2008ha () as well as the four-quark model Achasov:1987ts (); Achasov:2008ut ().

The description of radiative scalar isoscalar meson decays was also considered in the linear -model Kleefeld:2001ds (); vanBeveren:2002mc (); vanBeveren:2008st (). However, in these papers the scalar meson and radiative decays were not considered.

The NJL model used here allows us to take into account both the quark-antiquark state which manifests itself in the form of quark loops, and the hidden four-quark state which shows up as meson loops. Let us emphasize that in the framework of the standard NJL model we can describe the radiative decays without any additional parameters.

The results obtain in this paper (see Table 1) are in agreement with the existing experimental data. Unfortunately, the data for the and decays are absent up to now and our results for these decays can be considered as a prediction.

###### Acknowledgements.
The authors wish to thank Prof. N. N. Achasov, Prof. S. B. Gerasimov and Prof. V. N. Pervushin for fruitful discussions. We also acknowledge the support of INTAS grant no. 05-1000008-8528.

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