$\eta-\eta^{\prime}$ Mixing Angle from Vector Meson Radiative Decays

The octet-singlet $\eta -\eta^{\prime}$ mixing mass term could have a derivative $O(p^{2})$ term as found in recent analysis of the $\eta -\eta^{\prime}$ system. This term gives rise to an additional momentum-dependent pole contribution which is suppressed by a factor $m_{\eta}^{2}/m_{\eta^{\prime}}^{2}$ for $\eta$ relative to the $\eta^{\prime}$ amplitude. The processes with $\eta$ meson can then be described, to a good approximation, by the momentum-independent mixing mass term which gives rise to a new $\eta -\eta^{\prime}$ mixing angle $\theta_{P}$, like the old $\eta -\eta^{\prime}$ mixing angle used in the past, but a momentum-dependent mixing term $d$, like $\sin(\theta_{0}-\theta_{8})$ in the two-angle mixing scheme used in the parametrization of the pseudo-scalar meson decay constants in the current literature, is needed to describe the amplitudes with $\eta^{\prime}$. In this paper, we obtain sum rules relating $\theta_{P}$ and $d$ to the physical vector meson radiative decays with $\eta$ and $\eta^{\prime}$, as done in our previous work for $\eta$ meson two-photon decay, and with nonet symmetry for the $\eta^{\prime}$ amplitude, we obtain a mixing angle $\theta_{P}=-(18.76\pm 3.4)^{\circ}$, $d=0.10\pm 0.03$ from $\rho\to\eta\gamma$ and $\eta^{\prime}\to\rho\gamma$ decays, for $\omega$, $\theta_{P}=-(15.81\pm 3.1)^{\circ}$, $d=0.02\pm 0.03$, and for $\phi$, $\theta_{P}=-(13.83\pm 2.1)^{\circ}$, $d=0.08\pm 0.03$. A larger value of $0.06\pm 0.02$ for $d$ is obtained directly from the nonet symmetry expression for the $\eta^{\prime}\to\omega\gamma$ amplitude. This indicates that more precise vector meson radiative decay measured branching ratios and higher order SU(3) breaking effects could bring these values for $\theta_{P}$ closer and allows a better determination of $d$.

The η − η ′ mixing angle plays an important role in physical processes involving the light pseudoscalar meson nonet, the η and η ′ mesons. In the presence of SU (3) breaking due to the large current s-quark mass compared to the light u and d current quark mass, with m s ≫ m u,d , the octet η 8 and the singlet η 0 could mix with each other through a small SU (3) symmetry breaking quark mass term and generate the two physical states, the η and η ′ . Since m s ≪ Λ QCD , and because of the U (1) QCD-anomaly, the η 0 mass is much larger compared to the η 8 mass, the η − η ′ mixing angle is O(m s /Λ QCD ) so that the physical η and η ′ are almost pure η 8 and η 0 eigenstate respectively, in contrast with the ideal mixing for the 1 − low-lying vector meson states. Assuming nonet symmetry for the off-diagonal mass term < η 0 |H SB |η 8 >, one would get a mixing angle θ P = −18 • [1] in good agreement with the value θ P ≈ −(22 ± 3) • in [2], or θ P ≈ −(18.4 ± 2) • in [3] obtained from the η and η ′ two-photon width. The large mixing angle obtained from the two-photon decay rate is consistent with nonet symmetry [4] (mixing angle with linear Gell-Mann-Okubo(GMO) mass formula is given in [5,6]). A previous phenomenological analysis many years ago [5] already found a large mixing angle θ P ≈ −(20 − 23) • in the pseudo-scalar meson twophoton widths, in J/ψ → γη(η ′ ), J/ψ → V P , in radiative decays of light vector mesons, and in π − p scattering a mixing angle ≈ −20 • is favored, but light tensor meson decays seem to favor a mixing angle of ≈ −10 • , the GMO mass formula value. Subsequently, a value between −13 • and −17 • , or an average θ P = −15.3 • ± 1.3 • is obtained [8] and θ P ≈ −11 • is obtained in [9]. Recent analysis [10,11] using the more precise V → P γ measured branching ratios [12] found a mixing angle θ P = −13.3 • ± 1.3 • . It appears that the mixing angle obtained in these recent theoretical calculations is a bit smaller than the nonet symmetry value [1] (the nonet symmetry value is very close to the mixing angle value we obtained from η two-photon decay rate using only the measured η ′ two-photon decay rate [3]). This could be due to various theoretical uncertainties, like the use of nonet symmetry in the treatment of radiative decays involving η ′ and possibly, the neglect of higher order SU (3) breaking in the radiative decay amplitudes. Also, most of the analysis in the past is based on the assumption that the off-diagonal octet-singlet transition mass term does not depend significantly on the energy of the state [13]. Recent works [14][15][16] show that a quadratic derivative off-diagonal octet-singlet transition of the form ∂ µ η 0 ∂ µ η 8 requires two angles θ 8 and θ 0 to describe the