Quadratic Mass Formula Research Articles

Black holes inN=8supergravity from SO(4,4) hidden symmetries

We detail the construction of the most general asymptotically flat, stationary, rotating, non-extremal, dyonic black hole of the four-dimensional N=2 supergravity coupled to 3 vector multiplets that describes the STU model. It generates through U-dualities the most general asymptotically flat, stationary black hole of N=8 supergravity. We develop the solution generating technique based on SO(4,4)/SL(2,R)^4 coset model symmetries, with an emphasis on the 4-fold permutation symmetry of the gauge fields. We indicate how previously known non-extremal and extremal solutions of the STU model are recovered as limiting cases. Several properties of the general black hole solution are discussed, including its thermodynamics, the quadratic mass formula, the Bogomolny-Prasad-Sommerfield limit, the slow and fast extremal rotating limits, its properties in regards to the Kerr/conformal field theory correspondence, its Killing tensors and the separability of geodesic motion and probe scalars.

On Nonlinear σ-Models Arising in (Super-)Gravity

In a previous paper with Gibbons [CMP 120 (1987) 295] we derived a list of three dimensional symmetric space $\sigma$-model obtained by dimensional reduction of a class of four dimensional gravity theories with abelian gauge fields and scalars. Here we give a detailed analysis of their group theoretical structure leading to an abstract parametrization in terms of `triangular' group elements. This allows for a uniform treatment of all these models. As an interesting application we give a simple derivation of a `Quadratic Mass Formula' for strictly stationary black holes.

Formation of spin-one mesons by photon-photon fusion.

We have observed the formation of two spin-one mesons in \ensuremath{\gamma}\ensuremath{\gamma} fusion reactions in which one photon was highly virtual. The first, consistent with being the ${J}^{\mathrm{PC}}$${=1}^{++}$ ${f}_{1}$(1285), was seen in the final state \ensuremath{\eta}${\ensuremath{\pi}}^{+}$${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$ (\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}), as was the \ensuremath{\eta}'(958). We have previously reported the observation of the second spin-one state, the X(1420), in the final state ${K}^{\ifmmode\pm\else\textpm\fi{}}$${K}_{S}^{0}$${\ensuremath{\pi}}^{\ensuremath{\mp}}$ (${K}_{S}^{0}$\ensuremath{\rightarrow}${\ensuremath{\pi}}^{+}$${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$). The formation of this state, which may be the ${f}_{1}$(1420) observed in hadronic interactions, is reanalyzed using a new model and more data. We consider whether the X(1420) could be the partner of the ${f}_{1}$(1285) in the ${J}^{\mathrm{PC}}$${=1}^{++}$ meson nonet. The \ensuremath{\gamma}${\ensuremath{\gamma}}^{\mathrm{*}}$ width of each resonance was determined in several ${Q}^{2}$ bins. Using a model due to Cahn with a \ensuremath{\rho} form factor, we obtained for the coupling parameter \ensuremath{\Gamma}${\ifmmode \tilde{}\else \~{}\fi{}}_{\ensuremath{\gamma}\ensuremath{\gamma}}$(${f}_{1}$(1285)) the value 2.4\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.5 keV.We also found B(X(1420)\ensuremath{\rightarrow}KK\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\pi})\ensuremath{\Gamma}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\ensuremath{\gamma}\ensuremath{\gamma}}$(X(1420))=1.3\ifmmode\pm\else\textpm\fi{}0.5\ifmmode\pm\else\textpm\fi{}0.3 keV using a \ensuremath{\rho} form factor, or 0.63\ifmmode\pm\else\textpm\fi{}0.24\ifmmode\pm\else\textpm\fi{}0.15 keV using a \ensuremath{\varphi} form factor. The decay distributions of the observed X(1420) events are consistent with a decay proceeding via ${K}^{\mathrm{*}}$(892)K, and favor positive, but do not exclude negative, parity. Assuming that the X(1420) and the ${f}_{1}$(1285) are members of the same qq\ifmmode\bar\else\textasciimacron\fi{} meson nonet and that B(X(1420)\ensuremath{\rightarrow}KK\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\pi})\ensuremath{\approxeq}1, and using a \ensuremath{\varphi} form factor in the model for X(1420) formation, we determined the singlet-octet mixing angle of that nonet. The result, (45.4\ifmmode\pm\else\textpm\fi{}6.2)\ifmmode^\circ\else\textdegree\fi{}, agrees well with the 42.2\ifmmode^\circ\else\textdegree\fi{} implied by the Gell-Mann--Okubo quadratic mass formula; it is farther from the 35.3\ifmmode^\circ\else\textdegree\fi{} ideal mixing angle. Our results are thus consistent with the X(1420) being the mostly ss\ifmmode\bar\else\textasciimacron\fi{} isoscalar member of the axial-vector qq\ifmmode\bar\else\textasciimacron\fi{} meson nonet; however, we have no evidence contrary to hypotheses that the X(1420) is an exotic ${q}^{2}$q\ifmmode\bar\else\textasciimacron\fi{} $^{2}$ or qq\ifmmode\bar\else\textasciimacron\fi{}g state.

