Radiative decays of light-quark mesons to a pion revisited in the covariant oscillator quark model
Kenji Yamada Tomohito Maeda
hep-ph




§ INTRODUCTION


In dealing with radiative decays some typical approximations, such as long wavelength
 and nonrelativistic approximations, which are not always justified, are usually used.
 For instance, the recoil effect of final-state mesons is
 neglected, though the momenta of those mesons are often comparable to their masses,
 especially in light-quark meson sectors.
 There is also ambiguity associated with a choice of the relativistic or nonrelativistic
 phase space in the nonrelativistic quark model. When the decays become relativistic,
 no one knows the rigorous way of deriving the relationship between nonrelativistic
 decay amplitudes and relativistic decay widths.

In the covariant oscillator quark model (COQM), on the other hand, hadrons themselves
 and their interactions are formulated in a manifestly covariant way. Since both the
 interaction operator and wave functions are relativistic, the gauge invariance is
 preserved and the conserved electromagnetic currents of hadrons are given explicitly
 in terms of the constituent quark variables <cit.>.




§ THE COVARIANT OSCILLATOR QUARK MODEL


Basic framework of the COQM.
 In the COQM quark-antiquark meson systems are described by the bilocal field

    Ψ(x_1,x_2)_α^β = Ψ(X,x)_α^β,

where x_1^μ (x_2^μ) is the space-time coordinate,
 α (β) the Dirac spinor index of the constituent quark (antiquark),
 and the center-of-mass and relative coordinates are defined, respectively, by

    X^μ=m_1x_1^μ+m_2x_2^μ/m_1+m_2,  
    	x^μ=x_1^μ-x_2^μ

with the constituent quark (antiquark) mass m_1 (m_2).
The bilocal meson field is required to satisfy the Klein-Gordon-type equation

    (-∂^2/∂ X_μ∂ X^μ-ℳ^2(x)) 
    	Ψ(X, x)_α^β=0

with the squared-mass operator, in the pure confining force limit,

    M^2(x) = 2(m_1+m_2) (1/2μ∂^2/∂ x_μ∂ x^μ+U(x) ),  
    	U(x)=-1/2Kx_μ x^μ+ const.

where μ = m_1m_2/(m_1+m_2) is the reduced mass and K is
 the spring constant. A solution of this equation can be written as

    Ψ (X, x)_α^β = N e^∓ i P_μX^μΦ (v, x)_α^β,  v^μ = P^μ/M,

where N is the normalization constant for the plane wave, P^μ and M
 are the center-of-mass momentum and mass, respectively, of the whole meson system,
 and Φ (v, x)_α^β is the internal wave function which is given
 by the direct product of eigenfunctions of the squared-mass operator and
 the Bargmann-Wigner spinor functions, defined by the direct tensor product of
 respective Dirac spinors, with the meson four velocity v^μ, for the constituent
 quark and antiquark.

Key features of the COQM.
 In order to freeze the redundant freedom of relative time for the four-dimensional
 harmonic oscillator, which gives here the squared-mass operator,
 the definite-metric-type subsidiary condition is adopted <cit.>.
 The space-time wave functions satisfying this condition
 is normalizable and leads to the desirable asymptotic behavior of
 electromagnetic form factors of hadrons. 
The eigenvalues of the squared-mass operator are given by

    M_N^2=M_0^2 + NΩ,  Ω = 2(m_1+m_2) √(K/μ),

where N = 2N_r+L, N_r (L) being the radial  (orbital angular momentum) quantum number.
 This squared-mass formula gives linear Regge trajectories with the slope Ω^-1,
 in accord with the well-known experimental fact.




§ ELECTROMAGNETIC CURRENTS FOR QUARK-ANTIQUARK MESON SYSTEMS



The above Klein-Gordon-type equation is rewritten in terms of the quark and
 antiquark coordinates as

    2(m_1 + m_2)( ∑_i=1^2 -1/2m_i∂^2/∂ x_iμ∂ x_i^μ-U(x_1,x_2) )
    	Ψ(x_1,x_2)_α^β = 0.