pseudo-scalar meson decay constants. Here we adopt a simple approach to describe the η − η ′ system. We consider the η − η ′ system with the non-derivative off-diagonal mass term diagonalized by the usual mixing angle θ P and the additional off-diagonal derivative SU (3) breaking mass term treated as a perturbation: where d is first order in SU (3) breaking parameter( O(m s /Λ QCD )). The two η and η ′ physical states are still the usual linear combinations of the pure singlet and octet SU (3) state with the monentum-independent mixing angle θ P , but the momentum-dependent off-diagonal mass term will give rise to an additional contribution to processes involving η and η ′ by the quadratic momentum dependent pole term( as in non-leptonic K → 3π decays [17], for which the K meson pole term is suppressed relative to the pion pole term by the factor m 2 π /m 2 K ). The η ′ pole contribution to the process with η on the mass shell is of the strength d (m 2 η /m 2 η ′ ), a second order SU (3) breaking effect and is suppressed by the factor m 2 η /m 2 η ′ . The η pole contribution to the η ′ amplitude is of a strength d, a first order SU (3) breaking mixing term, like the sin θ P term. Thus the quadratic momentum-dependent off-diagonal mixing mass term, while leaves the amplitude with η almost unaffected, could enhance or suppress the η ′ amplitude. This seems to be the origin of the two-angle description of the pseudo-scalar meson decay constants introduced in the literature as mentioned above. The angle θ 8 , like the new mixing angle in our scheme ( denoted by θ P in the following), behaves like the old mixing angle and effectively describes the mixing of η 0 with η 8 to make the physical η meson while sin θ 0 would effectively give the admixture of η 8 in η ′ . There have been recent calculations of vector meson radiative decays [18,19], using the two-angle mixing scheme with the result that the angle θ 8 is quite close to the nonet symmetry value in the one-mixing angle analysis, while θ 0 is found to be rather small, implying a smaller admixture of the η 8 component in η ′ than the case with one mixing angle . If one neglects second order in SU (3) breaking parameters, the determination of the old and new mixing angle would give essentially the same result and the results for the mixing angle obtained in the past still apply, in particular our previous result from the two-photon η meson decay rates [3]. We now apply our method to vector meson radiative decays to obtain first θ P with the sum rules for η and then determine both θ P and d using both sum rules for η and η ′ and nonet symmetry for the pure singlet V → η 0 γ amplitude. The sum rules for η Since the η −η ′ mixing is an additional SU (3) breaking effect not present in the decay amplitude for the pure octet η 8 state, the difference between the decay involving the physical η meson and the η 8 state is a measure of the SU (3) octet-singlet mixing effect, it is thus possible to express this difference in terms of the measured radiative decay branching ratios and a minimum theoretical input without involving the pure singlet η 0 state. This method has been used in a determination of the η − η ′ mixing angle without involving the pure singlet η 0 → γγ amplitude which is usually obtained with nonet symmetry. We have, without the momentum-dependent mixing mass term: where δ = −0.27 as estimated in [3] from the continuum contribution of the SU (3) breaking effects to the anomaly term, similar to SU (2) breaking terms for two-photon π 0 decay [20]. The expressions with the momentum-dependent η − η ′ transition included are obtained by making a substitution in Eq. (2) : These additional mixing terms will contribute to the l.h.s of Eq.
(2) terms second order in SU (3) breaking parameters. Since second order in SU (3) breaking in the r.h.s of Eq. (2) is not known at present, for example, in the two-angle mixing scheme, the quantity sin(θ 0 − θ 8 ) is given to leading order in SU (3) breaking mass term [15], to be consistent, one has to drop all second order terms in the Eq. (2). This allow a determination of the new mixing angle from the measured pseudo-scalar two-photon and vector meson radiative decays without large theoretical uncertainties which could be due to possible second order SU (3) breaking terms in vector meson radiative decays. This seems to be the price to pay for the presence of the momentum-dependent mixing mass term which now should be determined from the amplitude with η ′ . This is also the reason to use the sum rules in Eq.
(2) which involves only the measured decay rates with η and η ′ .