Quadratic Mass Relations in Topological Bootstrap Theory

From the requirement of reality of discontinuities of scattering amplitudes at the spherical level of the topological bootstrap theory, a large number of mass relations for hadrons is derived. Quadratic mass formulas for the symmetry-breaking pattern of both mesons and baryons is obtained and their relation to conventional models of symmetry breaking is briefly discussed.

Determination of the chiral-SU(4) × SU(4)-breaking parameters

We consider broken chiral SU(4) \ifmmode\times\else\texttimes\fi{} SU(4) symmetry. From the observed mass spectrum of pseudoscalar charmed mesons, we are able to solve for the symmetry-breaking parameters of the theory. We find that both vacuum and Hamiltonian breaking play an important role as far as charmed states are concerned. Purely from the masses of $D$ and $F$ mesons we deduce the current-algebra mass ratio $\frac{{m}_{c}}{{m}_{s}}l5$. This differs greatly from values obtained using linear or quadratic mass formulas. Considering $\ensuremath{\eta}$, ${\ensuremath{\eta}}^{\ensuremath{'}}$, and ${\ensuremath{\eta}}_{c}$ mixing we further obtain a good solution with $\frac{{m}_{c}}{{m}_{s}}\ensuremath{\simeq}3.2$ and $\frac{〈\overline{c}c〉}{〈\overline{u}u〉}\ensuremath{\simeq}5.67$.

X0(958)→ηππdecay andηX0mixing

Within the framework of a broken-SU(6)\ifmmode\times\else\texttimes\fi{}O(3) quark model of Lorentz-invariant hadron-coupling structures, the three-body decay process ${X}^{0}(958)\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$ is investigated with a view to assessing the value of the $\ensuremath{\eta}{X}^{0}$ mixing angle. The model assumes that the amplitude for this process is dominated by a $\ensuremath{\delta}(970)\ensuremath{\pi}$ intermediate state so that ${X}^{0}\ensuremath{\rightarrow}\ensuremath{\delta}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$. The results obtained seem to favor a quadratic mass formula for pseudoscalar mesons with an $\ensuremath{\eta}{X}^{0}$ mixing angle of ---10\ifmmode^\circ\else\textdegree\fi{}, in agreement with recent analyses.

Radiative decays of mesons in a relativistic quark model

Radiative decays of mesons, both $V\ensuremath{\rightarrow}P\ensuremath{\gamma}$ and $P\ensuremath{\rightarrow}V\ensuremath{\gamma}$, are calculated in a relativistic quark model of hadrons proposed earlier by one of the authors. The matrix elements for the above processes are calculated directly by considering appropriately Lorentz-boosted hadronic states. The electromagnetic current operator in terms of the appropriately Lorentz-boosted quark field operators are utilized. Only the wave functions of hadrons enter the picture. No vector dominance is assumed. Mixing angles of mesons are taken from quadratic mass formula in SU(3). Thus with no adjustable free parameters or concepts, there is reasonable agreement with experimental results.