This equation is derived from either of  the actions

    S_free^(KG,S) = ∫ d^4 x_1 ∫ d^4 x_2  ℒ_free^(KG,S)(Ψ, ∂_1μΨ, ∂_2μΨ)

with the Klein-Gordon-like Lagrangian

    ℒ_free^(KG) = Ψ(x_1,x_2)
    	2(m_1 + m_2)(∑_i=1^2 -1/2m_i∂/∂ x_iμ∂/∂ x_i^μ
    	-U(x_1,x_2) )Ψ(x_1,x_2)

and the Schrödinger-like Lagrangian

    ℒ_free^(S)=Ψ(x_1,x_2)
    	(∑_i=1^2 -1/2m_i∂/∂ x_iμ∂/∂ x_i^μ
    	-U(x_1,x_2) )Ψ(x_1,x_2),

where the bilocal field Ψ(x_1,x_2) has the dimension of bosons [M^1]
 and fermions [M^3/2] for ℒ_free^(KG) and
 ℒ_free^(S), respectively, except for the dimension
 of internal wave functions [M^2].


The interaction of quark-antiquark meson systems with an electromagnetic field
 can be obtained <cit.> by the minimal substitutions

    ∂/∂ x_i^μ→∂/∂ x_i^μ + ieQ_iA_μ(x_i)

in the free Lagrangians ℒ_free^(KG,S), in which
 the heuristic prescription by Feynman, Kislinger and Ravndal <cit.>
 with some extension is adopted as the following replacements

    ∂/∂ x_iμ∂/∂ x_i^μ→
    	(1-g_M^(i))∂/∂ x_iμ∂/∂ x_i^μ
    	+ g_M^(i)∂/∂ x_i^μγ^μγ^ν∂/∂ x_i^ν,

where Q_i (i=1,2) are the quark and antiquark charges in units of e and
 g_M^(i) are the parameters related to the anomalous magnetic moments
 of constituent quarks. Then the action for the electromagnetic interaction of
 meson systems is obtained, up to the first order of e, as

    S_int^(KG,S) = ∫ d^4 x_1 ∫ d^4 x_2 ∑_i=1^2 j_i^(KG,S)μ(x_1,x_2) A_μ(x_i)
    	≡∫ d^4 X J^(KG,S)μ(X) A_μ(X)

with the conserved currents

    j_i^(KG)μ(x_1, x_2) = 2(m_1+m_2)
    	⟨Ψ(x_1, x_2)-ieQ_i/2m_i(
    	∂/∂ x_iμ
    	-ig_M^(i)σ^μν(
    	∂/∂ x_i^ν
    	+ ∂/∂ x_i^ν)
    	) Ψ(x_1, x_2)⟩

and

    j_i^(S)μ(x_1, x_2) = j_i^(KG)μ(x_1, x_2)/2(m_1+m_2)

where ⟨⋯⟩ means taking trace concerning the Dirac indices.
 The electric charges of meson systems are given by the diagonal elements

    Q_ meson^(KG,S)
    	 = ∫ d^3 X  ⟨ i|J^(KG,S)0(X)|i ⟩.

The respective features of the Klein-Gordon-like and Schrödinger-like currents
 are as follows:
 

 
  * The Klein-Gordon-like current.
The meson charge is given by

    Q_ meson^(KG) = (Q_1+Q_2)e,

which reproduces the physical one correctly.
 The currents j_i^(KG)μ(x_1, x_2) have no absolute mass scales
 of constituent quarks. This would seem to be natural for light-quark systems
 from the viewpoint of QCD in the chiral limit.
 
 
  * The Schrödinger-like current. 
In this case the meson charge becomes

    Q_ meson^(S) = M/m_1+m_2(Q_1+Q_2)e,

where M is the meson mass. This expression does not generally coincide with
 the physical meson charge. However, it gives the correct charge in the heavy
 quark limit. This means that the Schrödinger-like current is applicable
 to heavy-light and heavy-heavy systems.
If the meson masses for nonrelativistic quark systems are written as

    M_n = (m_1 + m_2) + ℰ_n

with the n-th excitation energy ℰ_n, then the meson charge can be
 expressed as

    Q_ meson^(S)
    	= (1 + ℰ_n/M_0)(Q_1+Q_2)e,

where M_0 is a mass of the ground-state meson. From this expression it is found
 that the applicability of the schrödinger-like current is estimated by
 the ratio of the excitation energy to the ground-state mass.
The currents j_i^(S)μ(x_1, x_2), unlike j_i^(KG)μ(x_1, x_2),
 have the absolute mass scales of constituent quarks, which is a desirable feature
 in describing the nonrelativistic quark systems.