The above sum rules shows clearly that the difference between the physical η and the pure η 8 two-photon decay amplitude is a direct measure of the mixing effect and hence give us the mixing angle using only the measured η ′ two-photon decay rate. Since our purpose is to extract only the mixing angle and not to make a theoretical calculation of η ′ → γγ, we do not need a theoretical expression for the pure η 0 two-photon decay amplitude. Eq. (2) gives [3] which is also practically the value obtained with the current measured η → γγ branching ratio [12] which has not changed over the years (θ P = −(18.1 ± 2) • ) with the current data. This value is in good agreement with the nonet symmetry value of −18 • obtained with the first order SU (3) breaking mass term in [1]. This shows that at least to first order in SU (3) breaking, one can use Eq.
(2) to determine the new mixing angle. We now apply this method to extract the η − η ′ mixing angle from radiative decays of light vector mesons V → P γ. In addition to SU (3) and nonet symmetry breaking effects in the magnetic coupling for V → η 8 γ and V → η 0 γ amplitude, there is also an SU (3) and nonet symmetry breaking O(p 2 ) derivative coupling term which requires a renormalization of K meson, η 8 and η 0 field operator [21,22] by the factor f π /f K , f π /f η 8 and f π /f η 0 to put the propagator in the canonical 1 (p 2 −m 2 ) form. Given these SU (3) and nonet symmetry breaking effects, similar expressions like Eq. (2) for V → η, η ′ γ, V = ρ, ω, φ are obtained and the η − η ′ mixing angle can be determined in a very simple manner.
Let |η 0 >, |η 8 > be the two SU (3) singlet and octet states of the pseudo-scalar I = 0 SU (3) nonet in terms of the flavor diagonal qq component: In the presence of SU (3) symmetry breaking quark mass term, the mixing of η 0 with η 8 will produce the two physical states, η and η ′ which are given by the linear superpositions of the pure η 0 and η 8 states obtained by an unitarity transformation to diagonalize the mass matrix.
in terms of the mixing angle θ P . By inverting Eq. (6) one can express η 0 and η 8 states in terms of the physical states η and η ′ as: Our basic idea is to compute the the V → η 8 γ amplitude and to derive a sum rules relating the θ P mixing angle to the measured V → ηγ and V → η ′ γ decay amplitude by expressing the pure octet η 8 amplitude in terms of the measured η and η ′ amplitudes using Eq. (7). This is possible as the radiative decay branching ratios are currently known with good accuracy [12]. Defining the radiative decay electromagnetic form factor V → P by: where J em µ the usual electromagnetic current in terms of quark field operators in SU (3) space and g V P γ is the on-shell V P γ coupling constant with dimension the inverse of energy. The radiative decay rates are then given by [7] For convenience, we give in Table. I the measured radiative branching ratios together with the extracted coupling constant g V P γ in unit of GeV −1 and its theoretical value derived either from an SU (3) effective Lagrangian with nonet symmetry for the V → η 0 γ amplitude or from the quark counting rule with the coupling constant g V P γ given in terms of the quark coupling constant g q , (q = u, d, s) for the magnetic transition (qq)(1 − ) → (qq)(0 − )γ [7,8,10]. The theoretical values for decay modes with η in the final state is obtained for the pure octet η 8 ( θ P = 0) and SU (3) breaking effects are taken into account with g s = k g u (g d = g u ) for the magnetic transition (qq)(1 − ) → (qq)(0 − )γ extracted from the ratio of the two measured K * 0 → K 0 γ to K * ± → K ± γ branching ratio with the magnetic coupling defined as [10] g where k =m/m s is the constituent quark mass ratio [10] in the quark model, but taken here as a parameter [10] and has a value k = 0.80 ± 0.06 obtained from the measured ratio [12] BR(K * 0 K 0 γ)/BR(K * + K + γ) which is sensitive to k . In addition to SU (3) and nonet symmetry breaking effects in the magnetic coupling, as mentioned earlier, the renormalization of K meson, η 8 and η 0 field operator in the K * → Kγ, V → η 8 γ and in V → η 0 γ amplitude is given by the factor f π /f K , f π /f η 8 , and f π /f η 0 . In particular, for K * → Kγ decay, the factor f π /f K with f K = 158 MeV and the SU (3) breaking factor k are needed to obtain agreement with experiments for the computed g K * Kπ coupling, as shown in Table. I . Thus the corresponding SU (3) and nonet symmetry breaking effect should also be present in V → η 8 γ and V → η 0 γ amplitude.
(17) above. From this we obtain a mixing angle of θ P = −18.04 • mentioned earlier.