Field-theoretic model of composite hadrons. I

We discuss here a field-theoretic model of composite hadrons with quarks. The quark field operators are assumed to be broken up into particle and antiparticle components at any time $t$ similar to the large and small components of a free-Dirac-field operator. This assumption is made consistent with equal-time anticommutators. This implies that the Dirac Hamiltonian for quark field operators has four components: the particle, antiparticle, pair-creation, and pair-annihilation components. For a simple ansatz for the field operators, the latter two components need not vanish. The particle and antiparticle components along with some potential-like interactions are assumed to generate the hadrons as composite states. With the usual form of weak and electromagnetic currents, this yields some corrections to the Van Royen-Weisskopf relations and gives excellent agreement for the static properties of the nucleons. It is seen that the pair-creation component of the Hamiltonian can generate $\ensuremath{\varphi}\ensuremath{\rightarrow}\mathrm{KK}$ decay, with a correct branching ratio for $\frac{\ensuremath{\Gamma}(\ensuremath{\varphi}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}})}{\ensuremath{\Gamma}(\ensuremath{\varphi}\ensuremath{\rightarrow}{K}^{0}{\overline{K}}^{0})}$. Thus, the pair-creation Hamiltonian seems to be the dynamical explanation of the Okubo-Zweig-Iizuka rule. Further, the pair-annihilation component of the Hamiltonian with minimal electromagnetic interaction also generates $\ensuremath{\Gamma}({\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma})$. With the mixing angle obtained from the quadratic mass formula, $\ensuremath{\Gamma}(\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma})$ also seems to have a reasonable prediction. We have considered only nonrelativistic hadrons hoping that a potential-like description is valid in such a frame of reference.

Radiativeη′′(2.8)decay and the charmed-quark content ofωandφ

We propose a sum rule connecting radiative decay modes of ${\ensuremath{\eta}}^{\ensuremath{'}\ensuremath{'}}$ (the charmed partner of $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$, and presumably the $2\ensuremath{\gamma}$ resonance at 2.8 GeV). The sum rule should allow a determination of the percentage of charmed quarks in the $\ensuremath{\omega}$ and $\ensuremath{\varphi}$ mesons. If the percentage of charmed quarks in the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ is large enough, the percentage of charmed quarks in the $\ensuremath{\omega}$ and $\ensuremath{\varphi}$ can also be determined by study of the decay modes $\ensuremath{\psi}\ensuremath{\rightarrow}(\ensuremath{\omega} or \ensuremath{\varphi})+(\ensuremath{\eta} or {\ensuremath{\eta}}^{\ensuremath{'}})$; this possibility is discussed. A phenomenological analysis suggests that the familiar octuplet-singlet pseudoscalar mixing angle ${\ensuremath{\theta}}_{P}$ for $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ varies from -8\ifmmode^\circ\else\textdegree\fi{} at the $\ensuremath{\eta}$ energy to approximately -38\ifmmode^\circ\else\textdegree\fi{} at the ${\ensuremath{\eta}}^{\ensuremath{'}}$ energy. The latter angle yields predictions $\ensuremath{\Gamma}({\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}\mathrm{all})=0.11$ MeV and $\frac{\ensuremath{\Gamma}({A}_{2}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\pi})}{\ensuremath{\Gamma}({A}_{2}\ensuremath{\rightarrow}\mathrm{all})}=0.12%$; both predictions are smaller than those obtained from either the quadratic or linear mass formula values for ${\ensuremath{\theta}}_{P}$. Finally, a relation $\frac{\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\leftarrow}\ensuremath{\eta}\ensuremath{\gamma})}{\ensuremath{\Gamma}(\ensuremath{\psi}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\gamma})}=(\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}\ensuremath{-}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s})\ifmmode\times\else\texttimes\fi{}\frac{\ensuremath{\Gamma}({\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\gamma})}{\ensuremath{\Gamma}({\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\gamma})}$ is shown to follow from the assumption that the $c\overline{c}$ content of $\ensuremath{\psi}$ and ${\ensuremath{\psi}}^{\ensuremath{'}}(3684)$ is negligible compared to the $c\overline{c}$ content of $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$.