From the above considerations, it is concluded that the dimension of bilocal meson
 fields is bosonic for light-quark systems, while fermionic for heavy-light and
 heavy-heavy systems, except for the dimension of internal wave functions
 [M^2]. Our recent studies of the pionic decays of excited heavy-light,
 charmed and charmed-strange, mesons support this conclusion for heavy-light
 systems <cit.>. 




§ RADIATIVE DECAY WIDTHS OF LIGHT-QUARK MESONS


The above effective electromagnetic interactions S_int^(KG,S)
 describe systematically all the electromagnetic interactions of quark-antiquark
 meson systems. For the radiative decays of light-quark mesons the decay width
 is obtained, following the usual procedure with the Klein-Gordon-like interaction
 S_int^(KG), as

    Γ = 1/2J_i+1|q|/8π M_i^2∑_spin|ℳ_fi|^2,

where M_i (J_i) are the mass (spin) of the initial-state meson and | q|
 is the photon three-momentum. In the actual applications of the above formula
 to radiative decay widths of light-quark mesons the physical masses are used
 for initial- and final-state mesons, except for the pion.


Here we restrict ourselves to a discussion on the radiative decays of light-quark mesons
 only with nonstrange quarks. Then the present radiative decay model has two parameters,
 Ω and g_M (≡ g_M^(u)=g_M^(d)), the inverse of the Regge slope
 and the parameter related to the anomalous magnetic moment of u and d quarks.
 We take the value of Ω = 1.14 GeV^2 and determine a value of g_M
 so as to fit the experimental width for ρ(770)^±→π^±γ.
 We also treat the pion mass as an additional parameter, which is determined
 in the following two ways:


 
  * Case A.
Assuming that the pion lies on the spin-singlet, b_1(1235)–π_2(1670),
 trajectory, the effective pion mass becomes 0.476 GeV. Calculating the numerical
 width for ρ(770)^±→π^±γ with this pion mass,
 g_M = 1.51 is obtained.
 
 
  * Case B.
In the case of the pion and the π_2(1670) lying on the same trajectory with
 Ω = 1.14 GeV^2, the effective pion mass is 0.719 GeV. In this case
 the width for ρ(770)^±→π^±γ is calculated with
 this pion mass only for the exponential factor of the formula, while
 the physical pion mass for the other part. This treatment gives g_M = 0.743.







Numerical results of the radiative decay widths are shown in comparison with experiment
 in Table <ref>. The agreement with the measured widths for πγ decays
 is satisfactory, though that for π_2(1670)^±→π^±γ
 less so in Case A, except for b_1(1235)^±→π^±γ.
 It should be noted that only the electric (convection) current, independent of
 g_M, contributes to the decay processes b_1(1235)^±→π^±γ
 and π_2(1670)^±→π^±γ.
 In addition, the width for the decay f_1(1285) →ρ(770)^0 γ can be
 calculated by just the same parameters, assuming that the f_1(1285) is
 a pure uu̅ + dd̅ state. The results are also shown in Table <ref>,
 where the present model predictions are found to be in good agreement with
 the VES <cit.> and CLAS <cit.> measurements, not the PDG average value.




§ SUMMARY


The dimension of bilocal meson fields should be bosonic for light-quark meson systems,
 while fermionic for heavy-light and heavy-heavy ones, except for the dimension
 of internal wave functions.
 The calculated results for the radiative decay widths of the a_1(1260), a_2(1320),
 b_1(1235) and π_2(1670) mesons to the pion are in fair agreement with experiment,
  aside from b_1(1235)^±→π^±γ.
 For the decay f_1(1285) →ρ(770)^0 γ the present model strongly
 supports the experimental results of VES and CLAS Collaborations.


99

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