It is clear from Eqs. (14)(15)(16), that SU (3)  Thus to within experimental error, it seems that our result could accommodate the value obtained from nonet symmetry [1] and from our previous value from η meson two-photon decay [3]. We note that the determination of θ P from ρ → ηγ decay is less precise than the determination by ω → ηγ , as the branching ratios for ρ → ηγ and ρ → π 0 γ are known with larger errors. Since the ω → π 0 γ branching ratio is currently known with an accuracy of about 3%, the main uncertainty in the determination of θ P comes from ω → ηγ branching ratio which is currently known with an accuracy at 10% level. Also some discrepancy with the current data could show up in new measurements of light vector meson radiative decays. In fact, the new KLOE [11] data, with the central value of BR(ω → π 0 γ) = 8.09% smaller by 10% than the current PDG value [12], would imply a mixing angle θ P = −17.00 • , slightly larger than the solution obtained here with the PDG value.
In ρ → ηγ and ω → ηγ decays, SU (3) breaking is due mainly to the factor f π /f η 8 , thus mixing angle obtained from ρ → ηγ and ω → ηγ decay suffers from less theoretical uncertainties than that from φ → ηγ decay which is rather sensitive to the SU (3) breaking effect for the s quark magnetic coupling given by g s = k g u . To obtain the value −(15.31 ± 2.1) • for φ → ηγ decay close to that from ρ → ηγ and ω → ηγ, we take k = 0.85, a bit larger than the value k = 0.80 ± 0.06 from the K * → Kγ branching ratios. This might not be a problem, since there could be other SU (3) breaking effects in φ → ηγ not accounted for by k alone and the K * → Kγ could have large experimental error as pointed out in [10] . Since cos ϕ V g ωη 8 γ + sin ϕ V g φη 8 γ = ( √ 3/9)g u cos ϕ V g φη 8 γ − sin ϕ V g ωη 8 γ = (2 √ 6/9)g s (18) one could then try to eliminate this uncertainty by using, instead of the ω → ηγ amplitude alone, a linear combination for an ideal mixing state, the ω 0 → ηγ amplitude. We have for the ideal mixing ω 0 state. We find S(ω 0 → ηγ) = 0.49 cos θ P + 0.42 sin θ P = 0.36 (20) which give θ P = −(15.52 ± 3.3) • consistent with the solution obtained with the ω → ηγ amplitude alone. Similarly, the linear combination amplitude for the pure ss state depends only on the s quark magnetic coupling and is given by which gives a solution θ P = −(15.37 ± 3.9) • , also consistent with the all the solutions obtained above.
We have obtained the mixing angle θ P by using the sum rules for V → ηγ alone. Since the derivative mixing term affects essentially the V → η ′ γ, by using both sum rules for V → ηγ and V → η ′ γ and nonet symmetry for the pure SU (3) singlet V → η 0 γ, one would be able to determine both θ P and d. The two sum rules, similar to Eq. (13), are then neglecting second order term d g V η ′ γ in Eq. (22). With k = 0.85, f π /f K = 0.85, f π /f η 8 = 0.78, f π /f η 0 = 0.80 and the nonet symmetry value for V → η 0 γ shown in Similarly, for ω and φ, we have: constants [15]. The value of d for η ′ → ωγ is rather small, but with large experimental errors. To reduce these errors, one could determine d directly from the V → η ′ γ amplitudes with the nonet symmetry V → η 0 γ amplitude and a mixing angle of −18 • obtained from nonet symmetry for the momentum-independent mixing mass term [1]. We find d : 0.09 ± 0.04, 0.06 ± 0.02 and 0.15 ± 0.03 for ρ, ω and φ respectively, comparable to the chiral perturbation results [15]. We note that a mixing angle of −22 • could produce a larger d : 0.16 ± 0.04, 0.13 ± 0.02 and 0.21 ± 0.03 for ρ, ω and φ , respectively, corresponding to a small θ 0 found in [18].
The above values for θ P are quite close to the values obtained from the sum rules S(V → ηγ) alone. Then, considering current theoretical and experimental uncertainties, one could just use the sum rules with η alone to obtain the new mixing angle for processes with η meson without involving the pure singlet V → η 0 γ, but for processes with η ′ one need to know d either from the two sum rules with theoretical input for the pure singlet η 0 amplitude or from some other method.
In conclusion, we have derived sum rules relating the new η − η ′ mixing angle to the measured V → ηγ and V → η ′ γ decay amplitude which allows a determination of the mixing angle using only the measured radiative decay branching ratios. With only the η sum rules, we find θ P in the range −14 • to −17 • within an error of (2.1 − 4.4) • , with two η, η ′ sum rules, we find a similar value in the range −14 • to −19 • and an evidence for the momentum-dependent mixing mass term d in ρ and φ radiative decays. We also obtain large d in η ′ → ωγ using nonet symmetry, comparable to d from ρ and φ decay. More precise vector meson radiative decay measured branching ratios and higher order SU (3) breaking effects could bring these extracted values for θ P closer and give us a better determination of the momentum-dependent mixing term d which is needed in processes with η ′ .