New Particles and Mass-Formulas

Both linear and quadratic mass-formulas are applied to vector-mesons and pseudoscalar­ mesons. We find that the linear mass-formula must be applied to vector-mesons in order to explain the small hadronic decay width of Jl¢ through the small mixture of ordinary quark-antiquark pairs in J I 1/J. If the linear mass-formula is also applied to pseucloscalar-mesons, we find (i) M(F) >lvi(F*) and (ii) M(n,)~2.80GeV. The hadronic decay widths of 'lc and r-decay widths of J I 1/J are also discussed.

Quark-loop anomalies and the SU(4) mixing angles

Three mixing angles are used to describe the mixing of the three isoscalar mesons assigned to the $15\ensuremath{\bigoplus}1$ representation of SU(4). The formalism is applied to the pseudoscalar ($P$), vector ($V$), and tensor ($T$) multiplets, the mixing angles being determined for both linear and quadratic mass formulas. The mixing angles can also be independently determined from decay rates. To reduce the number of parameters involved, the effective $\mathrm{PVV}$, $\mathrm{PV}\ensuremath{\gamma}$, and $P\ensuremath{\gamma}\ensuremath{\gamma}$ interactions are derived from the $\mathrm{VPP}$ interaction using anomalies in the partial conservation of the axial-vector current. The $\mathrm{VVV}$ vertex is also incorporated without introducing an additional parameter since a U(4) Yang-Mills effective Lagrangian is used. The mixing angles so determined in the case of the vectors and the pseudoscalars are used to reduce from five to two the number of parameters in the mass matrix. By fitting the remaining parameters to the observed masses it is found that the quadratic mass formula is strongly favored in both cases.

Decay Modes of the Mesonη′(958)and Chiral-Symmetry Breaking

Using the simple model that the amplitude for ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$ is dominated by the $\ensuremath{\delta}(970)$ resonance we find a satisfactory fit to the data. The quadratic mass formula for $\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{\ensuremath{'}}$ mixing with a positive mixing angle (i.e. reduction of $s\overline{s}$ content in the physical $\ensuremath{\eta}$ state) is preferred in agreement with the requirement for ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}2\ensuremath{\gamma}$. Failure of the current-algebra calculation for ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$ is explained and is shown not to be evidence against the (3*, 3) \ensuremath{\bigoplus} (3, 3*) model of chiral-symmetry breaking.

Validity of the Talmi's mass formula for single closed shell nuclei

A quadratic mass formula proposed by Talmi for identical nucleons in an open and non degenerate shell is discussed. Corrections coming from the breaking of generalized seniority are examined.

A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation

A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound spin- 1 2 quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons. Numerical solutions to the Bethe-Salpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic. Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The λ quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula. The decay amplitudes for π, K → μν are calculated, and it is found, independent of parameters, that ƒ π ≈ ƒ K for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for ϱ o , ω, φ → e + e −, μ + μ − are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment. The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as ω → π o γ. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model. The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters. The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with J PC = 0 +−) with masses of about 2 BeV.

A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation: Alan H. Guth. Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540

Abstract A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound spin - 1 2 quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons. Numerical solutions to the Bethe-Salpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic. Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The λ quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula. The decay amplitudes for π, K → μν are calculated, and it is found, independent of parameters, that ƒ π ≈ ƒ K for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for ϱo, ω, φ → e+e−, μ+μ− are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment. The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as ω → πoγ. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model. The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters. The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with JPC = 0+−) with masses of about 2 BeV.

Δ++η and Δ++X° production in π+p interactions at 5.45 GeV/ c and η−X° mixing

The production of η and X° mesons has been investigated in four and six prong events from π +p interactions at 5.45 GeV/ c. The cross sections for the quasi two body states Δ ++ η and Δ ++X° were found to be 0.076±0.013 mb and 0.017±0.006 mb respectively. A comparison of the matrix elements for these reactions yields an η−X° mixing angle different from that predicted by the quadratic mass formula by about 20°, but within 6° of the linear mass formula result.

Mass and Half-Life ofC9

The mass of $^{9}\mathrm{C}$ has been measured by counting delayed protons near the $^{7}\mathrm{Be}(^{3}\mathrm{He},n)^{9}\mathrm{C}$ threshold. The data are consistent with an $s$-wave threshold at ${E}_{{3}_{\mathrm{He}}}=8980\ifmmode\pm\else\textpm\fi{}5$ keV, giving a $^{9}\mathrm{C}$ mass excess of 28 907\ifmmode\pm\else\textpm\fi{}4 keV, and confirming the reported deviation of the $A=9$, lowest $T=\frac{3}{2}$ quartet from the quadratic mass formula. The half-life of $^{9}\mathrm{C}$, measured with a multiscaler, is 126.5 \ifmmode\pm\else\textpm\fi{} 2 msec.

Asymptotic Symmetry, Lagrangian Gauge Model, and thePVVVertex

We study a model in which $\mathrm{SU}(3)$ breaking in the $\mathrm{PVV}$ vertex is determined by the requirement of asymptotic nonet symmetry. A vector-meson gauge model is constructed which satisfies this asymptotic symmetry condition. The model also incorporates the usual asymptotic symmetry result for the $\mathrm{VPP}$ vertex, the field-current identities, and the algebra of fields. Nevertheless, we obtain a second Weinberg sum rule of the Das-Mathur-Okubo form, and consequently, a quadratic mass formula as in a mass-mixing model. The predictions of the model are compared with available experimental data on meson decays involving the $\mathrm{PVV}$ vertex. The predicted rates for radiative decays of vector mesons are also given.

SU3Mass Formula for Bosons andX0−ηMixing

The possibility of a linear ${\mathrm{SU}}_{3}$ mass formula for bosons and consequent non-negligible ${X}^{0}\ensuremath{-}\ensuremath{\eta}$ mixing is considered. The existence of ${X}^{0}\ensuremath{-}\ensuremath{\eta}$ mixing is indicated by the experimental ratio of the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ and $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ rates. A phenomenological ${\mathrm{SU}}_{3}$ analysis, with a minimum of dynamical assumptions, of the electromagnetic decay modes $P\ensuremath{\rightarrow}2\ensuremath{\gamma}$, $V\ensuremath{\rightarrow}P+\ensuremath{\gamma}$, and $P\ensuremath{\rightarrow}V+\ensuremath{\gamma}$ is made with the purpose of distinguishing between the linear and quadratic mass formulas.

WidthΓX0→2γas a Test of the Mass Formula for Boson Nonets

It is generally believed that the physical $\ensuremath{\eta}$ and ${X}^{0}$ particles are not pure $S{U}_{3}$ states, but result from octet-singlet mixing. Depending on the assumption of a linear or quadratic mass formula, different values can be obtained for the mixing angle. The latter can be used in turn to estimate the width of ${X}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ using the experimental values of ${\ensuremath{\Gamma}}_{\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}}$ and ${\ensuremath{\Gamma}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}}$. Note that the quadratic mass formula predicts values considerably larger than the linear one. An upper bound of 20-30 keV for the ${X}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ width, which is not outside the experimental possibilities---e.g., with the Primakoff-effect technique---would be evidence against the use of a quadratic mass formula within the standard $S{U}_{3}$ scheme.